Optimal. Leaf size=114 \[ -\frac{\left (a^2-8 a b+4 b^2\right ) \tanh (c+d x)}{4 d}+\frac{1}{8} x \left (3 a^2-24 a b+8 b^2\right )+\frac{a^2 \sinh ^4(c+d x) \tanh (c+d x)}{4 d}-\frac{a (a-8 b) \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac{b^2 \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.138374, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4132, 463, 455, 1153, 206} \[ -\frac{\left (a^2-8 a b+4 b^2\right ) \tanh (c+d x)}{4 d}+\frac{1}{8} x \left (3 a^2-24 a b+8 b^2\right )+\frac{a^2 \sinh ^4(c+d x) \tanh (c+d x)}{4 d}-\frac{a (a-8 b) \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac{b^2 \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4132
Rule 463
Rule 455
Rule 1153
Rule 206
Rubi steps
\begin{align*} \int \left (a+b \text{sech}^2(c+d x)\right )^2 \sinh ^4(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (a+b-b x^2\right )^2}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{a^2 \sinh ^4(c+d x) \tanh (c+d x)}{4 d}-\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (5 a^2-4 (a+b)^2+4 b^2 x^2\right )}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=-\frac{a (a-8 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{a^2 \sinh ^4(c+d x) \tanh (c+d x)}{4 d}-\frac{\operatorname{Subst}\left (\int \frac{-a (a-8 b)-2 a (a-8 b) x^2-8 b^2 x^4}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac{a (a-8 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{a^2 \sinh ^4(c+d x) \tanh (c+d x)}{4 d}-\frac{\operatorname{Subst}\left (\int \left (2 \left (a^2-8 a b+4 b^2\right )+8 b^2 x^2+\frac{-3 a^2+24 a b-8 b^2}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac{a (a-8 b) \cosh (c+d x) \sinh (c+d x)}{8 d}-\frac{\left (a^2-8 a b+4 b^2\right ) \tanh (c+d x)}{4 d}+\frac{a^2 \sinh ^4(c+d x) \tanh (c+d x)}{4 d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}+\frac{\left (3 a^2-24 a b+8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac{1}{8} \left (3 a^2-24 a b+8 b^2\right ) x-\frac{a (a-8 b) \cosh (c+d x) \sinh (c+d x)}{8 d}-\frac{\left (a^2-8 a b+4 b^2\right ) \tanh (c+d x)}{4 d}+\frac{a^2 \sinh ^4(c+d x) \tanh (c+d x)}{4 d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 1.6206, size = 153, normalized size = 1.34 \[ \frac{\text{sech}^3(c+d x) \left (a \cosh ^2(c+d x)+b\right )^2 \left (3 \cosh ^3(c+d x) \left (4 d x \left (3 a^2-24 a b+8 b^2\right )+a^2 \sinh (4 (c+d x))-8 a (a-2 b) \sinh (2 (c+d x))\right )+64 b (3 a-2 b) \text{sech}(c) \sinh (d x) \cosh ^2(c+d x)+32 b^2 \tanh (c) \cosh (c+d x)+32 b^2 \text{sech}(c) \sinh (d x)\right )}{24 d (a \cosh (2 (c+d x))+a+2 b)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 109, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{3\,\sinh \left ( dx+c \right ) }{8}} \right ) \cosh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +2\,ab \left ( 1/2\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{\cosh \left ( dx+c \right ) }}-3/2\,dx-3/2\,c+3/2\,\tanh \left ( dx+c \right ) \right ) +{b}^{2} \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 = 1.05235, size = 285, normalized size = 2.5 \begin{align*} \frac{1}{64} \, a^{2}{\left (24 \, x + \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac{8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac{1}{3} \, b^{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 )} - \frac{1}{4} \, a b{\left (\frac{12 \,{\left (d x + c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )}\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.70513, size = 880, normalized size = 7.72 \begin{align*} \frac{3 \, a^{2} \sinh \left (d x + c\right )^{7} + 3 \,{\left (21 \, a^{2} \cosh \left (d x + c\right )^{2} - 5 \, a^{2} + 16 \, a b\right )} \sinh \left (d x + c\right )^{5} + 8 \,{\left (3 \,{\left (3 \, a^{2} - 24 \, a b + 8 \, b^{2}\right )} d x - 48 \, a b + 32 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 24 \,{\left (3 \,{\left (3 \, a^{2} - 24 \, a b + 8 \, b^{2}\right )} d x - 48 \, a b + 32 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} +{\left (105 \, a^{2} \cosh \left (d x + c\right )^{4} - 30 \,{\left (5 \, a^{2} - 16 \, a b\right )} \cosh \left (d x + c\right )^{2} - 63 \, a^{2} + 528 \, a b - 256 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 24 \,{\left (3 \,{\left (3 \, a^{2} - 24 \, a b + 8 \, b^{2}\right )} d x - 48 \, a b + 32 \, b^{2}\right )} \cosh \left (d x + c\right ) + 3 \,{\left (7 \, a^{2} \cosh \left (d x + c\right )^{6} - 5 \,{\left (5 \, a^{2} - 16 \, a b\right )} \cosh \left (d x + c\right )^{4} -{\left (63 \, a^{2} - 528 \, a b + 256 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} - 15 \, a^{2} + 160 \, a b\right )} \sinh \left (d x + c\right )}{192 \,{\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \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 = 1.182, size = 328, normalized size = 2.88 \begin{align*} \frac{{\left (3 \, a^{2} - 24 \, a b + 8 \, b^{2}\right )}{\left (d x + c\right )}}{8 \, d} - \frac{{\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 144 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 16 \, a b e^{\left (2 \, d x + 2 \, c\right )} + a^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} + \frac{a^{2} d e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + 16 \, a b d e^{\left (2 \, d x + 2 \, c\right )}}{64 \, d^{2}} - \frac{4 \,{\left (3 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b - 2 \, b^{2}\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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