Optimal. Leaf size=34 \[ \frac {\text {csch}^3(x)}{3}+\frac {1}{8} \tanh ^{-1}(\cosh (x))-\frac {1}{4} \coth (x) \text {csch}^3(x)-\frac {1}{8} \coth (x) \text {csch}(x) \]
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Rubi [A] time = 0.19, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3518, 3108, 3107, 2606, 30, 2611, 3768, 3770} \[ \frac {\text {csch}^3(x)}{3}+\frac {1}{8} \tanh ^{-1}(\cosh (x))-\frac {1}{4} \coth (x) \text {csch}^3(x)-\frac {1}{8} \coth (x) \text {csch}(x) \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2611
Rule 3107
Rule 3108
Rule 3518
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}^5(x)}{1+\tanh (x)} \, dx &=\int \frac {\coth (x) \text {csch}^4(x)}{\cosh (x)+\sinh (x)} \, dx\\ &=i \int \coth (x) \text {csch}^4(x) (-i \cosh (x)+i \sinh (x)) \, dx\\ &=-\int \left (\coth (x) \text {csch}^3(x)-\coth ^2(x) \text {csch}^3(x)\right ) \, dx\\ &=-\int \coth (x) \text {csch}^3(x) \, dx+\int \coth ^2(x) \text {csch}^3(x) \, dx\\ &=-\frac {1}{4} \coth (x) \text {csch}^3(x)-i \operatorname {Subst}\left (\int x^2 \, dx,x,-i \text {csch}(x)\right )+\frac {1}{4} \int \text {csch}^3(x) \, dx\\ &=-\frac {1}{8} \coth (x) \text {csch}(x)+\frac {\text {csch}^3(x)}{3}-\frac {1}{4} \coth (x) \text {csch}^3(x)-\frac {1}{8} \int \text {csch}(x) \, dx\\ &=\frac {1}{8} \tanh ^{-1}(\cosh (x))-\frac {1}{8} \coth (x) \text {csch}(x)+\frac {\text {csch}^3(x)}{3}-\frac {1}{4} \coth (x) \text {csch}^3(x)\\ \end {align*}
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Mathematica [A] time = 0.12, size = 49, normalized size = 1.44 \[ -\frac {1}{192} \text {csch}^4(x) \left (42 \cosh (x)+6 \cosh (3 x)+2 \sinh (x) \left (-9 \sinh (x) \log \left (\tanh \left (\frac {x}{2}\right )\right )+3 \sinh (3 x) \log \left (\tanh \left (\frac {x}{2}\right )\right )-32\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 640, normalized size = 18.82 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 49, normalized size = 1.44 \[ -\frac {3 \, e^{\left (7 \, x\right )} - 11 \, e^{\left (5 \, x\right )} + 53 \, e^{\left (3 \, x\right )} + 3 \, e^{x}}{12 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{4}} + \frac {1}{8} \, \log \left (e^{x} + 1\right ) - \frac {1}{8} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 55, normalized size = 1.62 \[ \frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{64}-\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{24}+\frac {\tanh \left (\frac {x}{2}\right )}{8}+\frac {1}{24 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {1}{64 \tanh \left (\frac {x}{2}\right )^{4}}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 74, normalized size = 2.18 \[ \frac {3 \, e^{\left (-x\right )} - 11 \, e^{\left (-3 \, x\right )} + 53 \, e^{\left (-5 \, x\right )} + 3 \, e^{\left (-7 \, x\right )}}{12 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} + \frac {1}{8} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {1}{8} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 117, normalized size = 3.44 \[ \frac {\ln \left (\frac {{\mathrm {e}}^x}{4}+\frac {1}{4}\right )}{8}-\frac {\ln \left (\frac {{\mathrm {e}}^x}{4}-\frac {1}{4}\right )}{8}-\frac {{\mathrm {e}}^x}{4\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{3\,x}+2\,{\mathrm {e}}^x}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {4\,{\mathrm {e}}^x}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}+\frac {{\mathrm {e}}^x}{6\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{5}{\relax (x )}}{\tanh {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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