Optimal. Leaf size=60 \[ \frac{\sinh (4 a+4 b x)}{256 b^3}-\frac{x \cosh (4 a+4 b x)}{64 b^2}+\frac{x^2 \sinh (4 a+4 b x)}{32 b}-\frac{x^3}{24} \]
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Rubi [A] time = 0.10117, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5448, 3296, 2637} \[ \frac{\sinh (4 a+4 b x)}{256 b^3}-\frac{x \cosh (4 a+4 b x)}{64 b^2}+\frac{x^2 \sinh (4 a+4 b x)}{32 b}-\frac{x^3}{24} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^2 \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac{x^2}{8}+\frac{1}{8} x^2 \cosh (4 a+4 b x)\right ) \, dx\\ &=-\frac{x^3}{24}+\frac{1}{8} \int x^2 \cosh (4 a+4 b x) \, dx\\ &=-\frac{x^3}{24}+\frac{x^2 \sinh (4 a+4 b x)}{32 b}-\frac{\int x \sinh (4 a+4 b x) \, dx}{16 b}\\ &=-\frac{x^3}{24}-\frac{x \cosh (4 a+4 b x)}{64 b^2}+\frac{x^2 \sinh (4 a+4 b x)}{32 b}+\frac{\int \cosh (4 a+4 b x) \, dx}{64 b^2}\\ &=-\frac{x^3}{24}-\frac{x \cosh (4 a+4 b x)}{64 b^2}+\frac{\sinh (4 a+4 b x)}{256 b^3}+\frac{x^2 \sinh (4 a+4 b x)}{32 b}\\ \end{align*}
Mathematica [A] time = 0.164927, size = 48, normalized size = 0.8 \[ \frac{3 \left (8 b^2 x^2+1\right ) \sinh (4 (a+b x))-12 b x \cosh (4 (a+b x))-32 b^3 x^3}{768 b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 232, normalized size = 3.9 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{ \left ( bx+a \right ) ^{2}\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{4}}-{\frac{ \left ( bx+a \right ) ^{2}\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{8}}-{\frac{ \left ( bx+a \right ) ^{3}}{24}}-{\frac{ \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{8}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }{32}}-{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{64}}-{\frac{bx}{64}}-{\frac{a}{64}}-2\,a \left ( 1/4\, \left ( bx+a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{3}-1/8\, \left ( bx+a \right ) \cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) -1/16\, \left ( bx+a \right ) ^{2}-1/16\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \right ) +{a}^{2} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }{4}}-{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{8}}-{\frac{bx}{8}}-{\frac{a}{8}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16365, size = 93, normalized size = 1.55 \begin{align*} -\frac{1}{24} \, x^{3} + \frac{{\left (8 \, b^{2} x^{2} e^{\left (4 \, a\right )} - 4 \, b x e^{\left (4 \, a\right )} + e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{512 \, b^{3}} - \frac{{\left (8 \, b^{2} x^{2} + 4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{512 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85527, size = 288, normalized size = 4.8 \begin{align*} -\frac{8 \, b^{3} x^{3} + 3 \, b x \cosh \left (b x + a\right )^{4} + 18 \, b x \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 3 \, b x \sinh \left (b x + a\right )^{4} - 3 \,{\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) - 3 \,{\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3}}{192 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.66419, size = 204, normalized size = 3.4 \begin{align*} \begin{cases} - \frac{x^{3} \sinh ^{4}{\left (a + b x \right )}}{24} + \frac{x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{12} - \frac{x^{3} \cosh ^{4}{\left (a + b x \right )}}{24} + \frac{x^{2} \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{8 b} + \frac{x^{2} \sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} - \frac{x \sinh ^{4}{\left (a + b x \right )}}{64 b^{2}} - \frac{3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{32 b^{2}} - \frac{x \cosh ^{4}{\left (a + b x \right )}}{64 b^{2}} + \frac{\sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{64 b^{3}} + \frac{\sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{64 b^{3}} & \text{for}\: b \neq 0 \\\frac{x^{3} \sinh ^{2}{\left (a \right )} \cosh ^{2}{\left (a \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29334, size = 84, normalized size = 1.4 \begin{align*} -\frac{1}{24} \, x^{3} + \frac{{\left (8 \, b^{2} x^{2} - 4 \, b x + 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{512 \, b^{3}} - \frac{{\left (8 \, b^{2} x^{2} + 4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{512 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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