Optimal. Leaf size=25 \[ -\frac{1}{9} \coth ^9(x)+\frac{2 \coth ^7(x)}{7}-\frac{\coth ^5(x)}{5} \]
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Rubi [A] time = 0.0293946, antiderivative size = 25, 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, 270} \[ -\frac{1}{9} \coth ^9(x)+\frac{2 \coth ^7(x)}{7}-\frac{\coth ^5(x)}{5} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 270
Rubi steps
\begin{align*} \int \coth ^4(x) \text{csch}^6(x) \, dx &=i \operatorname{Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,i \coth (x)\right )\\ &=i \operatorname{Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,i \coth (x)\right )\\ &=-\frac{1}{5} \coth ^5(x)+\frac{2 \coth ^7(x)}{7}-\frac{\coth ^9(x)}{9}\\ \end{align*}
Mathematica [A] time = 0.027158, size = 47, normalized size = 1.88 \[ -\frac{8 \coth (x)}{315}-\frac{1}{9} \coth (x) \text{csch}^8(x)-\frac{10}{63} \coth (x) \text{csch}^6(x)-\frac{1}{105} \coth (x) \text{csch}^4(x)+\frac{4}{315} \coth (x) \text{csch}^2(x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 50, normalized size = 2. \begin{align*} -{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{3}}{6\, \left ( \sinh \left ( x \right ) \right ) ^{9}}}+{\frac{\cosh \left ( x \right ) }{16\, \left ( \sinh \left ( x \right ) \right ) ^{9}}}+{\frac{{\rm coth} \left (x\right )}{16} \left ( -{\frac{128}{315}}-{\frac{ \left ({\rm csch} \left (x\right ) \right ) ^{8}}{9}}+{\frac{8\, \left ({\rm csch} \left (x\right ) \right ) ^{6}}{63}}-{\frac{16\, \left ({\rm csch} \left (x\right ) \right ) ^{4}}{105}}+{\frac{64\, \left ({\rm csch} \left (x\right ) \right ) ^{2}}{315}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.25276, size = 582, normalized size = 23.28 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.90838, size = 1516, normalized size = 60.64 \begin{align*} \text{result too large to display} \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 [B] time = 1.17559, size = 65, normalized size = 2.6 \begin{align*} -\frac{16 \,{\left (210 \, e^{\left (12 \, x\right )} + 315 \, e^{\left (10 \, x\right )} + 441 \, e^{\left (8 \, x\right )} + 126 \, e^{\left (6 \, x\right )} + 36 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} + 1\right )}}{315 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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