Optimal. Leaf size=129 \[ -\frac{\sinh \left (a+b x^3\right ) \cosh ^7\left (a+b x^3\right )}{192 b^2}-\frac{7 \sinh \left (a+b x^3\right ) \cosh ^5\left (a+b x^3\right )}{1152 b^2}-\frac{35 \sinh \left (a+b x^3\right ) \cosh ^3\left (a+b x^3\right )}{4608 b^2}-\frac{35 \sinh \left (a+b x^3\right ) \cosh \left (a+b x^3\right )}{3072 b^2}+\frac{x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac{35 x^3}{3072 b} \]
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Rubi [A] time = 0.140261, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {5373, 5321, 2635, 8} \[ -\frac{\sinh \left (a+b x^3\right ) \cosh ^7\left (a+b x^3\right )}{192 b^2}-\frac{7 \sinh \left (a+b x^3\right ) \cosh ^5\left (a+b x^3\right )}{1152 b^2}-\frac{35 \sinh \left (a+b x^3\right ) \cosh ^3\left (a+b x^3\right )}{4608 b^2}-\frac{35 \sinh \left (a+b x^3\right ) \cosh \left (a+b x^3\right )}{3072 b^2}+\frac{x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac{35 x^3}{3072 b} \]
Antiderivative was successfully verified.
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Rule 5373
Rule 5321
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int x^5 \cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right ) \, dx &=\frac{x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac{\int x^2 \cosh ^8\left (a+b x^3\right ) \, dx}{8 b}\\ &=\frac{x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac{\operatorname{Subst}\left (\int \cosh ^8(a+b x) \, dx,x,x^3\right )}{24 b}\\ &=\frac{x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac{\cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{192 b^2}-\frac{7 \operatorname{Subst}\left (\int \cosh ^6(a+b x) \, dx,x,x^3\right )}{192 b}\\ &=\frac{x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac{7 \cosh ^5\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{1152 b^2}-\frac{\cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{192 b^2}-\frac{35 \operatorname{Subst}\left (\int \cosh ^4(a+b x) \, dx,x,x^3\right )}{1152 b}\\ &=\frac{x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac{35 \cosh ^3\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{4608 b^2}-\frac{7 \cosh ^5\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{1152 b^2}-\frac{\cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{192 b^2}-\frac{35 \operatorname{Subst}\left (\int \cosh ^2(a+b x) \, dx,x,x^3\right )}{1536 b}\\ &=\frac{x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac{35 \cosh \left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{3072 b^2}-\frac{35 \cosh ^3\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{4608 b^2}-\frac{7 \cosh ^5\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{1152 b^2}-\frac{\cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{192 b^2}-\frac{35 \operatorname{Subst}\left (\int 1 \, dx,x,x^3\right )}{3072 b}\\ &=-\frac{35 x^3}{3072 b}+\frac{x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac{35 \cosh \left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{3072 b^2}-\frac{35 \cosh ^3\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{4608 b^2}-\frac{7 \cosh ^5\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{1152 b^2}-\frac{\cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{192 b^2}\\ \end{align*}
