Optimal. Leaf size=67 \[ \frac{3 \sinh (2 a+2 b x)}{128 b^2}-\frac{\sinh (6 a+6 b x)}{1152 b^2}-\frac{3 x \cosh (2 a+2 b x)}{64 b}+\frac{x \cosh (6 a+6 b x)}{192 b} \]
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Rubi [A] time = 0.0726938, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5448, 3296, 2637} \[ \frac{3 \sinh (2 a+2 b x)}{128 b^2}-\frac{\sinh (6 a+6 b x)}{1152 b^2}-\frac{3 x \cosh (2 a+2 b x)}{64 b}+\frac{x \cosh (6 a+6 b x)}{192 b} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{3}{32} x \sinh (2 a+2 b x)+\frac{1}{32} x \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac{1}{32} \int x \sinh (6 a+6 b x) \, dx-\frac{3}{32} \int x \sinh (2 a+2 b x) \, dx\\ &=-\frac{3 x \cosh (2 a+2 b x)}{64 b}+\frac{x \cosh (6 a+6 b x)}{192 b}-\frac{\int \cosh (6 a+6 b x) \, dx}{192 b}+\frac{3 \int \cosh (2 a+2 b x) \, dx}{64 b}\\ &=-\frac{3 x \cosh (2 a+2 b x)}{64 b}+\frac{x \cosh (6 a+6 b x)}{192 b}+\frac{3 \sinh (2 a+2 b x)}{128 b^2}-\frac{\sinh (6 a+6 b x)}{1152 b^2}\\ \end{align*}
Mathematica [A] time = 0.143971, size = 50, normalized size = 0.75 \[ -\frac{-27 \sinh (2 (a+b x))+\sinh (6 (a+b x))+54 b x \cosh (2 (a+b x))-6 b x \cosh (6 (a+b x))}{1152 b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 170, normalized size = 2.5 \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{ \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}{6}}-{\frac{ \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{12}}-{\frac{ \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{12}}-{\frac{\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{36}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }{36}}+{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{24}}+{\frac{bx}{24}}+{\frac{a}{24}}-a \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 = 1.0983, size = 123, normalized size = 1.84 \begin{align*} \frac{{\left (6 \, b x e^{\left (6 \, a\right )} - e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x\right )}}{2304 \, b^{2}} - \frac{3 \,{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{256 \, b^{2}} - \frac{3 \,{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{2}} + \frac{{\left (6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{2304 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.80394, size = 408, normalized size = 6.09 \begin{align*} \frac{3 \, b x \cosh \left (b x + a\right )^{6} + 45 \, b x \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} + 3 \, b x \sinh \left (b x + a\right )^{6} - 10 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} - 27 \, b x \cosh \left (b x + a\right )^{2} + 9 \,{\left (5 \, b x \cosh \left (b x + a\right )^{4} - 3 \, b x\right )} \sinh \left (b x + a\right )^{2} - 3 \,{\left (\cosh \left (b x + a\right )^{5} - 9 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{576 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.62976, size = 148, normalized size = 2.21 \begin{align*} \begin{cases} - \frac{x \sinh ^{6}{\left (a + b x \right )}}{24 b} + \frac{x \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8 b} + \frac{x \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{8 b} - \frac{x \cosh ^{6}{\left (a + b x \right )}}{24 b} + \frac{\sinh ^{5}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{24 b^{2}} - \frac{\sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac{\sinh{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{24 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \sinh ^{3}{\left (a \right )} \cosh ^{3}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15879, size = 109, normalized size = 1.63 \begin{align*} \frac{{\left (6 \, b x - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{2304 \, b^{2}} - \frac{3 \,{\left (2 \, b x - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{256 \, b^{2}} - \frac{3 \,{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{2}} + \frac{{\left (6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{2304 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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