Optimal. Leaf size=65 \[ -\frac{\sinh (a+b x) \cosh ^3(a+b x)}{16 b^2}-\frac{3 \sinh (a+b x) \cosh (a+b x)}{32 b^2}+\frac{x \cosh ^4(a+b x)}{4 b}-\frac{3 x}{32 b} \]
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Rubi [A] time = 0.042589, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {5373, 2635, 8} \[ -\frac{\sinh (a+b x) \cosh ^3(a+b x)}{16 b^2}-\frac{3 \sinh (a+b x) \cosh (a+b x)}{32 b^2}+\frac{x \cosh ^4(a+b x)}{4 b}-\frac{3 x}{32 b} \]
Antiderivative was successfully verified.
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Rule 5373
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int x \cosh ^3(a+b x) \sinh (a+b x) \, dx &=\frac{x \cosh ^4(a+b x)}{4 b}-\frac{\int \cosh ^4(a+b x) \, dx}{4 b}\\ &=\frac{x \cosh ^4(a+b x)}{4 b}-\frac{\cosh ^3(a+b x) \sinh (a+b x)}{16 b^2}-\frac{3 \int \cosh ^2(a+b x) \, dx}{16 b}\\ &=\frac{x \cosh ^4(a+b x)}{4 b}-\frac{3 \cosh (a+b x) \sinh (a+b x)}{32 b^2}-\frac{\cosh ^3(a+b x) \sinh (a+b x)}{16 b^2}-\frac{3 \int 1 \, dx}{32 b}\\ &=-\frac{3 x}{32 b}+\frac{x \cosh ^4(a+b x)}{4 b}-\frac{3 \cosh (a+b x) \sinh (a+b x)}{32 b^2}-\frac{\cosh ^3(a+b x) \sinh (a+b x)}{16 b^2}\\ \end{align*}
Mathematica [A] time = 0.154184, size = 50, normalized size = 0.77 \[ -\frac{8 \sinh (2 (a+b x))+\sinh (4 (a+b x))-16 b x \cosh (2 (a+b x))-4 b x \cosh (4 (a+b x))}{128 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 113, normalized size = 1.7 \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 ) ^{2}}{4}}+{\frac{ \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{4}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }{16}}-{\frac{3\,\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{32}}-{\frac{3\,bx}{32}}-{\frac{3\,a}{32}}-a \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{4}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{4}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1851, size = 123, normalized size = 1.89 \begin{align*} \frac{{\left (4 \, b x e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{256 \, b^{2}} + \frac{{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{32 \, b^{2}} + \frac{{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{2}} + \frac{{\left (4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80991, size = 289, normalized size = 4.45 \begin{align*} \frac{b x \cosh \left (b x + a\right )^{4} + b x \sinh \left (b x + a\right )^{4} + 4 \, b x \cosh \left (b x + a\right )^{2} - \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 2 \,{\left (3 \, b x \cosh \left (b x + a\right )^{2} + 2 \, b x\right )} \sinh \left (b x + a\right )^{2} -{\left (\cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{32 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.23581, size = 110, normalized size = 1.69 \begin{align*} \begin{cases} - \frac{3 x \sinh ^{4}{\left (a + b x \right )}}{32 b} + \frac{3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16 b} + \frac{5 x \cosh ^{4}{\left (a + b x \right )}}{32 b} + \frac{3 \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{32 b^{2}} - \frac{5 \sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{32 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \sinh{\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 = 1.15961, size = 109, normalized size = 1.68 \begin{align*} \frac{{\left (4 \, b x - 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{256 \, b^{2}} + \frac{{\left (2 \, b x - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b^{2}} + \frac{{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{2}} + \frac{{\left (4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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