Optimal. Leaf size=172 \[ -\frac{3 x^2 \cosh ^4(a+b x)}{16 b^2}-\frac{9 x^2 \cosh ^2(a+b x)}{16 b^2}-\frac{3 \cosh ^4(a+b x)}{128 b^4}-\frac{45 \cosh ^2(a+b x)}{128 b^4}+\frac{3 x \sinh (a+b x) \cosh ^3(a+b x)}{32 b^3}+\frac{45 x \sinh (a+b x) \cosh (a+b x)}{64 b^3}+\frac{x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac{3 x^3 \sinh (a+b x) \cosh (a+b x)}{8 b}+\frac{45 x^2}{128 b^2}+\frac{3 x^4}{32} \]
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Rubi [A] time = 0.145787, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3311, 30, 3310} \[ -\frac{3 x^2 \cosh ^4(a+b x)}{16 b^2}-\frac{9 x^2 \cosh ^2(a+b x)}{16 b^2}-\frac{3 \cosh ^4(a+b x)}{128 b^4}-\frac{45 \cosh ^2(a+b x)}{128 b^4}+\frac{3 x \sinh (a+b x) \cosh ^3(a+b x)}{32 b^3}+\frac{45 x \sinh (a+b x) \cosh (a+b x)}{64 b^3}+\frac{x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac{3 x^3 \sinh (a+b x) \cosh (a+b x)}{8 b}+\frac{45 x^2}{128 b^2}+\frac{3 x^4}{32} \]
Antiderivative was successfully verified.
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Rule 3311
Rule 30
Rule 3310
Rubi steps
\begin{align*} \int x^3 \cosh ^4(a+b x) \, dx &=-\frac{3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac{x^3 \cosh ^3(a+b x) \sinh (a+b x)}{4 b}+\frac{3}{4} \int x^3 \cosh ^2(a+b x) \, dx+\frac{3 \int x \cosh ^4(a+b x) \, dx}{8 b^2}\\ &=-\frac{9 x^2 \cosh ^2(a+b x)}{16 b^2}-\frac{3 \cosh ^4(a+b x)}{128 b^4}-\frac{3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac{3 x^3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac{3 x \cosh ^3(a+b x) \sinh (a+b x)}{32 b^3}+\frac{x^3 \cosh ^3(a+b x) \sinh (a+b x)}{4 b}+\frac{3 \int x^3 \, dx}{8}+\frac{9 \int x \cosh ^2(a+b x) \, dx}{32 b^2}+\frac{9 \int x \cosh ^2(a+b x) \, dx}{8 b^2}\\ &=\frac{3 x^4}{32}-\frac{45 \cosh ^2(a+b x)}{128 b^4}-\frac{9 x^2 \cosh ^2(a+b x)}{16 b^2}-\frac{3 \cosh ^4(a+b x)}{128 b^4}-\frac{3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac{45 x \cosh (a+b x) \sinh (a+b x)}{64 b^3}+\frac{3 x^3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac{3 x \cosh ^3(a+b x) \sinh (a+b x)}{32 b^3}+\frac{x^3 \cosh ^3(a+b x) \sinh (a+b x)}{4 b}+\frac{9 \int x \, dx}{64 b^2}+\frac{9 \int x \, dx}{16 b^2}\\ &=\frac{45 x^2}{128 b^2}+\frac{3 x^4}{32}-\frac{45 \cosh ^2(a+b x)}{128 b^4}-\frac{9 x^2 \cosh ^2(a+b x)}{16 b^2}-\frac{3 \cosh ^4(a+b x)}{128 b^4}-\frac{3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac{45 x \cosh (a+b x) \sinh (a+b x)}{64 b^3}+\frac{3 x^3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac{3 x \cosh ^3(a+b x) \sinh (a+b x)}{32 b^3}+\frac{x^3 \cosh ^3(a+b x) \sinh (a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.404904, size = 100, normalized size = 0.58 \[ \frac{4 b x \left (32 \left (2 b^2 x^2+3\right ) \sinh (2 (a+b x))+\left (8 b^2 x^2+3\right ) \sinh (4 (a+b x))+24 b^3 x^3\right )-192 \left (2 b^2 x^2+1\right ) \cosh (2 (a+b x))-3 \left (8 b^2 x^2+1\right ) \cosh (4 (a+b x))}{1024 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 432, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04323, size = 238, normalized size = 1.38 \begin{align*} \frac{3}{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 (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}} - \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 [A] time = 2.05032, size = 486, normalized size = 2.83 \begin{align*} \frac{96 \, 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 ) \sinh \left (b x + a\right )^{3} - 3 \,{\left (8 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{4} - 192 \,{\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} - 6 \,{\left (64 \, b^{2} x^{2} + 3 \,{\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} + 32\right )} \sinh \left (b x + a\right )^{2} + 16 \,{\left ({\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{3} + 16 \,{\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{1024 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.10609, size = 262, normalized size = 1.52 \begin{align*} \begin{cases} \frac{3 x^{4} \sinh ^{4}{\left (a + b x \right )}}{32} - \frac{3 x^{4} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} + \frac{3 x^{4} \cosh ^{4}{\left (a + b x \right )}}{32} - \frac{3 x^{3} \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{8 b} + \frac{5 x^{3} \sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} + \frac{45 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{51 x^{2} \cosh ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac{45 x \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{64 b^{3}} + \frac{51 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 )}}{8 b^{4}} - \frac{51 \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{128 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \cosh ^{4}{\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 = 1.23744, size = 203, normalized size = 1.18 \begin{align*} \frac{3}{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 (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}} - \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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