Optimal. Leaf size=82 \[ \frac{1}{8} x \left (3 a^2+8 a b+8 b^2\right )+\frac{3 a (a+2 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{a \sinh (c+d x) \cosh ^3(c+d x) \left (a-b \tanh ^2(c+d x)+b\right )}{4 d} \]
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Rubi [A] time = 0.0929244, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {4146, 413, 385, 206} \[ \frac{1}{8} x \left (3 a^2+8 a b+8 b^2\right )+\frac{3 a (a+2 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{a \sinh (c+d x) \cosh ^3(c+d x) \left (a-b \tanh ^2(c+d x)+b\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 4146
Rule 413
Rule 385
Rule 206
Rubi steps
\begin{align*} \int \cosh ^4(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-b x^2\right )^2}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{a \cosh ^3(c+d x) \sinh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )}{4 d}-\frac{\operatorname{Subst}\left (\int \frac{-(a+b) (3 a+4 b)+b (a+4 b) x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=\frac{3 a (a+2 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{a \cosh ^3(c+d x) \sinh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )}{4 d}+\frac{\left (3 a^2+8 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+8 a b+8 b^2\right ) x+\frac{3 a (a+2 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{a \cosh ^3(c+d x) \sinh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 0.129441, size = 58, normalized size = 0.71 \[ \frac{4 \left (3 a^2+8 a b+8 b^2\right ) (c+d x)+a^2 \sinh (4 (c+d x))+8 a (a+2 b) \sinh (2 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 79, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( \left ({\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( dx+c \right ) }{8}} \right ) \sinh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +2\,ab \left ( 1/2\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) +1/2\,dx+c/2 \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 [A] time = 1.04198, size = 142, normalized size = 1.73 \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}{4} \, a b{\left (4 \, x + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + b^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20434, size = 193, normalized size = 2.35 \begin{align*} \frac{a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} d x +{\left (a^{2} \cosh \left (d x + c\right )^{3} + 4 \,{\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, d} \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 [A] time = 1.15788, size = 204, normalized size = 2.49 \begin{align*} \frac{a^{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 )} + 8 \,{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )}{\left (d x + c\right )} -{\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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