Optimal. Leaf size=36 \[ \frac {\left (a+b \sinh ^2(x)\right )^5}{10 b^2}-\frac {a \left (a+b \sinh ^2(x)\right )^4}{8 b^2} \]
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Rubi [A] time = 0.09, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 = {3198, 266, 43} \[ \frac {\left (a+b \sinh ^2(x)\right )^5}{10 b^2}-\frac {a \left (a+b \sinh ^2(x)\right )^4}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 3198
Rubi steps
\begin {align*} \int \cosh (x) \sinh ^3(x) \left (a+b \sinh ^2(x)\right )^3 \, dx &=\operatorname {Subst}\left (\int x^3 \left (a+b x^2\right )^3 \, dx,x,\sinh (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int x (a+b x)^3 \, dx,x,\sinh ^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,\sinh ^2(x)\right )\\ &=-\frac {a \left (a+b \sinh ^2(x)\right )^4}{8 b^2}+\frac {\left (a+b \sinh ^2(x)\right )^5}{10 b^2}\\ \end {align*}
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Mathematica [B] time = 0.62, size = 114, normalized size = 3.17 \[ \frac {-20 \left (64 a^3+24 a b^2-7 b^3\right ) \cosh (2 x)+20 \left (16 a^3+18 a b^2-5 b^3\right ) \cosh (4 x)+b \left (320 \sinh ^6(x) \left ((b-4 a)^2-b^2 \cosh (2 x)\right )-10 b (16 a-5 b) \cosh (6 x)+15 b (2 a-b) \cosh (8 x)+2 b^2 \cosh (10 x)\right )}{10240} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 386, normalized size = 10.72 \[ \frac {1}{5120} \, b^{3} \cosh \relax (x)^{10} + \frac {1}{5120} \, b^{3} \sinh \relax (x)^{10} + \frac {1}{1024} \, {\left (3 \, a b^{2} - 2 \, b^{3}\right )} \cosh \relax (x)^{8} + \frac {1}{1024} \, {\left (9 \, b^{3} \cosh \relax (x)^{2} + 3 \, a b^{2} - 2 \, b^{3}\right )} \sinh \relax (x)^{8} + \frac {1}{1024} \, {\left (16 \, a^{2} b - 24 \, a b^{2} + 9 \, b^{3}\right )} \cosh \relax (x)^{6} + \frac {1}{1024} \, {\left (42 \, b^{3} \cosh \relax (x)^{4} + 16 \, a^{2} b - 24 \, a b^{2} + 9 \, b^{3} + 28 \, {\left (3 \, a b^{2} - 2 \, b^{3}\right )} \cosh \relax (x)^{2}\right )} \sinh \relax (x)^{6} + \frac {1}{256} \, {\left (8 \, a^{3} - 24 \, a^{2} b + 21 \, a b^{2} - 6 \, b^{3}\right )} \cosh \relax (x)^{4} + \frac {1}{1024} \, {\left (42 \, b^{3} \cosh \relax (x)^{6} + 70 \, {\left (3 \, a b^{2} - 2 \, b^{3}\right )} \cosh \relax (x)^{4} + 32 \, a^{3} - 96 \, a^{2} b + 84 \, a b^{2} - 24 \, b^{3} + 15 \, {\left (16 \, a^{2} b - 24 \, a b^{2} + 9 \, b^{3}\right )} \cosh \relax (x)^{2}\right )} \sinh \relax (x)^{4} - \frac {1}{512} \, {\left (64 \, a^{3} - 120 \, a^{2} b + 84 \, a b^{2} - 21 \, b^{3}\right )} \cosh \relax (x)^{2} + \frac {1}{1024} \, {\left (9 \, b^{3} \cosh \relax (x)^{8} + 28 \, {\left (3 \, a b^{2} - 2 \, b^{3}\right )} \cosh \relax (x)^{6} + 15 \, {\left (16 \, a^{2} b - 24 \, a b^{2} + 9 \, b^{3}\right )} \cosh \relax (x)^{4} - 128 \, a^{3} + 240 \, a^{2} b - 168 \, a b^{2} + 42 \, b^{3} + 24 \, {\left (8 \, a^{3} - 24 \, a^{2} b + 21 \, a b^{2} - 6 \, b^{3}\right )} \cosh \relax (x)^{2}\right )} \sinh \relax (x)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 224, normalized size = 6.22 \[ \frac {1}{10240} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{5} + \frac {3}{2048} \, a b^{2} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{4} - \frac {1}{1024} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{4} + \frac {1}{128} \, a^{2} b {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{3} - \frac {3}{256} \, a b^{2} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{3} + \frac {1}{256} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{3} + \frac {1}{64} \, a^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} - \frac {3}{64} \, a^{2} b {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac {9}{256} \, a b^{2} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} - \frac {1}{128} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} - \frac {1}{16} \, a^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + \frac {3}{32} \, a^{2} b {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} - \frac {3}{64} \, a b^{2} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + \frac {1}{128} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 40, normalized size = 1.11 \[ \frac {b^{3} \left (\sinh ^{10}\relax (x )\right )}{10}+\frac {3 a \,b^{2} \left (\sinh ^{8}\relax (x )\right )}{8}+\frac {a^{2} b \left (\sinh ^{6}\relax (x )\right )}{2}+\frac {a^{3} \left (\sinh ^{4}\relax (x )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 39, normalized size = 1.08 \[ \frac {1}{10} \, b^{3} \sinh \relax (x)^{10} + \frac {3}{8} \, a b^{2} \sinh \relax (x)^{8} + \frac {1}{2} \, a^{2} b \sinh \relax (x)^{6} + \frac {1}{4} \, a^{3} \sinh \relax (x)^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 39, normalized size = 1.08 \[ \frac {a^3\,{\mathrm {sinh}\relax (x)}^4}{4}+\frac {a^2\,b\,{\mathrm {sinh}\relax (x)}^6}{2}+\frac {3\,a\,b^2\,{\mathrm {sinh}\relax (x)}^8}{8}+\frac {b^3\,{\mathrm {sinh}\relax (x)}^{10}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.84, size = 44, normalized size = 1.22 \[ \frac {a^{3} \sinh ^{4}{\relax (x )}}{4} + \frac {a^{2} b \sinh ^{6}{\relax (x )}}{2} + \frac {3 a b^{2} \sinh ^{8}{\relax (x )}}{8} + \frac {b^{3} \sinh ^{10}{\relax (x )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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