Optimal. Leaf size=38 \[ \frac {1}{16} \tanh ^{-1}(\cosh (x))-\frac {1}{6} \coth ^3(x) \text {csch}^3(x)-\frac {1}{8} \coth (x) \text {csch}^3(x)-\frac {1}{16} \coth (x) \text {csch}(x) \]
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Rubi [A] time = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2611, 3768, 3770} \[ \frac {1}{16} \tanh ^{-1}(\cosh (x))-\frac {1}{6} \coth ^3(x) \text {csch}^3(x)-\frac {1}{8} \coth (x) \text {csch}^3(x)-\frac {1}{16} \coth (x) \text {csch}(x) \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \coth ^4(x) \text {csch}^3(x) \, dx &=-\frac {1}{6} \coth ^3(x) \text {csch}^3(x)+\frac {1}{2} \int \coth ^2(x) \text {csch}^3(x) \, dx\\ &=-\frac {1}{8} \coth (x) \text {csch}^3(x)-\frac {1}{6} \coth ^3(x) \text {csch}^3(x)+\frac {1}{8} \int \text {csch}^3(x) \, dx\\ &=-\frac {1}{16} \coth (x) \text {csch}(x)-\frac {1}{8} \coth (x) \text {csch}^3(x)-\frac {1}{6} \coth ^3(x) \text {csch}^3(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}{8} \coth (x) \text {csch}^3(x)-\frac {1}{6} \coth ^3(x) \text {csch}^3(x)\\ \end {align*}
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Mathematica [B] time = 0.02, size = 84, normalized size = 2.21 \[ -\frac {1}{384} \text {csch}^6\left (\frac {x}{2}\right )-\frac {1}{64} \text {csch}^4\left (\frac {x}{2}\right )-\frac {1}{64} \text {csch}^2\left (\frac {x}{2}\right )-\frac {1}{384} \text {sech}^6\left (\frac {x}{2}\right )+\frac {1}{64} \text {sech}^4\left (\frac {x}{2}\right )-\frac {1}{64} \text {sech}^2\left (\frac {x}{2}\right )-\frac {1}{16} \log \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 1260, normalized size = 33.16 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 71, normalized size = 1.87 \[ -\frac {3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{5} + 32 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 48 \, e^{\left (-x\right )} - 48 \, e^{x}}{24 \, {\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{3}} + \frac {1}{32} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) - \frac {1}{32} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 46, normalized size = 1.21 \[ -\frac {\cosh ^{3}\relax (x )}{3 \sinh \relax (x )^{6}}+\frac {\cosh \relax (x )}{5 \sinh \relax (x )^{6}}+\frac {\left (-\frac {\mathrm {csch}\relax (x )^{5}}{6}+\frac {5 \mathrm {csch}\relax (x )^{3}}{24}-\frac {5 \,\mathrm {csch}\relax (x )}{16}\right ) \coth \relax (x )}{5}+\frac {\arctanh \left ({\mathrm e}^{x}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 98, normalized size = 2.58 \[ \frac {3 \, e^{\left (-x\right )} + 47 \, e^{\left (-3 \, x\right )} + 78 \, e^{\left (-5 \, x\right )} + 78 \, e^{\left (-7 \, x\right )} + 47 \, e^{\left (-9 \, x\right )} + 3 \, e^{\left (-11 \, x\right )}}{24 \, {\left (6 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} - e^{\left (-12 \, x\right )} - 1\right )}} + \frac {1}{16} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {1}{16} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 214, normalized size = 5.63 \[ \frac {\ln \left (\frac {{\mathrm {e}}^x}{8}+\frac {1}{8}\right )}{16}-\frac {\ln \left (\frac {{\mathrm {e}}^x}{8}-\frac {1}{8}\right )}{16}-\frac {10\,{\mathrm {e}}^x}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {{\mathrm {e}}^x}{8\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {7\,{\mathrm {e}}^x}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}-\frac {\frac {8\,{\mathrm {e}}^{3\,x}}{3}+4\,{\mathrm {e}}^{5\,x}+\frac {8\,{\mathrm {e}}^{7\,x}}{3}+\frac {2\,{\mathrm {e}}^{9\,x}}{3}+\frac {2\,{\mathrm {e}}^x}{3}}{15\,{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}-20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}-6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1}-\frac {16\,{\mathrm {e}}^x}{3\,\left (5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1\right )}-\frac {23\,{\mathrm {e}}^x}{12\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )} \]
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 \[ \int \coth ^{4}{\relax (x )} \operatorname {csch}^{3}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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