Optimal. Leaf size=59 \[ -\frac{a^3 \coth (c+d x)}{d}-\frac{b^2 (3 a+b) \tanh (c+d x)}{d}+x (a+b)^3-\frac{b^3 \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0800658, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3670, 461, 207} \[ -\frac{a^3 \coth (c+d x)}{d}-\frac{b^2 (3 a+b) \tanh (c+d x)}{d}+x (a+b)^3-\frac{b^3 \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 461
Rule 207
Rubi steps
\begin{align*} \int \coth ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^3}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-b^2 (3 a+b)+\frac{a^3}{x^2}-b^3 x^2-\frac{(a+b)^3}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{a^3 \coth (c+d x)}{d}-\frac{b^2 (3 a+b) \tanh (c+d x)}{d}-\frac{b^3 \tanh ^3(c+d x)}{3 d}-\frac{(a+b)^3 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=(a+b)^3 x-\frac{a^3 \coth (c+d x)}{d}-\frac{b^2 (3 a+b) \tanh (c+d x)}{d}-\frac{b^3 \tanh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 2.18137, size = 81, normalized size = 1.37 \[ \frac{\tanh (c+d x) \left (-3 a^3 \coth ^2(c+d x)-b^2 \left (9 a+b \tanh ^2(c+d x)+3 b\right )+3 (a+b)^3 \sqrt{\coth ^2(c+d x)} \tanh ^{-1}\left (\sqrt{\coth ^2(c+d x)}\right )\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 80, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( dx+c-{\rm coth} \left (dx+c\right ) \right ) +3\,{a}^{2}b \left ( dx+c \right ) +3\,a{b}^{2} \left ( dx+c-\tanh \left ( dx+c \right ) \right ) +{b}^{3} \left ( dx+c-\tanh \left ( dx+c \right ) -{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{3}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15813, size = 198, normalized size = 3.36 \begin{align*} \frac{1}{3} \, b^{3}{\left (3 \, x + \frac{3 \, c}{d} - \frac{4 \,{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + 3 \, a b^{2}{\left (x + \frac{c}{d} - \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + a^{3}{\left (x + \frac{c}{d} + \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + 3 \, a^{2} b x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.951, size = 811, normalized size = 13.75 \begin{align*} -\frac{{\left (3 \, a^{3} + 9 \, a b^{2} + 4 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} - 4 \,{\left (3 \, a^{3} + 9 \, a b^{2} + 4 \, b^{3} + 3 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (3 \, a^{3} + 9 \, a b^{2} + 4 \, b^{3}\right )} \sinh \left (d x + c\right )^{4} + 9 \, a^{3} - 9 \, a b^{2} + 4 \,{\left (3 \, a^{3} - b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (6 \, a^{3} - 2 \, b^{3} + 3 \,{\left (3 \, a^{3} + 9 \, a b^{2} + 4 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 4 \,{\left ({\left (3 \, a^{3} + 9 \, a b^{2} + 4 \, b^{3} + 3 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{3} +{\left (3 \, a^{3} + 9 \, a b^{2} + 4 \, b^{3} + 3 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{12 \,{\left (d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \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 = 1.42829, size = 186, normalized size = 3.15 \begin{align*} \frac{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}{\left (d x + c\right )}}{d} - \frac{2 \, a^{3}}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}} + \frac{2 \,{\left (9 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 6 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 18 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a b^{2} + 4 \, b^{3}\right )}}{3 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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