Optimal. Leaf size=34 \[ \frac {1}{8} \tanh ^{-1}(\cosh (x))-\frac {1}{8} \coth (x) \text {csch}(x)+\frac {\text {csch}^3(x)}{3}-\frac {1}{4} \coth (x) \text {csch}^3(x) \]
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Rubi [A]
time = 0.14, 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 = {3599, 3187,
3186, 2686, 30, 2691, 3853, 3855} \begin {gather*} \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) \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2686
Rule 2691
Rule 3186
Rule 3187
Rule 3599
Rule 3853
Rule 3855
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 \text {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 = 0.09, size = 49, normalized size = 1.44 \begin {gather*} -\frac {1}{192} \text {csch}^4(x) \left (42 \cosh (x)+6 \cosh (3 x)+2 \sinh (x) \left (-32-9 \log \left (\tanh \left (\frac {x}{2}\right )\right ) \sinh (x)+3 \log \left (\tanh \left (\frac {x}{2}\right )\right ) \sinh (3 x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs.
\(2(26)=52\).
time = 0.63, size = 55, normalized size = 1.62
method | result | size |
risch | \(-\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{6 x}-11 \,{\mathrm e}^{4 x}+53 \,{\mathrm e}^{2 x}+3\right )}{12 \left ({\mathrm e}^{2 x}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{x}+1\right )}{8}-\frac {\ln \left ({\mathrm e}^{x}-1\right )}{8}\) | \(48\) |
default | \(\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{64}-\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{24}+\frac {\tanh \left (\frac {x}{2}\right )}{8}-\frac {1}{64 \tanh \left (\frac {x}{2}\right )^{4}}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8}+\frac {1}{24 \tanh \left (\frac {x}{2}\right )^{3}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs.
\(2 (26) = 52\).
time = 0.27, size = 74, normalized size = 2.18 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 640 vs.
\(2 (26) = 52\).
time = 0.44, size = 640, normalized size = 18.82 \begin {gather*} -\frac {6 \, \cosh \left (x\right )^{7} + 42 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + 6 \, \sinh \left (x\right )^{7} + 2 \, {\left (63 \, \cosh \left (x\right )^{2} - 11\right )} \sinh \left (x\right )^{5} - 22 \, \cosh \left (x\right )^{5} + 10 \, {\left (21 \, \cosh \left (x\right )^{3} - 11 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + 2 \, {\left (105 \, \cosh \left (x\right )^{4} - 110 \, \cosh \left (x\right )^{2} + 53\right )} \sinh \left (x\right )^{3} + 106 \, \cosh \left (x\right )^{3} + 2 \, {\left (63 \, \cosh \left (x\right )^{5} - 110 \, \cosh \left (x\right )^{3} + 159 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 3 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{6} - 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - 3 \, \cosh \left (x\right )^{5} + 3 \, \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 ) + 3 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{6} - 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - 3 \, \cosh \left (x\right )^{5} + 3 \, \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 ) + 2 \, {\left (21 \, \cosh \left (x\right )^{6} - 55 \, \cosh \left (x\right )^{4} + 159 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )}{24 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{6} - 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - 3 \, \cosh \left (x\right )^{5} + 3 \, \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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}^{5}{\left (x \right )}}{\tanh {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 49, normalized size = 1.44 \begin {gather*} -\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 ) \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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