Optimal. Leaf size=28 \[ \frac{4}{1-\cosh (x)}-\frac{2}{(1-\cosh (x))^2}+\log (1-\cosh (x)) \]
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Rubi [A] time = 0.0670859, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, 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 4392
Rule 2667
Rule 43
Rubi steps
\begin{align*} \int (\coth (x)+\text{csch}(x))^5 \, dx &=-\left (i \int (i+i \cosh (x))^5 \text{csch}^5(x) \, dx\right )\\ &=-\operatorname{Subst}\left (\int \frac{(i+x)^2}{(i-x)^3} \, dx,x,i \cosh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{i-x}+\frac{4}{(-i+x)^3}-\frac{4 i}{(-i+x)^2}\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*}
Mathematica [A] time = 0.0870054, size = 53, normalized size = 1.89 \[ -\frac{1}{2} \text{csch}^4\left (\frac{x}{2}\right )-2 \text{csch}^2\left (\frac{x}{2}\right )+6 \log \left (\sinh \left (\frac{x}{2}\right )\right )+\log (\sinh (x))-5 \log \left (\tanh \left (\frac{x}{2}\right )\right )-6 \log \left (\cosh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.021, size = 75, normalized size = 2.7 \begin{align*} \ln \left ( \sinh \left ( x \right ) \right ) -{\frac{ \left ({\rm coth} \left (x\right ) \right ) ^{2}}{2}}-{\frac{ \left ({\rm coth} \left (x\right ) \right ) ^{4}}{4}}-5\,{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{3}}{ \left ( \sinh \left ( x \right ) \right ) ^{4}}}+{\frac{5\,\cosh \left ( x \right ) }{3\, \left ( \sinh \left ( x \right ) \right ) ^{4}}}+{\frac{8\,{\rm coth} \left (x\right )}{3} \left ( -{\frac{ \left ({\rm csch} \left (x\right ) \right ) ^{3}}{4}}+{\frac{3\,{\rm csch} \left (x\right )}{8}} \right ) }-2\,{\it Artanh} \left ({{\rm e}^{x}} \right ) -{\frac{15\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{4\, \left ( \sinh \left ( x \right ) \right ) ^{4}}}-{\frac{5\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{4\, \left ( \sinh \left ( x \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.27624, size = 319, normalized size = 11.39 \begin{align*} -\frac{5}{2} \, \coth \left (x\right )^{4} + x + \frac{5 \,{\left (5 \, e^{\left (-x\right )} + 3 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-5 \, x\right )} + 5 \, e^{\left (-7 \, x\right )}\right )}}{4 \,{\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{3 \, e^{\left (-x\right )} - 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} + 3 \, e^{\left (-7 \, x\right )}}{4 \,{\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{5 \,{\left (e^{\left (-x\right )} + 7 \, e^{\left (-3 \, x\right )} + 7 \, e^{\left (-5 \, x\right )} + e^{\left (-7 \, x\right )}\right )}}{2 \,{\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{4 \,{\left (e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )}\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} - \frac{20}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{4}} + 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.08206, size = 925, normalized size = 33.04 \begin{align*} -\frac{x \cosh \left (x\right )^{4} + x \sinh \left (x\right )^{4} - 4 \,{\left (x - 2\right )} \cosh \left (x\right )^{3} + 4 \,{\left (x \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right )^{3} + 2 \,{\left (3 \, x - 4\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, x \cosh \left (x\right )^{2} - 6 \,{\left (x - 2\right )} \cosh \left (x\right ) + 3 \, x - 4\right )} \sinh \left (x\right )^{2} - 4 \,{\left (x - 2\right )} \cosh \left (x\right ) - 2 \,{\left (\cosh \left (x\right )^{4} + 4 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{3} + 6 \,{\left (\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) - 4 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 4 \,{\left (x \cosh \left (x\right )^{3} - 3 \,{\left (x - 2\right )} \cosh \left (x\right )^{2} +{\left (3 \, x - 4\right )} \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{4} + 4 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{3} + 6 \,{\left (\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) - 4 \, \cosh \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12696, size = 45, normalized size = 1.61 \begin{align*} -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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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