Optimal. Leaf size=8 \[ \cosh (x)-\tanh ^{-1}(\cosh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3770, 2638} \[ \cosh (x)-\tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2638
Rule 3770
Rubi steps
\begin {align*} \int (\text {csch}(x)+\sinh (x)) \, dx &=\int \text {csch}(x) \, dx+\int \sinh (x) \, dx\\ &=-\tanh ^{-1}(\cosh (x))+\cosh (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 10, normalized size = 1.25 \[ \cosh (x)+\log \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.40, size = 53, normalized size = 6.62 \[ \frac {\cosh \relax (x)^{2} - 2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1}{2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.11, size = 24, normalized size = 3.00 \[ \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 9, normalized size = 1.12 \[ \ln \left (\tanh \left (\frac {x}{2}\right )\right )+\cosh \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 8, normalized size = 1.00 \[ \cosh \relax (x) + \log \left (\tanh \left (\frac {1}{2} \, x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 27, normalized size = 3.38 \[ \ln \left (2-2\,{\mathrm {e}}^x\right )-\ln \left (-2\,{\mathrm {e}}^x-2\right )+\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\sinh {\relax (x )} + \operatorname {csch}{\relax (x )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________