Optimal. Leaf size=18 \[ -\frac {1}{2} \tanh ^{-1}(\cosh (x))+\text {csch}(x)-\frac {1}{2} \coth (x) \text {csch}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3599, 3187,
3186, 2686, 8, 2691, 3855} \begin {gather*} \text {csch}(x)-\frac {1}{2} \tanh ^{-1}(\cosh (x))-\frac {1}{2} \coth (x) \text {csch}(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2686
Rule 2691
Rule 3186
Rule 3187
Rule 3599
Rule 3855
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(x)}{1+\tanh (x)} \, dx &=\int \frac {\coth (x) \text {csch}^2(x)}{\cosh (x)+\sinh (x)} \, dx\\ &=i \int \coth (x) \text {csch}^2(x) (-i \cosh (x)+i \sinh (x)) \, dx\\ &=\int \left (-\coth (x) \text {csch}(x)+\coth ^2(x) \text {csch}(x)\right ) \, dx\\ &=-\int \coth (x) \text {csch}(x) \, dx+\int \coth ^2(x) \text {csch}(x) \, dx\\ &=-\frac {1}{2} \coth (x) \text {csch}(x)+i \text {Subst}(\int 1 \, dx,x,-i \text {csch}(x))+\frac {1}{2} \int \text {csch}(x) \, dx\\ &=-\frac {1}{2} \tanh ^{-1}(\cosh (x))+\text {csch}(x)-\frac {1}{2} \coth (x) \text {csch}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 20, normalized size = 1.11 \begin {gather*} \frac {1}{2} \left (-((-2+\coth (x)) \text {csch}(x))+\log \left (\tanh \left (\frac {x}{2}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs.
\(2(14)=28\).
time = 0.61, size = 39, normalized size = 2.17
method | result | size |
risch | \(\frac {{\mathrm e}^{x} \left ({\mathrm e}^{2 x}-3\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2}\) | \(33\) |
default | \(\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}-\frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (14) = 28\).
time = 0.26, size = 48, normalized size = 2.67 \begin {gather*} -\frac {e^{\left (-x\right )} - 3 \, e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac {1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 209 vs.
\(2 (14) = 28\).
time = 0.36, size = 209, normalized size = 11.61 \begin {gather*} \frac {2 \, \cosh \left (x\right )^{3} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{3} - {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 6 \, {\left (\cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - 6 \, \cosh \left (x\right )}{2 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}^{3}{\left (x \right )}}{\tanh {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs.
\(2 (14) = 28\).
time = 0.40, size = 34, normalized size = 1.89 \begin {gather*} \frac {e^{\left (3 \, x\right )} - 3 \, e^{x}}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.09, size = 48, normalized size = 2.67 \begin {gather*} \frac {\ln \left (1-{\mathrm {e}}^x\right )}{2}-\frac {\ln \left (-{\mathrm {e}}^x-1\right )}{2}+\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________