Optimal. Leaf size=117 \[ -\frac{2 x^2 \sinh (a+b x)}{3 b^2}-\frac{x^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b^2}-\frac{2 \sinh ^3(a+b x)}{27 b^4}-\frac{14 \sinh (a+b x)}{9 b^4}+\frac{2 x \cosh ^3(a+b x)}{9 b^3}+\frac{4 x \cosh (a+b x)}{3 b^3}+\frac{x^3 \cosh ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.120894, 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 = {5373, 3311, 3296, 2637, 2633} \[ -\frac{2 x^2 \sinh (a+b x)}{3 b^2}-\frac{x^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b^2}-\frac{2 \sinh ^3(a+b x)}{27 b^4}-\frac{14 \sinh (a+b x)}{9 b^4}+\frac{2 x \cosh ^3(a+b x)}{9 b^3}+\frac{4 x \cosh (a+b x)}{3 b^3}+\frac{x^3 \cosh ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 5373
Rule 3311
Rule 3296
Rule 2637
Rule 2633
Rubi steps
\begin{align*} \int x^3 \cosh ^2(a+b x) \sinh (a+b x) \, dx &=\frac{x^3 \cosh ^3(a+b x)}{3 b}-\frac{\int x^2 \cosh ^3(a+b x) \, dx}{b}\\ &=\frac{2 x \cosh ^3(a+b x)}{9 b^3}+\frac{x^3 \cosh ^3(a+b x)}{3 b}-\frac{x^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b^2}-\frac{2 \int \cosh ^3(a+b x) \, dx}{9 b^3}-\frac{2 \int x^2 \cosh (a+b x) \, dx}{3 b}\\ &=\frac{2 x \cosh ^3(a+b x)}{9 b^3}+\frac{x^3 \cosh ^3(a+b x)}{3 b}-\frac{2 x^2 \sinh (a+b x)}{3 b^2}-\frac{x^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b^2}-\frac{(2 i) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (a+b x)\right )}{9 b^4}+\frac{4 \int x \sinh (a+b x) \, dx}{3 b^2}\\ &=\frac{4 x \cosh (a+b x)}{3 b^3}+\frac{2 x \cosh ^3(a+b x)}{9 b^3}+\frac{x^3 \cosh ^3(a+b x)}{3 b}-\frac{2 \sinh (a+b x)}{9 b^4}-\frac{2 x^2 \sinh (a+b x)}{3 b^2}-\frac{x^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b^2}-\frac{2 \sinh ^3(a+b x)}{27 b^4}-\frac{4 \int \cosh (a+b x) \, dx}{3 b^3}\\ &=\frac{4 x \cosh (a+b x)}{3 b^3}+\frac{2 x \cosh ^3(a+b x)}{9 b^3}+\frac{x^3 \cosh ^3(a+b x)}{3 b}-\frac{14 \sinh (a+b x)}{9 b^4}-\frac{2 x^2 \sinh (a+b x)}{3 b^2}-\frac{x^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b^2}-\frac{2 \sinh ^3(a+b x)}{27 b^4}\\ \end{align*}
Mathematica [A] time = 0.380855, size = 86, normalized size = 0.74 \[ \frac{27 b x \left (b^2 x^2+6\right ) \cosh (a+b x)+\left (9 b^3 x^3+6 b x\right ) \cosh (3 (a+b x))-2 \sinh (a+b x) \left (\left (9 b^2 x^2+2\right ) \cosh (2 (a+b x))+45 b^2 x^2+82\right )}{108 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 334, normalized size = 2.9 \begin{align*}{\frac{1}{{b}^{4}} \left ({\frac{ \left ( bx+a \right ) ^{3}\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{3}}+{\frac{ \left ( bx+a \right ) ^{3}\cosh \left ( bx+a \right ) }{3}}-{\frac{ \left ( bx+a \right ) ^{2}\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}}-{\frac{2\, \left ( bx+a \right ) ^{2}\sinh \left ( bx+a \right ) }{3}}+{\frac{ \left ( 2\,bx+2\,a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) }{9}}+{\frac{ \left ( 14\,bx+14\,a \right ) \cosh \left ( bx+a \right ) }{9}}-{\frac{2\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{27}}-{\frac{40\,\sinh \left ( bx+a \right ) }{27}}-3\,a \left ( 1/3\, \left ( bx+a \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) +1/3\, \left ( bx+a \right ) ^{2}\cosh \left ( bx+a \right ) -2/9\, \left ( bx+a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-4/9\, \left ( bx+a \right ) \sinh \left ( bx+a \right ) +{\frac{2\,\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{27}}+{\frac{14\,\cosh \left ( bx+a \right ) }{27}} \right ) +3\,{a}^{2} \left ( 1/3\, \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) +1/3\, \left ( bx+a \right ) \cosh \left ( bx+a \right ) -1/9\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-2/9\,\sinh \left ( bx+a \right ) \right ) -{a}^{3} \left ({\frac{\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{3}}+{\frac{\cosh \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 = 1.16696, 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 = 1.99839, size = 332, normalized size = 2.84 \begin{align*} \frac{3 \,{\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )^{3} + 9 \,{\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} -{\left (9 \, b^{2} x^{2} + 2\right )} \sinh \left (b x + a\right )^{3} + 27 \,{\left (b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right ) - 3 \,{\left (27 \, b^{2} x^{2} +{\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} + 54\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 = 4.19104, size = 146, normalized size = 1.25 \begin{align*} \begin{cases} \frac{x^{3} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac{2 x^{2} \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} - \frac{x^{2} \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} - \frac{4 x \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{3 b^{3}} + \frac{14 x \cosh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac{40 \sinh ^{3}{\left (a + b x \right )}}{27 b^{4}} - \frac{14 \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{9 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \sinh{\left (a \right )} \cosh ^{2}{\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 = 1.16019, 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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