Optimal. Leaf size=44 \[ -\frac {1}{16} \tanh ^{-1}(\cosh (x))+\frac {1}{16} \coth (x) \text {csch}(x)-\frac {1}{24} \coth (x) \text {csch}^3(x)+\frac {\text {csch}^5(x)}{5}-\frac {1}{6} \coth (x) \text {csch}^5(x) \]
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Rubi [A]
time = 0.15, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 10, 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}^5(x)}{5}-\frac {1}{16} \tanh ^{-1}(\cosh (x))-\frac {1}{6} \coth (x) \text {csch}^5(x)-\frac {1}{24} \coth (x) \text {csch}^3(x)+\frac {1}{16} \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}^7(x)}{1+\tanh (x)} \, dx &=\int \frac {\coth (x) \text {csch}^6(x)}{\cosh (x)+\sinh (x)} \, dx\\ &=i \int \coth (x) \text {csch}^6(x) (-i \cosh (x)+i \sinh (x)) \, dx\\ &=\int \left (-\coth (x) \text {csch}^5(x)+\coth ^2(x) \text {csch}^5(x)\right ) \, dx\\ &=-\int \coth (x) \text {csch}^5(x) \, dx+\int \coth ^2(x) \text {csch}^5(x) \, dx\\ &=-\frac {1}{6} \coth (x) \text {csch}^5(x)+i \text {Subst}\left (\int x^4 \, dx,x,-i \text {csch}(x)\right )+\frac {1}{6} \int \text {csch}^5(x) \, dx\\ &=-\frac {1}{24} \coth (x) \text {csch}^3(x)+\frac {\text {csch}^5(x)}{5}-\frac {1}{6} \coth (x) \text {csch}^5(x)-\frac {1}{8} \int \text {csch}^3(x) \, dx\\ &=\frac {1}{16} \coth (x) \text {csch}(x)-\frac {1}{24} \coth (x) \text {csch}^3(x)+\frac {\text {csch}^5(x)}{5}-\frac {1}{6} \coth (x) \text {csch}^5(x)+\frac {1}{16} \int \text {csch}(x) \, dx\\ &=-\frac {1}{16} \tanh ^{-1}(\cosh (x))+\frac {1}{16} \coth (x) \text {csch}(x)-\frac {1}{24} \coth (x) \text {csch}^3(x)+\frac {\text {csch}^5(x)}{5}-\frac {1}{6} \coth (x) \text {csch}^5(x)\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 68, normalized size = 1.55 \begin {gather*} \frac {\text {csch}^6(x) \left (-1140 \cosh (x)-170 \cosh (3 x)+30 \cosh (5 x)+6 \sinh (x) \left (256+50 \log \left (\tanh \left (\frac {x}{2}\right )\right ) \sinh (x)-25 \log \left (\tanh \left (\frac {x}{2}\right )\right ) \sinh (3 x)+5 \log \left (\tanh \left (\frac {x}{2}\right )\right ) \sinh (5 x)\right )\right )}{7680} \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. \(102\) vs.
\(2(34)=68\).
time = 0.63, size = 103, normalized size = 2.34
method | result | size |
risch | \(\frac {{\mathrm e}^{x} \left (15 \,{\mathrm e}^{10 x}-85 \,{\mathrm e}^{8 x}+198 \,{\mathrm e}^{6 x}-1338 \,{\mathrm e}^{4 x}-85 \,{\mathrm e}^{2 x}+15\right )}{120 \left ({\mathrm e}^{2 x}-1\right )^{6}}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{16}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{16}\) | \(60\) |
default | \(\frac {\left (\tanh ^{6}\left (\frac {x}{2}\right )\right )}{384}-\frac {\left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{160}-\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{128}+\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{32}-\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{128}-\frac {\tanh \left (\frac {x}{2}\right )}{16}+\frac {1}{128 \tanh \left (\frac {x}{2}\right )^{4}}-\frac {1}{32 \tanh \left (\frac {x}{2}\right )^{3}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{16}+\frac {1}{128 \tanh \left (\frac {x}{2}\right )^{2}}-\frac {1}{384 \tanh \left (\frac {x}{2}\right )^{6}}+\frac {1}{16 \tanh \left (\frac {x}{2}\right )}+\frac {1}{160 \tanh \left (\frac {x}{2}\right )^{5}}\) | \(103\) |
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. 98 vs.
\(2 (34) = 68\).
time = 0.27, size = 98, normalized size = 2.23 \begin {gather*} -\frac {15 \, e^{\left (-x\right )} - 85 \, e^{\left (-3 \, x\right )} + 198 \, e^{\left (-5 \, x\right )} - 1338 \, e^{\left (-7 \, x\right )} - 85 \, e^{\left (-9 \, x\right )} + 15 \, e^{\left (-11 \, x\right )}}{120 \, {\left (6 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} - e^{\left (-12 \, x\right )} - 1\right )}} - \frac {1}{16} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{16} \, \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. 1260 vs.
\(2 (34) = 68\).
time = 0.36, size = 1260, normalized size = 28.64 \begin {gather*} \text {Too large to display} \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}^{7}{\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 = 0.40, size = 61, normalized size = 1.39 \begin {gather*} \frac {15 \, e^{\left (11 \, x\right )} - 85 \, e^{\left (9 \, x\right )} + 198 \, e^{\left (7 \, x\right )} - 1338 \, e^{\left (5 \, x\right )} - 85 \, e^{\left (3 \, x\right )} + 15 \, e^{x}}{120 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{6}} - \frac {1}{16} \, \log \left (e^{x} + 1\right ) + \frac {1}{16} \, \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 = 1.17, size = 207, normalized size = 4.70 \begin {gather*} \frac {\ln \left (\frac {1}{8}-\frac {{\mathrm {e}}^x}{8}\right )}{16}-\frac {\ln \left (-\frac {{\mathrm {e}}^x}{8}-\frac {1}{8}\right )}{16}-\frac {\frac {16\,{\mathrm {e}}^{3\,x}}{3}+\frac {16\,{\mathrm {e}}^{5\,x}}{3}}{15\,{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}-20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}-6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1}-\frac {\frac {8\,{\mathrm {e}}^{3\,x}}{3}+\frac {8\,{\mathrm {e}}^x}{5}}{5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1}-\frac {6\,{\mathrm {e}}^x}{5\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {{\mathrm {e}}^x}{8\,\left ({\mathrm {e}}^{2\,x}-1\right )}+\frac {{\mathrm {e}}^x}{15\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {{\mathrm {e}}^x}{12\,\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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