Optimal. Leaf size=117 \[ \frac{2 x^2 \cosh (a+b x)}{3 b^2}-\frac{x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b^2}+\frac{2 x \sinh ^3(a+b x)}{9 b^3}-\frac{4 x \sinh (a+b x)}{3 b^3}-\frac{2 \cosh ^3(a+b x)}{27 b^4}+\frac{14 \cosh (a+b x)}{9 b^4}+\frac{x^3 \sinh ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.14488, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {5372, 3311, 3296, 2638, 2633} \[ \frac{2 x^2 \cosh (a+b x)}{3 b^2}-\frac{x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b^2}+\frac{2 x \sinh ^3(a+b x)}{9 b^3}-\frac{4 x \sinh (a+b x)}{3 b^3}-\frac{2 \cosh ^3(a+b x)}{27 b^4}+\frac{14 \cosh (a+b x)}{9 b^4}+\frac{x^3 \sinh ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 5372
Rule 3311
Rule 3296
Rule 2638
Rule 2633
Rubi steps
\begin{align*} \int x^3 \cosh (a+b x) \sinh ^2(a+b x) \, dx &=\frac{x^3 \sinh ^3(a+b x)}{3 b}-\frac{\int x^2 \sinh ^3(a+b x) \, dx}{b}\\ &=-\frac{x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac{2 x \sinh ^3(a+b x)}{9 b^3}+\frac{x^3 \sinh ^3(a+b x)}{3 b}-\frac{2 \int \sinh ^3(a+b x) \, dx}{9 b^3}+\frac{2 \int x^2 \sinh (a+b x) \, dx}{3 b}\\ &=\frac{2 x^2 \cosh (a+b x)}{3 b^2}-\frac{x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac{2 x \sinh ^3(a+b x)}{9 b^3}+\frac{x^3 \sinh ^3(a+b x)}{3 b}+\frac{2 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (a+b x)\right )}{9 b^4}-\frac{4 \int x \cosh (a+b x) \, dx}{3 b^2}\\ &=\frac{2 \cosh (a+b x)}{9 b^4}+\frac{2 x^2 \cosh (a+b x)}{3 b^2}-\frac{2 \cosh ^3(a+b x)}{27 b^4}-\frac{4 x \sinh (a+b x)}{3 b^3}-\frac{x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac{2 x \sinh ^3(a+b x)}{9 b^3}+\frac{x^3 \sinh ^3(a+b x)}{3 b}+\frac{4 \int \sinh (a+b x) \, dx}{3 b^3}\\ &=\frac{14 \cosh (a+b x)}{9 b^4}+\frac{2 x^2 \cosh (a+b x)}{3 b^2}-\frac{2 \cosh ^3(a+b x)}{27 b^4}-\frac{4 x \sinh (a+b x)}{3 b^3}-\frac{x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac{2 x \sinh ^3(a+b x)}{9 b^3}+\frac{x^3 \sinh ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.398752, size = 84, normalized size = 0.72 \[ \frac{81 \left (b^2 x^2+2\right ) \cosh (a+b x)-\left (9 b^2 x^2+2\right ) \cosh (3 (a+b x))+6 b x \sinh (a+b x) \left (\left (3 b^2 x^2+2\right ) \cosh (2 (a+b x))-3 b^2 x^2-26\right )}{108 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 334, normalized size = 2.9 \begin{align*}{\frac{1}{{b}^{4}} \left ({\frac{ \left ( bx+a \right ) ^{3}\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}}-{\frac{ \left ( bx+a \right ) ^{3}\sinh \left ( bx+a \right ) }{3}}-{\frac{ \left ( bx+a \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) }{3}}+{\frac{2\, \left ( bx+a \right ) ^{2}\cosh \left ( bx+a \right ) }{3}}+{\frac{ \left ( 2\,bx+2\,a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{9}}-{\frac{ \left ( 14\,bx+14\,a \right ) \sinh \left ( bx+a \right ) }{9}}-{\frac{2\,\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{27}}+{\frac{40\,\cosh \left ( bx+a \right ) }{27}}-3\,a \left ( 1/3\, \left ( bx+a \right ) ^{2}\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-1/3\, \left ( bx+a \right ) ^{2}\sinh \left ( bx+a \right ) -2/9\, \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) +4/9\, \left ( bx+a \right ) \cosh \left ( bx+a \right ) +{\frac{2\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{27}}-{\frac{14\,\sinh \left ( bx+a \right ) }{27}} \right ) +3\,{a}^{2} \left ( 1/3\, \left ( bx+a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-1/3\, \left ( bx+a \right ) \sinh \left ( bx+a \right ) -1/9\,\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}+2/9\,\cosh \left ( bx+a \right ) \right ) -{a}^{3} \left ({\frac{\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}}-{\frac{\sinh \left ( bx+a \right ) }{3}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13066, size = 216, normalized size = 1.85 \begin{align*} \frac{{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 9 \, b^{2} x^{2} e^{\left (3 \, a\right )} + 6 \, b x e^{\left (3 \, a\right )} - 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{216 \, b^{4}} - \frac{{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{8 \, b^{4}} + \frac{{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} - \frac{{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76391, size = 332, normalized size = 2.84 \begin{align*} -\frac{{\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} + 3 \,{\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \,{\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \sinh \left (b x + a\right )^{3} - 81 \,{\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) + 9 \,{\left (3 \, b^{3} x^{3} -{\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )^{2} + 18 \, b x\right )} \sinh \left (b x + a\right )}{108 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.19061, size = 146, normalized size = 1.25 \begin{align*} \begin{cases} \frac{x^{3} \sinh ^{3}{\left (a + b x \right )}}{3 b} - \frac{x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{b^{2}} + \frac{2 x^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac{14 x \sinh ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac{4 x \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac{14 \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{9 b^{4}} + \frac{40 \cosh ^{3}{\left (a + b x \right )}}{27 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \sinh ^{2}{\left (a \right )} \cosh{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.162, size = 189, normalized size = 1.62 \begin{align*} \frac{{\left (9 \, b^{3} x^{3} - 9 \, b^{2} x^{2} + 6 \, b x - 2\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{4}} - \frac{{\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} e^{\left (b x + a\right )}}{8 \, b^{4}} + \frac{{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} - \frac{{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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