Optimal. Leaf size=13 \[ \cosh (x) \log \left (\cosh ^2(x)\right )-2 \cosh (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2638, 2554, 12} \[ \cosh (x) \log \left (\cosh ^2(x)\right )-2 \cosh (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2554
Rule 2638
Rubi steps
\begin {align*} \int \log \left (\cosh ^2(x)\right ) \sinh (x) \, dx &=\cosh (x) \log \left (\cosh ^2(x)\right )-\int 2 \sinh (x) \, dx\\ &=\cosh (x) \log \left (\cosh ^2(x)\right )-2 \int \sinh (x) \, dx\\ &=-2 \cosh (x)+\cosh (x) \log \left (\cosh ^2(x)\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ \cosh (x) \log \left (\cosh ^2(x)\right )-2 \cosh (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 62, normalized size = 4.77 \[ -\frac {2 \, \cosh \relax (x)^{2} - {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (\frac {1}{2} \, \cosh \relax (x)^{2} + \frac {1}{2} \, \sinh \relax (x)^{2} + \frac {1}{2}\right ) + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} + 2}{2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.18, size = 31, normalized size = 2.38 \[ {\left (e^{\left (-x\right )} + e^{x}\right )} \log \left (\frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x}\right ) - e^{\left (-x\right )} - e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.29, size = 14, normalized size = 1.08 \[ \cosh \relax (x ) \ln \left (\cosh ^{2}\relax (x )\right )-2 \cosh \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 12, normalized size = 0.92 \[ 2 \, \cosh \relax (x) \log \left (\cosh \relax (x)\right ) - 2 \, \cosh \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.38, size = 9, normalized size = 0.69 \[ 2\,\mathrm {cosh}\relax (x)\,\left (\ln \left (\mathrm {cosh}\relax (x)\right )-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.91, size = 14, normalized size = 1.08 \[ 2 \log {\left (\cosh {\relax (x )} \right )} \cosh {\relax (x )} - 2 \cosh {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________