Optimal. Leaf size=15 \[ -\frac{\coth ^4(a+b x)}{4 b} \]
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Rubi [A] time = 0.0290491, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2607, 30} \[ -\frac{\coth ^4(a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \coth ^3(a+b x) \text{csch}^2(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,i \coth (a+b x)\right )}{b}\\ &=-\frac{\coth ^4(a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.004639, size = 15, normalized size = 1. \[ -\frac{\coth ^4(a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 42, normalized size = 2.8 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{4\, \left ( \sinh \left ( bx+a \right ) \right ) ^{4}}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{4\, \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04203, size = 18, normalized size = 1.2 \begin{align*} -\frac{\coth \left (b x + a\right )^{4}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85037, size = 568, normalized size = 37.87 \begin{align*} -\frac{2 \,{\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} +{\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{5} + 5 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + b \sinh \left (b x + a\right )^{5} - 3 \, b \cosh \left (b x + a\right )^{3} + 5 \,{\left (2 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{3} +{\left (10 \, b \cosh \left (b x + a\right )^{3} - 9 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) + 5 \,{\left (b \cosh \left (b x + a\right )^{4} - 3 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + 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*} \int \coth ^{3}{\left (a + b x \right )} \operatorname{csch}^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25696, size = 50, normalized size = 3.33 \begin{align*} -\frac{2 \,{\left (e^{\left (6 \, b x + 6 \, a\right )} + e^{\left (2 \, b x + 2 \, a\right )}\right )}}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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