Optimal. Leaf size=94 \[ -\frac{3 x^2 \sinh (a+b x) \cosh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}-\frac{3 \sinh (a+b x) \cosh (a+b x)}{8 b^4}+\frac{x^3 \sinh ^2(a+b x)}{2 b}+\frac{3 x}{8 b^3}+\frac{x^3}{4 b} \]
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Rubi [A] time = 0.0691723, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5372, 3311, 30, 2635, 8} \[ -\frac{3 x^2 \sinh (a+b x) \cosh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}-\frac{3 \sinh (a+b x) \cosh (a+b x)}{8 b^4}+\frac{x^3 \sinh ^2(a+b x)}{2 b}+\frac{3 x}{8 b^3}+\frac{x^3}{4 b} \]
Antiderivative was successfully verified.
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Rule 5372
Rule 3311
Rule 30
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int x^3 \cosh (a+b x) \sinh (a+b x) \, dx &=\frac{x^3 \sinh ^2(a+b x)}{2 b}-\frac{3 \int x^2 \sinh ^2(a+b x) \, dx}{2 b}\\ &=-\frac{3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}+\frac{x^3 \sinh ^2(a+b x)}{2 b}-\frac{3 \int \sinh ^2(a+b x) \, dx}{4 b^3}+\frac{3 \int x^2 \, dx}{4 b}\\ &=\frac{x^3}{4 b}-\frac{3 \cosh (a+b x) \sinh (a+b x)}{8 b^4}-\frac{3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}+\frac{x^3 \sinh ^2(a+b x)}{2 b}+\frac{3 \int 1 \, dx}{8 b^3}\\ &=\frac{3 x}{8 b^3}+\frac{x^3}{4 b}-\frac{3 \cosh (a+b x) \sinh (a+b x)}{8 b^4}-\frac{3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}+\frac{x^3 \sinh ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.103852, size = 50, normalized size = 0.53 \[ \frac{\left (4 b^3 x^3+6 b x\right ) \cosh (2 (a+b x))-3 \left (2 b^2 x^2+1\right ) \sinh (2 (a+b x))}{16 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 203, normalized size = 2.2 \begin{align*}{\frac{1}{{b}^{4}} \left ({\frac{ \left ( bx+a \right ) ^{3} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{2}}-{\frac{3\, \left ( bx+a \right ) ^{2}\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{4}}-{\frac{ \left ( bx+a \right ) ^{3}}{4}}+{\frac{ \left ( 3\,bx+3\,a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{4}}-{\frac{3\,\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{8}}-{\frac{3\,bx}{8}}-{\frac{3\,a}{8}}-3\,a \left ( 1/2\, \left ( bx+a \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-1/2\, \left ( bx+a \right ) \cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) -1/4\, \left ( bx+a \right ) ^{2}+1/4\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \right ) +3\,{a}^{2} \left ( 1/2\, \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-1/4\,\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) -1/4\,bx-a/4 \right ) -{\frac{{a}^{3} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10756, size = 116, normalized size = 1.23 \begin{align*} \frac{{\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{32 \, b^{4}} + \frac{{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05681, size = 180, normalized size = 1.91 \begin{align*} \frac{{\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{2} - 3 \,{\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) +{\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \sinh \left (b x + a\right )^{2}}{8 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.38946, size = 119, normalized size = 1.27 \begin{align*} \begin{cases} \frac{x^{3} \sinh ^{2}{\left (a + b x \right )}}{4 b} + \frac{x^{3} \cosh ^{2}{\left (a + b x \right )}}{4 b} - \frac{3 x^{2} \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{4 b^{2}} + \frac{3 x \sinh ^{2}{\left (a + b x \right )}}{8 b^{3}} + \frac{3 x \cosh ^{2}{\left (a + b x \right )}}{8 b^{3}} - \frac{3 \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{8 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \sinh{\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 = 1.16116, size = 99, normalized size = 1.05 \begin{align*} \frac{{\left (4 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, b x - 3\right )} e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b^{4}} + \frac{{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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