Optimal. Leaf size=24 \[ -\frac{4}{\cosh (x)+1}+\frac{2}{(\cosh (x)+1)^2}-\log (\cosh (x)+1) \]
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Rubi [A] time = 0.0626796, antiderivative size = 24, normalized size of antiderivative = 1., 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} \[ -\frac{4}{\cosh (x)+1}+\frac{2}{(\cosh (x)+1)^2}-\log (\cosh (x)+1) \]
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 [B] time = 0.0846704, size = 55, normalized size = 2.29 \[ \frac{1}{2} \text{sech}^4\left (\frac{x}{2}\right )-2 \text{sech}^2\left (\frac{x}{2}\right )+6 \log \left (\sinh \left (\frac{x}{2}\right )\right )-\log (\sinh (x))-5 \log \left (\tanh \left (\frac{x}{2}\right )\right )-6 \log \left (\cosh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 77, normalized size = 3.2 \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 = 1.23135, size = 321, normalized size = 13.38 \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 = 2.04957, size = 923, normalized size = 38.46 \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] time = 0., size = 0, normalized size = 0. \begin{align*} - \int 5 \coth{\left (x \right )} \operatorname{csch}^{4}{\left (x \right )}\, dx - \int - 10 \coth ^{2}{\left (x \right )} \operatorname{csch}^{3}{\left (x \right )}\, dx - \int 10 \coth ^{3}{\left (x \right )} \operatorname{csch}^{2}{\left (x \right )}\, dx - \int - 5 \coth ^{4}{\left (x \right )} \operatorname{csch}{\left (x \right )}\, dx - \int \coth ^{5}{\left (x \right )}\, dx - \int - \operatorname{csch}^{5}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12839, size = 38, normalized size = 1.58 \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 (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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