Optimal. Leaf size=64 \[ -\frac{3 a^2 b \text{sech}(c+d x)}{d}+\frac{a^3 \cosh (c+d x)}{d}-\frac{a b^2 \text{sech}^3(c+d x)}{d}-\frac{b^3 \text{sech}^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0592691, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4133, 270} \[ -\frac{3 a^2 b \text{sech}(c+d x)}{d}+\frac{a^3 \cosh (c+d x)}{d}-\frac{a b^2 \text{sech}^3(c+d x)}{d}-\frac{b^3 \text{sech}^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 4133
Rule 270
Rubi steps
\begin{align*} \int \left (a+b \text{sech}^2(c+d x)\right )^3 \sinh (c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b+a x^2\right )^3}{x^6} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^3+\frac{b^3}{x^6}+\frac{3 a b^2}{x^4}+\frac{3 a^2 b}{x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{a^3 \cosh (c+d x)}{d}-\frac{3 a^2 b \text{sech}(c+d x)}{d}-\frac{a b^2 \text{sech}^3(c+d x)}{d}-\frac{b^3 \text{sech}^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.247026, size = 93, normalized size = 1.45 \[ \frac{8 \text{sech}^5(c+d x) \left (a \cosh ^2(c+d x)+b\right )^3 \left (-15 a^2 b \cosh ^4(c+d x)+5 a^3 \cosh ^6(c+d x)-5 a b^2 \cosh ^2(c+d x)-b^3\right )}{5 d (a \cosh (2 (c+d x))+a+2 b)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 58, normalized size = 0.9 \begin{align*} -{\frac{1}{d} \left ({\frac{{b}^{3} \left ({\rm sech} \left (dx+c\right ) \right ) ^{5}}{5}}+a{b}^{2} \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}+3\,{a}^{2}b{\rm sech} \left (dx+c\right )-{\frac{{a}^{3}}{{\rm sech} \left (dx+c\right )}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05862, size = 127, normalized size = 1.98 \begin{align*} \frac{a^{3} \cosh \left (d x + c\right )}{d} - \frac{6 \, a^{2} b}{d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}} - \frac{8 \, a b^{2}}{d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}} - \frac{32 \, b^{3}}{5 \, d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.52159, size = 702, normalized size = 10.97 \begin{align*} \frac{5 \, a^{3} \cosh \left (d x + c\right )^{6} + 5 \, a^{3} \sinh \left (d x + c\right )^{6} + 30 \,{\left (a^{3} - 2 \, a^{2} b\right )} \cosh \left (d x + c\right )^{4} + 15 \,{\left (5 \, a^{3} \cosh \left (d x + c\right )^{2} + 2 \, a^{3} - 4 \, a^{2} b\right )} \sinh \left (d x + c\right )^{4} + 50 \, a^{3} - 180 \, a^{2} b - 80 \, a b^{2} - 32 \, b^{3} + 5 \,{\left (15 \, a^{3} - 48 \, a^{2} b - 16 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2} + 5 \,{\left (15 \, a^{3} \cosh \left (d x + c\right )^{4} + 15 \, a^{3} - 48 \, a^{2} b - 16 \, a b^{2} + 36 \,{\left (a^{3} - 2 \, a^{2} b\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2}}{10 \,{\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + 5 \,{\left (2 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, 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 [A] time = 1.19049, size = 138, normalized size = 2.16 \begin{align*} \frac{a^{3}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{2 \, d} - \frac{2 \,{\left (15 \, a^{2} b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} + 20 \, a b^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 16 \, b^{3}\right )}}{5 \, d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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