Optimal. Leaf size=105 \[ \frac{3 x \sinh (2 a+2 b x)}{64 b^2}-\frac{x \sinh (6 a+6 b x)}{576 b^2}-\frac{3 \cosh (2 a+2 b x)}{128 b^3}+\frac{\cosh (6 a+6 b x)}{3456 b^3}-\frac{3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac{x^2 \cosh (6 a+6 b x)}{192 b} \]
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Rubi [A] time = 0.140026, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5448, 3296, 2638} \[ \frac{3 x \sinh (2 a+2 b x)}{64 b^2}-\frac{x \sinh (6 a+6 b x)}{576 b^2}-\frac{3 \cosh (2 a+2 b x)}{128 b^3}+\frac{\cosh (6 a+6 b x)}{3456 b^3}-\frac{3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac{x^2 \cosh (6 a+6 b x)}{192 b} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^2 \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{3}{32} x^2 \sinh (2 a+2 b x)+\frac{1}{32} x^2 \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac{1}{32} \int x^2 \sinh (6 a+6 b x) \, dx-\frac{3}{32} \int x^2 \sinh (2 a+2 b x) \, dx\\ &=-\frac{3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac{x^2 \cosh (6 a+6 b x)}{192 b}-\frac{\int x \cosh (6 a+6 b x) \, dx}{96 b}+\frac{3 \int x \cosh (2 a+2 b x) \, dx}{32 b}\\ &=-\frac{3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac{x^2 \cosh (6 a+6 b x)}{192 b}+\frac{3 x \sinh (2 a+2 b x)}{64 b^2}-\frac{x \sinh (6 a+6 b x)}{576 b^2}+\frac{\int \sinh (6 a+6 b x) \, dx}{576 b^2}-\frac{3 \int \sinh (2 a+2 b x) \, dx}{64 b^2}\\ &=-\frac{3 \cosh (2 a+2 b x)}{128 b^3}-\frac{3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac{\cosh (6 a+6 b x)}{3456 b^3}+\frac{x^2 \cosh (6 a+6 b x)}{192 b}+\frac{3 x \sinh (2 a+2 b x)}{64 b^2}-\frac{x \sinh (6 a+6 b x)}{576 b^2}\\ \end{align*}
Mathematica [A] time = 0.219113, size = 72, normalized size = 0.69 \[ \frac{-81 \left (2 b^2 x^2+1\right ) \cosh (2 (a+b x))+\left (18 b^2 x^2+1\right ) \cosh (6 (a+b x))+6 b x (27 \sinh (2 (a+b x))-\sinh (6 (a+b x)))}{3456 b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 358, normalized size = 3.4 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{ \left ( bx+a \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}{6}}-{\frac{ \left ( bx+a \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{12}}-{\frac{ \left ( bx+a \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{12}}-{\frac{ \left ( bx+a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{18}}+{\frac{ \left ( bx+a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{18}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{108}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{216}}-{\frac{5\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{108}}+{\frac{ \left ( bx+a \right ) \cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{12}}+{\frac{ \left ( bx+a \right ) ^{2}}{24}}-2\,a \left ( 1/6\, \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{4}-1/12\, \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-1/12\, \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-1/36\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{5}+1/36\, \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) +1/24\,\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) +1/24\,bx+a/24 \right ) +{a}^{2} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{6}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{12}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{12}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06654, size = 171, normalized size = 1.63 \begin{align*} \frac{{\left (18 \, b^{2} x^{2} e^{\left (6 \, a\right )} - 6 \, b x e^{\left (6 \, a\right )} + e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x\right )}}{6912 \, b^{3}} - \frac{3 \,{\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{256 \, b^{3}} - \frac{3 \,{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{3}} + \frac{{\left (18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{6912 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75665, size = 531, normalized size = 5.06 \begin{align*} -\frac{120 \, b x \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 36 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} -{\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{6} - 15 \,{\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} -{\left (18 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{6} + 81 \,{\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} - 3 \,{\left (5 \,{\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} - 54 \, b^{2} x^{2} - 27\right )} \sinh \left (b x + a\right )^{2} + 36 \,{\left (b x \cosh \left (b x + a\right )^{5} - 9 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{3456 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.4604, size = 223, normalized size = 2.12 \begin{align*} \begin{cases} - \frac{x^{2} \sinh ^{6}{\left (a + b x \right )}}{24 b} + \frac{x^{2} \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8 b} + \frac{x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{8 b} - \frac{x^{2} \cosh ^{6}{\left (a + b x \right )}}{24 b} + \frac{x \sinh ^{5}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{12 b^{2}} - \frac{2 x \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac{x \sinh{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{12 b^{2}} - \frac{5 \sinh ^{6}{\left (a + b x \right )}}{108 b^{3}} + \frac{7 \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{72 b^{3}} - \frac{\sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{24 b^{3}} & \text{for}\: b \neq 0 \\\frac{x^{3} \sinh ^{3}{\left (a \right )} \cosh ^{3}{\left (a \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22813, size = 153, normalized size = 1.46 \begin{align*} \frac{{\left (18 \, b^{2} x^{2} - 6 \, b x + 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{6912 \, b^{3}} - \frac{3 \,{\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{256 \, b^{3}} - \frac{3 \,{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{3}} + \frac{{\left (18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{6912 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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