Optimal. Leaf size=41 \[ \frac{\coth ^5(x)}{5 a}-\frac{\text{csch}^5(x)}{5 a}-\frac{2 \text{csch}^3(x)}{3 a}-\frac{\text{csch}(x)}{a} \]
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Rubi [A] time = 0.0807787, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2706, 2607, 30, 2606, 194} \[ \frac{\coth ^5(x)}{5 a}-\frac{\text{csch}^5(x)}{5 a}-\frac{2 \text{csch}^3(x)}{3 a}-\frac{\text{csch}(x)}{a} \]
Antiderivative was successfully verified.
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Rule 2706
Rule 2607
Rule 30
Rule 2606
Rule 194
Rubi steps
\begin{align*} \int \frac{\coth ^4(x)}{a+a \cosh (x)} \, dx &=\frac{\int \coth ^5(x) \text{csch}(x) \, dx}{a}-\frac{\int \coth ^4(x) \text{csch}^2(x) \, dx}{a}\\ &=-\frac{i \operatorname{Subst}\left (\int x^4 \, dx,x,i \coth (x)\right )}{a}-\frac{i \operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,-i \text{csch}(x)\right )}{a}\\ &=\frac{\coth ^5(x)}{5 a}-\frac{i \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-i \text{csch}(x)\right )}{a}\\ &=\frac{\coth ^5(x)}{5 a}-\frac{\text{csch}(x)}{a}-\frac{2 \text{csch}^3(x)}{3 a}-\frac{\text{csch}^5(x)}{5 a}\\ \end{align*}
Mathematica [A] time = 0.0728887, size = 41, normalized size = 1. \[ -\frac{(8 \cosh (x)+36 \cosh (2 x)+24 \cosh (3 x)-3 \cosh (4 x)-25) \text{csch}^3(x)}{120 a (\cosh (x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 45, normalized size = 1.1 \begin{align*}{\frac{1}{16\,a} \left ({\frac{1}{5} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}+{\frac{4}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+6\,\tanh \left ( x/2 \right ) -4\, \left ( \tanh \left ( x/2 \right ) \right ) ^{-1}-{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0675, size = 633, normalized size = 15.44 \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 = 1.85135, size = 728, normalized size = 17.76 \begin{align*} -\frac{2 \,{\left (15 \, \cosh \left (x\right )^{4} + 6 \,{\left (10 \, \cosh \left (x\right ) + 3\right )} \sinh \left (x\right )^{3} + 15 \, \sinh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{3} + 2 \,{\left (45 \, \cosh \left (x\right )^{2} + 18 \, \cosh \left (x\right ) + 2\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 2 \,{\left (30 \, \cosh \left (x\right )^{3} + 27 \, \cosh \left (x\right )^{2} - 14 \, \cosh \left (x\right ) - 23\right )} \sinh \left (x\right ) - 4 \, \cosh \left (x\right ) + 13\right )}}{15 \,{\left (a \cosh \left (x\right )^{5} + a \sinh \left (x\right )^{5} + 2 \, a \cosh \left (x\right )^{4} +{\left (5 \, a \cosh \left (x\right ) + 2 \, a\right )} \sinh \left (x\right )^{4} - 3 \, a \cosh \left (x\right )^{3} +{\left (10 \, a \cosh \left (x\right )^{2} + 8 \, a \cosh \left (x\right ) - a\right )} \sinh \left (x\right )^{3} - 8 \, a \cosh \left (x\right )^{2} +{\left (10 \, a \cosh \left (x\right )^{3} + 12 \, a \cosh \left (x\right )^{2} - 9 \, a \cosh \left (x\right ) - 8 \, a\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) +{\left (5 \, a \cosh \left (x\right )^{4} + 8 \, a \cosh \left (x\right )^{3} - 3 \, a \cosh \left (x\right )^{2} - 8 \, a \cosh \left (x\right ) - 2 \, a\right )} \sinh \left (x\right ) + 6 \, a\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*} \frac{\int \frac{\coth ^{4}{\left (x \right )}}{\cosh{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31358, size = 80, normalized size = 1.95 \begin{align*} -\frac{15 \, e^{\left (2 \, x\right )} - 24 \, e^{x} + 13}{24 \, a{\left (e^{x} - 1\right )}^{3}} - \frac{165 \, e^{\left (4 \, x\right )} + 480 \, e^{\left (3 \, x\right )} + 650 \, e^{\left (2 \, x\right )} + 400 \, e^{x} + 113}{120 \, a{\left (e^{x} + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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