Optimal. Leaf size=72 \[ \frac{1}{2} a^2 x (a+6 b)+\frac{a^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac{b^2 (3 a+b) \tanh (c+d x)}{d}-\frac{b^3 \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0923167, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {4146, 390, 385, 206} \[ \frac{1}{2} a^2 x (a+6 b)+\frac{a^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac{b^2 (3 a+b) \tanh (c+d x)}{d}-\frac{b^3 \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4146
Rule 390
Rule 385
Rule 206
Rubi steps
\begin{align*} \int \cosh ^2(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-b x^2\right )^3}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (b^2 (3 a+b)-b^3 x^2+\frac{a^2 (a+3 b)-3 a^2 b x^2}{\left (1-x^2\right )^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b^2 (3 a+b) \tanh (c+d x)}{d}-\frac{b^3 \tanh ^3(c+d x)}{3 d}+\frac{\operatorname{Subst}\left (\int \frac{a^2 (a+3 b)-3 a^2 b x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{a^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{b^2 (3 a+b) \tanh (c+d x)}{d}-\frac{b^3 \tanh ^3(c+d x)}{3 d}+\frac{\left (a^2 (a+6 b)\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac{1}{2} a^2 (a+6 b) x+\frac{a^3 \cosh (c+d x) \sinh (c+d x)}{2 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 = 0.479235, size = 64, normalized size = 0.89 \[ \frac{6 a^2 (a+6 b) (c+d x)+3 a^3 \sinh (2 (c+d x))+4 b^2 \tanh (c+d x) \left (9 a+b \text{sech}^2(c+d x)+2 b\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 77, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +3\,{a}^{2}b \left ( dx+c \right ) +3\,a{b}^{2}\tanh \left ( dx+c \right ) +{b}^{3} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{3}} \right ) \tanh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13754, size = 216, normalized size = 3. \begin{align*} \frac{1}{8} \, a^{3}{\left (4 \, x + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + 3 \, a^{2} b x + \frac{4}{3} \, b^{3}{\left (\frac{3 \, e^{\left (-2 \, d x - 2 \, c\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 )}} + \frac{1}{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{6 \, a b^{2}}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.30101, size = 670, normalized size = 9.31 \begin{align*} \frac{3 \, a^{3} \sinh \left (d x + c\right )^{5} - 4 \,{\left (18 \, a b^{2} + 4 \, b^{3} - 3 \,{\left (a^{3} + 6 \, a^{2} b\right )} d x\right )} \cosh \left (d x + c\right )^{3} - 12 \,{\left (18 \, a b^{2} + 4 \, b^{3} - 3 \,{\left (a^{3} + 6 \, a^{2} b\right )} d x\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} +{\left (30 \, a^{3} \cosh \left (d x + c\right )^{2} + 9 \, a^{3} + 72 \, a b^{2} + 16 \, b^{3}\right )} \sinh \left (d x + c\right )^{3} - 12 \,{\left (18 \, a b^{2} + 4 \, b^{3} - 3 \,{\left (a^{3} + 6 \, a^{2} b\right )} d x\right )} \cosh \left (d x + c\right ) + 3 \,{\left (5 \, a^{3} \cosh \left (d x + c\right )^{4} + 2 \, a^{3} + 24 \, a b^{2} + 16 \, b^{3} +{\left (9 \, a^{3} + 72 \, a b^{2} + 16 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{24 \,{\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 = 1.18982, size = 215, normalized size = 2.99 \begin{align*} \frac{a^{3} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} + \frac{{\left (a^{3} + 6 \, a^{2} b\right )}{\left (d x + c\right )}}{2 \, d} - \frac{{\left (2 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + a^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} - \frac{2 \,{\left (9 \, a b^{2} 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} + 2 \, 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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