Optimal. Leaf size=79 \[ -\frac{3 x^2 \cosh (4 a+4 b x)}{128 b^2}+\frac{3 x \sinh (4 a+4 b x)}{256 b^3}-\frac{3 \cosh (4 a+4 b x)}{1024 b^4}+\frac{x^3 \sinh (4 a+4 b x)}{32 b}-\frac{x^4}{32} \]
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Rubi [A] time = 0.114185, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5448, 3296, 2638} \[ -\frac{3 x^2 \cosh (4 a+4 b x)}{128 b^2}+\frac{3 x \sinh (4 a+4 b x)}{256 b^3}-\frac{3 \cosh (4 a+4 b x)}{1024 b^4}+\frac{x^3 \sinh (4 a+4 b x)}{32 b}-\frac{x^4}{32} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^3 \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac{x^3}{8}+\frac{1}{8} x^3 \cosh (4 a+4 b x)\right ) \, dx\\ &=-\frac{x^4}{32}+\frac{1}{8} \int x^3 \cosh (4 a+4 b x) \, dx\\ &=-\frac{x^4}{32}+\frac{x^3 \sinh (4 a+4 b x)}{32 b}-\frac{3 \int x^2 \sinh (4 a+4 b x) \, dx}{32 b}\\ &=-\frac{x^4}{32}-\frac{3 x^2 \cosh (4 a+4 b x)}{128 b^2}+\frac{x^3 \sinh (4 a+4 b x)}{32 b}+\frac{3 \int x \cosh (4 a+4 b x) \, dx}{64 b^2}\\ &=-\frac{x^4}{32}-\frac{3 x^2 \cosh (4 a+4 b x)}{128 b^2}+\frac{3 x \sinh (4 a+4 b x)}{256 b^3}+\frac{x^3 \sinh (4 a+4 b x)}{32 b}-\frac{3 \int \sinh (4 a+4 b x) \, dx}{256 b^3}\\ &=-\frac{x^4}{32}-\frac{3 \cosh (4 a+4 b x)}{1024 b^4}-\frac{3 x^2 \cosh (4 a+4 b x)}{128 b^2}+\frac{3 x \sinh (4 a+4 b x)}{256 b^3}+\frac{x^3 \sinh (4 a+4 b x)}{32 b}\\ \end{align*}
Mathematica [A] time = 0.198099, size = 58, normalized size = 0.73 \[ \frac{4 b x \left (8 b^2 x^2+3\right ) \sinh (4 (a+b x))-3 \left (8 b^2 x^2+1\right ) \cosh (4 (a+b x))-32 b^4 x^4}{1024 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 384, normalized size = 4.9 \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 ) ^{3}}{4}}-{\frac{ \left ( bx+a \right ) ^{3}\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{8}}-{\frac{ \left ( bx+a \right ) ^{4}}{32}}-{\frac{3\, \left ( bx+a \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{16}}+{\frac{ \left ( 3\,bx+3\,a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{32}}-{\frac{ \left ( 3\,bx+3\,a \right ) \cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{64}}-{\frac{3\, \left ( bx+a \right ) ^{2}}{128}}-{\frac{3\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{128}}-3\,a \left ( 1/4\, \left ( bx+a \right ) ^{2}\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{3}-1/8\, \left ( bx+a \right ) ^{2}\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) -1/24\, \left ( bx+a \right ) ^{3}-1/8\, \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}+1/32\, \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) -{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{64}}-{\frac{bx}{64}}-{\frac{a}{64}} \right ) +3\,{a}^{2} \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}^{3} \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 = 1.17843, size = 123, normalized size = 1.56 \begin{align*} -\frac{1}{32} \, x^{4} + \frac{{\left (32 \, b^{3} x^{3} e^{\left (4 \, a\right )} - 24 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 12 \, b x e^{\left (4 \, a\right )} - 3 \, e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{2048 \, b^{4}} - \frac{{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85244, size = 352, normalized size = 4.46 \begin{align*} -\frac{32 \, b^{4} x^{4} + 3 \,{\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} - 16 \,{\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 18 \,{\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} - 16 \,{\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 3 \,{\left (8 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{4}}{1024 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.077, size = 250, normalized size = 3.16 \begin{align*} \begin{cases} - \frac{x^{4} \sinh ^{4}{\left (a + b x \right )}}{32} + \frac{x^{4} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} - \frac{x^{4} \cosh ^{4}{\left (a + b x \right )}}{32} + \frac{x^{3} \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{8 b} + \frac{x^{3} \sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} - \frac{3 x^{2} \sinh ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac{9 x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{64 b^{2}} - \frac{3 x^{2} \cosh ^{4}{\left (a + b x \right )}}{128 b^{2}} + \frac{3 x \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{64 b^{3}} + \frac{3 x \sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{64 b^{3}} - \frac{3 \sinh ^{4}{\left (a + b x \right )}}{256 b^{4}} - \frac{3 \cosh ^{4}{\left (a + b x \right )}}{256 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \sinh ^{2}{\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 = 1.17165, size = 105, normalized size = 1.33 \begin{align*} -\frac{1}{32} \, x^{4} + \frac{{\left (32 \, b^{3} x^{3} - 24 \, b^{2} x^{2} + 12 \, b x - 3\right )} e^{\left (4 \, b x + 4 \, a\right )}}{2048 \, b^{4}} - \frac{{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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