Optimal. Leaf size=17 \[ \frac{\coth ^3(x)}{3}-\frac{\coth ^5(x)}{5} \]
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Rubi [A] time = 0.0263057, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2607, 14} \[ \frac{\coth ^3(x)}{3}-\frac{\coth ^5(x)}{5} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \coth ^2(x) \text{csch}^4(x) \, dx &=i \operatorname{Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,i \coth (x)\right )\\ &=i \operatorname{Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,i \coth (x)\right )\\ &=\frac{\coth ^3(x)}{3}-\frac{\coth ^5(x)}{5}\\ \end{align*}
Mathematica [A] time = 0.0225496, size = 27, normalized size = 1.59 \[ \frac{2 \coth (x)}{15}-\frac{1}{5} \coth (x) \text{csch}^4(x)-\frac{1}{15} \coth (x) \text{csch}^2(x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 28, normalized size = 1.7 \begin{align*} -{\frac{\cosh \left ( x \right ) }{4\, \left ( \sinh \left ( x \right ) \right ) ^{5}}}-{\frac{{\rm coth} \left (x\right )}{4} \left ( -{\frac{8}{15}}-{\frac{ \left ({\rm csch} \left (x\right ) \right ) ^{4}}{5}}+{\frac{4\, \left ({\rm csch} \left (x\right ) \right ) ^{2}}{15}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01819, size = 201, normalized size = 11.82 \begin{align*} \frac{4 \, e^{\left (-2 \, x\right )}}{3 \,{\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} + \frac{4 \, e^{\left (-4 \, x\right )}}{3 \,{\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} + \frac{4 \, e^{\left (-6 \, x\right )}}{5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1} - \frac{4}{15 \,{\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.83057, size = 555, normalized size = 32.65 \begin{align*} -\frac{8 \,{\left (7 \, \cosh \left (x\right )^{3} + 24 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 21 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 8 \, \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )}}{15 \,{\left (\cosh \left (x\right )^{7} + 7 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + \sinh \left (x\right )^{7} +{\left (21 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{5} - 5 \, \cosh \left (x\right )^{5} + 5 \,{\left (7 \, \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} +{\left (35 \, \cosh \left (x\right )^{4} - 50 \, \cosh \left (x\right )^{2} + 11\right )} \sinh \left (x\right )^{3} + 9 \, \cosh \left (x\right )^{3} +{\left (21 \, \cosh \left (x\right )^{5} - 50 \, \cosh \left (x\right )^{3} + 27 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} +{\left (7 \, \cosh \left (x\right )^{6} - 25 \, \cosh \left (x\right )^{4} + 33 \, \cosh \left (x\right )^{2} - 15\right )} \sinh \left (x\right ) - 5 \, \cosh \left (x\right )\right )}} \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 \coth ^{2}{\left (x \right )} \operatorname{csch}^{4}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18252, size = 41, normalized size = 2.41 \begin{align*} -\frac{4 \,{\left (15 \, e^{\left (6 \, x\right )} + 5 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )}}{15 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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