Mathematica [A] time = 0.498366, size = 120, normalized size = 0.93 \[ \frac{-672 \sinh \left (2 \left (a+b x^3\right )\right )-168 \sinh \left (4 \left (a+b x^3\right )\right )-32 \sinh \left (6 \left (a+b x^3\right )\right )-3 \sinh \left (8 \left (a+b x^3\right )\right )+1344 b x^3 \cosh \left (2 \left (a+b x^3\right )\right )+672 b x^3 \cosh \left (4 \left (a+b x^3\right )\right )+192 b x^3 \cosh \left (6 \left (a+b x^3\right )\right )+24 b x^3 \cosh \left (8 \left (a+b x^3\right )\right )}{73728 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 194, normalized size = 1.5 \begin{align*}{\frac{ \left ( 8\,b{x}^{3}-1 \right ){{\rm e}^{8\,b{x}^{3}+8\,a}}}{49152\,{b}^{2}}}+{\frac{ \left ( 6\,b{x}^{3}-1 \right ){{\rm e}^{6\,b{x}^{3}+6\,a}}}{4608\,{b}^{2}}}+{\frac{ \left ( 28\,b{x}^{3}-7 \right ){{\rm e}^{4\,b{x}^{3}+4\,a}}}{6144\,{b}^{2}}}+{\frac{ \left ( 14\,b{x}^{3}-7 \right ){{\rm e}^{2\,b{x}^{3}+2\,a}}}{1536\,{b}^{2}}}+{\frac{ \left ( 14\,b{x}^{3}+7 \right ){{\rm e}^{-2\,b{x}^{3}-2\,a}}}{1536\,{b}^{2}}}+{\frac{ \left ( 28\,b{x}^{3}+7 \right ){{\rm e}^{-4\,b{x}^{3}-4\,a}}}{6144\,{b}^{2}}}+{\frac{ \left ( 6\,b{x}^{3}+1 \right ){{\rm e}^{-6\,b{x}^{3}-6\,a}}}{4608\,{b}^{2}}}+{\frac{ \left ( 8\,b{x}^{3}+1 \right ){{\rm e}^{-8\,b{x}^{3}-8\,a}}}{49152\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10528, size = 288, normalized size = 2.23 \begin{align*} \frac{{\left (8 \, b x^{3} e^{\left (8 \, a\right )} - e^{\left (8 \, a\right )}\right )} e^{\left (8 \, b x^{3}\right )}}{49152 \, b^{2}} + \frac{{\left (6 \, b x^{3} e^{\left (6 \, a\right )} - e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x^{3}\right )}}{4608 \, b^{2}} + \frac{7 \,{\left (4 \, b x^{3} e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x^{3}\right )}}{6144 \, b^{2}} + \frac{7 \,{\left (2 \, b x^{3} e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x^{3}\right )}}{1536 \, b^{2}} + \frac{7 \,{\left (2 \, b x^{3} + 1\right )} e^{\left (-2 \, b x^{3} - 2 \, a\right )}}{1536 \, b^{2}} + \frac{7 \,{\left (4 \, b x^{3} + 1\right )} e^{\left (-4 \, b x^{3} - 4 \, a\right )}}{6144 \, b^{2}} + \frac{{\left (6 \, b x^{3} + 1\right )} e^{\left (-6 \, b x^{3} - 6 \, a\right )}}{4608 \, b^{2}} + \frac{{\left (8 \, b x^{3} + 1\right )} e^{\left (-8 \, b x^{3} - 8 \, a\right )}}{49152 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.24002, size = 976, normalized size = 7.57 \begin{align*} \frac{3 \, b x^{3} \cosh \left (b x^{3} + a\right )^{8} + 3 \, b x^{3} \sinh \left (b x^{3} + a\right )^{8} + 24 \, b x^{3} \cosh \left (b x^{3} + a\right )^{6} + 84 \, b x^{3} \cosh \left (b x^{3} + a\right )^{4} - 3 \, \cosh \left (b x^{3} + a\right ) \sinh \left (b x^{3} + a\right )^{7} + 12 \,{\left (7 \, b x^{3} \cosh \left (b x^{3} + a\right )^{2} + 2 \, b x^{3}\right )} \sinh \left (b x^{3} + a\right )^{6} + 168 \, b x^{3} \cosh \left (b x^{3} + a\right )^{2} - 3 \,{\left (7 \, \cosh \left (b x^{3} + a\right )^{3} + 8 \, \cosh \left (b x^{3} + a\right )\right )} \sinh \left (b x^{3} + a\right )^{5} + 6 \,{\left (35 \, b x^{3} \cosh \left (b x^{3} + a\right )^{4} + 60 \, b x^{3} \cosh \left (b x^{3} + a\right )^{2} + 14 \, b x^{3}\right )} \sinh \left (b x^{3} + a\right )^{4} -{\left (21 \, \cosh \left (b x^{3} + a\right )^{5} + 80 \, \cosh \left (b x^{3} + a\right )^{3} + 84 \, \cosh \left (b