Optimal. Leaf size=30 \[ -\frac {4}{1-\cosh (x)}+\frac {2}{(1-\cosh (x))^2}-\log (1-\cosh (x)) \]
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Rubi [A] time = 0.06, antiderivative size = 30, 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 = {4392, 2667, 43} \[ -\frac {4}{1-\cosh (x)}+\frac {2}{(1-\cosh (x))^2}-\log (1-\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rule 4392
Rubi steps
\begin {align*} \int \frac {1}{(-\coth (x)+\text {csch}(x))^5} \, dx &=i \int \frac {\sinh ^5(x)}{(i-i \cosh (x))^5} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {(i-x)^2}{(i+x)^3} \, dx,x,-i \cosh (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {4}{(i+x)^3}-\frac {4 i}{(i+x)^2}+\frac {1}{i+x}\right ) \, dx,x,-i \cosh (x)\right )\\ &=-\frac {2}{(i-i \cosh (x))^2}-\frac {4 i}{i-i \cosh (x)}-\log (1-\cosh (x))\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 1.07 \[ \frac {1}{2} \text {csch}^4\left (\frac {x}{2}\right )+2 \text {csch}^2\left (\frac {x}{2}\right )-2 \log \left (\sinh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 269, normalized size = 8.97 \[ \frac {x \cosh \relax (x)^{4} + x \sinh \relax (x)^{4} - 4 \, {\left (x - 2\right )} \cosh \relax (x)^{3} + 4 \, {\left (x \cosh \relax (x) - x + 2\right )} \sinh \relax (x)^{3} + 2 \, {\left (3 \, x - 4\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, x \cosh \relax (x)^{2} - 6 \, {\left (x - 2\right )} \cosh \relax (x) + 3 \, x - 4\right )} \sinh \relax (x)^{2} - 4 \, {\left (x - 2\right )} \cosh \relax (x) - 2 \, {\left (\cosh \relax (x)^{4} + 4 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 4 \, \cosh \relax (x)^{3} + 6 \, {\left (\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1\right )} \sinh \relax (x)^{2} + 6 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - 3 \, \cosh \relax (x)^{2} + 3 \, \cosh \relax (x) - 1\right )} \sinh \relax (x) - 4 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 4 \, {\left (x \cosh \relax (x)^{3} - 3 \, {\left (x - 2\right )} \cosh \relax (x)^{2} + {\left (3 \, x - 4\right )} \cosh \relax (x) - x + 2\right )} \sinh \relax (x) + x}{\cosh \relax (x)^{4} + 4 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 4 \, \cosh \relax (x)^{3} + 6 \, {\left (\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1\right )} \sinh \relax (x)^{2} + 6 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - 3 \, \cosh \relax (x)^{2} + 3 \, \cosh \relax (x) - 1\right )} \sinh \relax (x) - 4 \, \cosh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 31, normalized size = 1.03 \[ x + \frac {8 \, {\left (e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} + e^{x}\right )}}{{\left (e^{x} - 1\right )}^{4}} - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 37, normalized size = 1.23 \[ \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {1}{2 \tanh \left (\frac {x}{2}\right )^{4}}+\frac {1}{\tanh \left (\frac {x}{2}\right )^{2}}-2 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 58, normalized size = 1.93 \[ -x - \frac {8 \, {\left (e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}\right )}}{4 \, e^{\left (-x\right )} - 6 \, e^{\left (-2 \, x\right )} + 4 \, e^{\left (-3 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.52, size = 79, normalized size = 2.63 \[ x-2\,\ln \left ({\mathrm {e}}^x-1\right )-\frac {16}{3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1}+\frac {16}{{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1}+\frac {8}{6\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^x+1}+\frac {8}{{\mathrm {e}}^x-1} \]
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 \[ - \int \frac {1}{\coth ^{5}{\relax (x )} - 5 \coth ^{4}{\relax (x )} \operatorname {csch}{\relax (x )} + 10 \coth ^{3}{\relax (x )} \operatorname {csch}^{2}{\relax (x )} - 10 \coth ^{2}{\relax (x )} \operatorname {csch}^{3}{\relax (x )} + 5 \coth {\relax (x )} \operatorname {csch}^{4}{\relax (x )} - \operatorname {csch}^{5}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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