Revision history [back]

I am trying to simplify the expression

1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))

under the assumptions that x is real, and x > 1. I expect

1/2 * acosh(x)

I tried

with assuming(s, x, 'real', x > 1):
    print(integrate(sqrt(abs(1 - s * s)), s, 1, x).full_simplify())

However, that just prints back the original expression.

I would do this by hand as follows:

$$ \begin{aligned} {} & 1/2 * \log(2) - 1/2\log(2 * x + 2\sqrt{x^2 - 1}) \\ = & 1/2 * \log(2) - 1/2\log(2 * (x + \sqrt{x^2 - 1})) \\ = & 1/2 * \log(2) - 1/2\log(2) - 1/2\log(x + \sqrt{x^2 - 1}) \\ = & 1/2\log(x + \sqrt{x^2 - 1}) \\ = & 1/2* \textrm{acosh}(x) \\ \end{aligned} $$

How can I get SageMath to perform this simplification?

I am trying to simplify the expression

1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))

under the assumptions that x is real, and x > 1. I expect

1/2 * acosh(x)

I tried

with assuming(s, x, 'real', x > 1):
    print(integrate(sqrt(abs(1 - s * s)), s, 1, x).full_simplify())
print((1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))).full_simplify())

However, that just prints back the original expression.

I would do this by hand as follows:

$$ \begin{aligned} {} & 1/2 * \log(2) - 1/2\log(2 * x + 2\sqrt{x^2 - 1}) \\ = & 1/2 * \log(2) - 1/2\log(2 * (x + \sqrt{x^2 - 1})) \\ = & 1/2 * \log(2) - 1/2\log(2) - 1/2\log(x + \sqrt{x^2 - 1}) \\ = & 1/2\log(x + \sqrt{x^2 - 1}) \\ = & 1/2* \textrm{acosh}(x) \\ \end{aligned} $$

How can I get SageMath to perform this simplification?

I am trying to simplify the expression

1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))

under the assumptions that x is real, and x > 1. I expect

1/2 * acosh(x)

I tried

with assuming(s, x, 'real', x > 1):
    print((1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))).full_simplify())

However, that just prints back the original expression.

I would do this by hand as follows:

 1/2 * \log(2) - 1/2\log(2 1/2*\log(2 * x + 2\sqrt{x^2 2*\sqrt{x^2 - 1}) \\
1})
 = & 1/2 * \log(2) - 1/2\log(2 1/2*\log(2 * (x + \sqrt{x^2 - 1})) \\
1}))
 = & 1/2 * \log(2) - 1/2\log(2) 1/2*\log(2) - 1/2\log(x 1/2*\log(x + \sqrt{x^2 - 1}) \\
1})
 = & 1/2\log(x 1/2*\log(x + \sqrt{x^2 - 1}) \\
1})
 = & 1/2* \textrm{acosh}(x) \\
\end{aligned}
$$
\textrm{acosh}(x)

I am trying to simplify the expression

1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))

under the assumptions that x is real, and x > 1. I expect

1/2 * acosh(x)

I tried

with assuming(s, x, 'real', x > 1):
    print((1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))).full_simplify())

However, that just prints back the original expression.

I would do this by hand as follows:

 1/2 * \log(2) - 1/2*\log(2 * x + 2*\sqrt{x^2 - 1})
 = 1/2 * \log(2) - 1/2*\log(2 log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))
 = 1/2 * log(2) - 1/2 * log(2 * (x + \sqrt{x^2 - 1}))
 = 1/2 * \log(2) - 1/2*\log(2) - 1/2*\log(x + \sqrt{x^2 - 1})
 = 1/2*\log(x + \sqrt{x^2 - 1})
sqrt(x^2 - 1)))
 = 1/2 * log(2) - 1/2 * log(2) - 1/2 * log(x + sqrt(x^2 - 1))
 = 1/2 * log(x + sqrt(x^2 - 1))
 = 1/2* \textrm{acosh}(x)
acosh(x)

How can I get SageMath to perform this simplification?

I am trying to simplify the expression

1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))

under the assumptions that x is real, and x > 1. I expect

1/2 * acosh(x)

I tried

with assuming(s, x, 'real', x > 1):
    print((1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))).full_simplify())

However, that just prints back the original expression.

I would do this by hand as follows:

 1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))
 = 1/2 * log(2) - 1/2 * log(2 * (x + sqrt(x^2 - 1)))
 = 1/2 * log(2) - 1/2 * log(2) - 1/2 * log(x + sqrt(x^2 - 1))
 = 1/2 * log(x + sqrt(x^2 - 1))
 = 1/2* acosh(x)

How can I get SageMath to perform this simplification?

I am trying to simplify the expression

1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))

under the assumptions that x is real, and x > 1. I expect

1/2 * acosh(x)

I tried

with assuming(s, x, assuming(x, 'real', x > 1):
    print((1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))).full_simplify())

However, that just prints back the original expression.

I would do this by hand as follows:

 1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))
 = 1/2 * log(2) - 1/2 * log(2 * (x + sqrt(x^2 - 1)))
 = 1/2 * log(2) - 1/2 * log(2) - 1/2 * log(x + sqrt(x^2 - 1))
 = 1/2 * log(x + sqrt(x^2 - 1))
 = 1/2* 1/2 * acosh(x)

How can I get SageMath to perform this simplification?

Even without the acosh simplification I would expect to get rid of the 1/2 * log(2) term.

I am trying to simplify the expression

1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))

under the assumptions that x is real, and x > 1. I expect

1/2 -1/2 * acosh(x)

I tried

with assuming(x, 'real', x > 1):
    print((1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))).full_simplify())

However, that just prints back the original expression.

I would do this by hand as follows:

 1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))
 = 1/2 * log(2) - 1/2 * log(2 * (x + sqrt(x^2 - 1)))
 = 1/2 * log(2) - 1/2 * log(2) - 1/2 * log(x + sqrt(x^2 - 1))
 = 1/2 -1/2 * log(x + sqrt(x^2 - 1))
 = 1/2 -1/2 * acosh(x)

How can I get SageMath to perform this simplification?

Even without the acosh simplification I would expect to get rid of the 1/2 * log(2) term.

I am trying to simplify the expression

1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))

under the assumptions that x is real, and x > 1. I expect

-1/2 * acosh(x)

I tried

with assuming(x, 'real', x > 1):
    print((1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))).full_simplify())

However, that just prints back the original expression.

I would do this by hand as follows:

 1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))
 = 1/2 * log(2) - 1/2 * log(2 * (x + sqrt(x^2 - 1)))
 = 1/2 * log(2) - 1/2 * log(2) - 1/2 * log(x + sqrt(x^2 - 1))
 = -1/2 * log(x + sqrt(x^2 - 1))
 = -1/2 * acosh(x)

How can I get SageMath to perform this simplification?

Even without the acosh simplification I would expect to get rid of the 1/2 * log(2) term.