Optimal. Leaf size=36 \[ \frac{\left (a+b \cosh ^2(x)\right )^5}{10 b^2}-\frac{a \left (a+b \cosh ^2(x)\right )^4}{8 b^2} \]
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Rubi [A] time = 0.0948361, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4335, 266, 43} \[ \frac{\left (a+b \cosh ^2(x)\right )^5}{10 b^2}-\frac{a \left (a+b \cosh ^2(x)\right )^4}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 4335
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \cosh ^3(x) \left (a+b \cosh ^2(x)\right )^3 \sinh (x) \, dx &=\operatorname{Subst}\left (\int x^3 \left (a+b x^2\right )^3 \, dx,x,\cosh (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^3 \, dx,x,\cosh ^2(x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^3}{b}+\frac{(a+b x)^4}{b}\right ) \, dx,x,\cosh ^2(x)\right )\\ &=-\frac{a \left (a+b \cosh ^2(x)\right )^4}{8 b^2}+\frac{\left (a+b \cosh ^2(x)\right )^5}{10 b^2}\\ \end{align*}
Mathematica [B] time = 0.248408, size = 136, normalized size = 3.78 \[ \frac{1}{32} \left (12 a^2 b \cosh ^4(x)+4 a^2 b \cosh (3 x) \cosh ^3(x)+4 a^3 \cosh (2 x)+a^3 \cosh (4 x)+8 a b^2 \cosh ^6(x)+\frac{1}{32} a b^2 (48 \cosh (2 x)+36 \cosh (4 x)+16 \cosh (6 x)+3 \cosh (8 x))+2 b^3 \cosh ^8(x)+\frac{1}{320} b^3 (140 \cosh (2 x)+100 \cosh (4 x)+50 \cosh (6 x)+15 \cosh (8 x)+2 \cosh (10 x))\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 40, normalized size = 1.1 \begin{align*}{\frac{{b}^{3} \left ( \cosh \left ( x \right ) \right ) ^{10}}{10}}+{\frac{3\,a{b}^{2} \left ( \cosh \left ( x \right ) \right ) ^{8}}{8}}+{\frac{{a}^{2}b \left ( \cosh \left ( x \right ) \right ) ^{6}}{2}}+{\frac{{a}^{3} \left ( \cosh \left ( x \right ) \right ) ^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11638, size = 53, normalized size = 1.47 \begin{align*} \frac{1}{10} \, b^{3} \cosh \left (x\right )^{10} + \frac{3}{8} \, a b^{2} \cosh \left (x\right )^{8} + \frac{1}{2} \, a^{2} b \cosh \left (x\right )^{6} + \frac{1}{4} \, a^{3} \cosh \left (x\right )^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.20736, size = 1041, normalized size = 28.92 \begin{align*} \frac{1}{5120} \, b^{3} \cosh \left (x\right )^{10} + \frac{1}{5120} \, b^{3} \sinh \left (x\right )^{10} + \frac{1}{1024} \,{\left (3 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (x\right )^{8} + \frac{1}{1024} \,{\left (9 \, b^{3} \cosh \left (x\right )^{2} + 3 \, a b^{2} + 2 \, b^{3}\right )} \sinh \left (x\right )^{8} + \frac{1}{1024} \,{\left (16 \, a^{2} b + 24 \, a b^{2} + 9 \, b^{3}\right )} \cosh \left (x\right )^{6} + \frac{1}{1024} \,{\left (42 \, b^{3} \cosh \left (x\right )^{4} + 16 \, a^{2} b + 24 \, a b^{2} + 9 \, b^{3} + 28 \,{\left (3 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{6} + \frac{1}{256} \,{\left (8 \, a^{3} + 24 \, a^{2} b + 21 \, a b^{2} + 6 \, b^{3}\right )} \cosh \left (x\right )^{4} + \frac{1}{1024} \,{\left (42 \, b^{3} \cosh \left (x\right )^{6} + 70 \,{\left (3 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (x\right )^{4} + 32 \, a^{3} + 96 \, a^{2} b + 84 \, a b^{2} + 24 \, b^{3} + 15 \,{\left (16 \, a^{2} b + 24 \, a b^{2} + 9 \, b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{4} + \frac{1}{512} \,{\left (64 \, a^{3} + 120 \, a^{2} b + 84 \, a b^{2} + 21 \, b^{3}\right )} \cosh \left (x\right )^{2} + \frac{1}{1024} \,{\left (9 \, b^{3} \cosh \left (x\right )^{8} + 28 \,{\left (3 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (x\right )^{6} + 15 \,{\left (16 \, a^{2} b + 24 \, a b^{2} + 9 \, b^{3}\right )} \cosh \left (x\right )^{4} + 128 \, a^{3} + 240 \, a^{2} b + 168 \, a b^{2} + 42 \, b^{3} + 24 \,{\left (8 \, a^{3} + 24 \, a^{2} b + 21 \, a b^{2} + 6 \, b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 24.0822, size = 100, normalized size = 2.78 \begin{align*} \frac{a^{3} \cosh ^{4}{\left (x \right )}}{4} + \frac{a^{2} b \cosh ^{6}{\left (x \right )}}{2} - \frac{3 a b^{2} \sinh ^{8}{\left (x \right )}}{8} + \frac{3 a b^{2} \sinh ^{6}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{2} - \frac{9 a b^{2} \sinh ^{4}{\left (x \right )} \cosh ^{4}{\left (x \right )}}{4} + \frac{3 a b^{2} \sinh ^{2}{\left (x \right )} \cosh ^{6}{\left (x \right )}}{2} + \frac{b^{3} \cosh ^{10}{\left (x \right )}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16365, size = 302, normalized size = 8.39 \begin{align*} \frac{1}{10240} \, b^{3}{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{5} + \frac{3}{2048} \, a b^{2}{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{4} + \frac{1}{1024} \, b^{3}{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{4} + \frac{1}{128} \, a^{2} b{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{3} + \frac{3}{256} \, a b^{2}{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{3} + \frac{1}{256} \, b^{3}{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{3} + \frac{1}{64} \, a^{3}{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac{3}{64} \, a^{2} b{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac{9}{256} \, a b^{2}{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac{1}{128} \, b^{3}{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac{1}{16} \, a^{3}{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + \frac{3}{32} \, a^{2} b{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + \frac{3}{64} \, a b^{2}{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + \frac{1}{128} \, b^{3}{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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