Optimal. Leaf size=43 \[ -\frac{a^2 \coth ^3(c+d x)}{3 d}-\frac{a (a+2 b) \coth (c+d x)}{d}+x (a+b)^2 \]
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Rubi [A] time = 0.0714495, antiderivative size = 43, 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} \[ -\frac{a^2 \coth ^3(c+d x)}{3 d}-\frac{a (a+2 b) \coth (c+d x)}{d}+x (a+b)^2 \]
Antiderivative was successfully verified.
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Rule 3670
Rule 461
Rule 207
Rubi steps
\begin{align*} \int \coth ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^2}{x^4 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x^4}+\frac{a (a+2 b)}{x^2}-\frac{(a+b)^2}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{a (a+2 b) \coth (c+d x)}{d}-\frac{a^2 \coth ^3(c+d x)}{3 d}-\frac{(a+b)^2 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=(a+b)^2 x-\frac{a (a+2 b) \coth (c+d x)}{d}-\frac{a^2 \coth ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.571053, size = 65, normalized size = 1.51 \[ -\frac{\coth (c+d x) \left (a \left (a \coth ^2(c+d x)+3 a+6 b\right )-3 (a+b)^2 \tanh ^{-1}\left (\sqrt{\tanh ^2(c+d x)}\right ) \sqrt{\tanh ^2(c+d x)}\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 59, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( dx+c-{\rm coth} \left (dx+c\right )-{\frac{ \left ({\rm coth} \left (dx+c\right ) \right ) ^{3}}{3}} \right ) +2\,ab \left ( dx+c-{\rm coth} \left (dx+c\right ) \right ) +{b}^{2} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0667, size = 154, normalized size = 3.58 \begin{align*} \frac{1}{3} \, a^{2}{\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 )} + 2 \, a b{\left (x + \frac{c}{d} + \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + b^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06659, size = 489, normalized size = 11.37 \begin{align*} -\frac{2 \,{\left (2 \, a^{2} + 3 \, a b\right )} \cosh \left (d x + c\right )^{3} + 6 \,{\left (2 \, a^{2} + 3 \, a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} -{\left (3 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 4 \, a^{2} + 6 \, a b\right )} \sinh \left (d x + c\right )^{3} - 6 \, a b \cosh \left (d x + c\right ) + 3 \,{\left (3 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} d x -{\left (3 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 4 \, a^{2} + 6 \, a b\right )} \cosh \left (d x + c\right )^{2} + 4 \, a^{2} + 6 \, a b\right )} \sinh \left (d x + c\right )}{3 \,{\left (d \sinh \left (d x + c\right )^{3} + 3 \,{\left (d \cosh \left (d x + c\right )^{2} - d\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 = 1.27391, size = 139, normalized size = 3.23 \begin{align*} \frac{{\left (a^{2} + 2 \, a b + b^{2}\right )}{\left (d x + c\right )}}{d} - \frac{4 \,{\left (3 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} + 3 \, a b\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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