x^{3} + a\right )\right )} \sinh \left (b x^{3} + a\right )^{3} + 12 \,{\left (7 \, b x^{3} \cosh \left (b x^{3} + a\right )^{6} + 30 \, b x^{3} \cosh \left (b x^{3} + a\right )^{4} + 42 \, b x^{3} \cosh \left (b x^{3} + a\right )^{2} + 14 \, b x^{3}\right )} \sinh \left (b x^{3} + a\right )^{2} - 3 \,{\left (\cosh \left (b x^{3} + a\right )^{7} + 8 \, \cosh \left (b x^{3} + a\right )^{5} + 28 \, \cosh \left (b x^{3} + a\right )^{3} + 56 \, \cosh \left (b x^{3} + a\right )\right )} \sinh \left (b x^{3} + a\right )}{9216 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 137.366, size = 241, normalized size = 1.87 \begin{align*} \begin{cases} - \frac{35 x^{3} \sinh ^{8}{\left (a + b x^{3} \right )}}{3072 b} + \frac{35 x^{3} \sinh ^{6}{\left (a + b x^{3} \right )} \cosh ^{2}{\left (a + b x^{3} \right )}}{768 b} - \frac{35 x^{3} \sinh ^{4}{\left (a + b x^{3} \right )} \cosh ^{4}{\left (a + b x^{3} \right )}}{512 b} + \frac{35 x^{3} \sinh ^{2}{\left (a + b x^{3} \right )} \cosh ^{6}{\left (a + b x^{3} \right )}}{768 b} + \frac{31 x^{3} \cosh ^{8}{\left (a + b x^{3} \right )}}{1024 b} + \frac{35 \sinh ^{7}{\left (a + b x^{3} \right )} \cosh{\left (a + b x^{3} \right )}}{3072 b^{2}} - \frac{385 \sinh ^{5}{\left (a + b x^{3} \right )} \cosh ^{3}{\left (a + b x^{3} \right )}}{9216 b^{2}} + \frac{511 \sinh ^{3}{\left (a + b x^{3} \right )} \cosh ^{5}{\left (a + b x^{3} \right )}}{9216 b^{2}} - \frac{31 \sinh{\left (a + b x^{3} \right )} \cosh ^{7}{\left (a + b x^{3} \right )}}{1024 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{6} \sinh{\left (a \right )} \cosh ^{7}{\left (a \right )}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16338, size = 516, normalized size = 4. \begin{align*} \frac{24 \,{\left (b x^{3} + a\right )} e^{\left (8 \, b x^{3} + 8 \, a\right )} - 24 \, a e^{\left (8 \, b x^{3} + 8 \, a\right )} + 192 \,{\left (b x^{3} + a\right )} e^{\left (6 \, b x^{3} + 6 \, a\right )} - 192 \, a e^{\left (6 \, b x^{3} + 6 \, a\right )} + 672 \,{\left (b x^{3} + a\right )} e^{\left (4 \, b x^{3} + 4 \, a\right )} - 672 \, a e^{\left (4 \, b x^{3} + 4 \, a\right )} + 1344 \,{\left (b x^{3} + a\right )} e^{\left (2 \, b x^{3} + 2 \, a\right )} - 1344 \, a e^{\left (2 \, b x^{3} + 2 \, a\right )} + 1344 \,{\left (b x^{3} + a\right )} e^{\left (-2 \, b x^{3} - 2 \, a\right )} - 1344 \, a e^{\left (-2 \, b x^{3} - 2 \, a\right )} + 672 \,{\left (b x^{3} + a\right )} e^{\left (-4 \, b x^{3} - 4 \, a\right )} - 672 \, a e^{\left (-4 \, b x^{3} - 4 \, a\right )} + 192 \,{\left (b x^{3} + a\right )} e^{\left (-6 \, b x^{3} - 6 \, a\right )} - 192 \, a e^{\left (-6 \, b x^{3} - 6 \, a\right )} + 24 \,{\left (b x^{3} + a\right )} e^{\left (-8 \, b x^{3} - 8 \, a\right )} - 24 \, a e^{\left (-8 \, b x^{3} - 8 \, a\right )} - 3 \, e^{\left (8 \, b x^{3} + 8 \, a\right )} - 32 \, e^{\left (6 \, b x^{3} + 6 \, a\right )} - 168 \, e^{\left (4 \, b x^{3} + 4 \, a\right )} - 672 \, e^{\left (2 \, b x^{3} + 2 \, a\right )} + 672 \, e^{\left (-2 \, b x^{3} - 2 \, a\right )} + 168 \, e^{\left (-4 \, b x^{3} - 4 \, a\right )} + 32 \, e^{\left (-6 \, b x^{3} - 6 \, a\right )} + 3 \, e^{\left (-8 \, b x^{3} - 8 \, a\right )}}{147456 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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