Optimal. Leaf size=225 \[ -\frac{8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}+\frac{160 d^3 (c+d x) \sinh (a+b x)}{9 b^4}-\frac{80 d^2 (c+d x)^2 \cosh (a+b x)}{9 b^3}+\frac{4 d^2 (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{9 b^3}-\frac{4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}+\frac{8 d (c+d x)^3 \sinh (a+b x)}{3 b^2}+\frac{8 d^4 \cosh ^3(a+b x)}{81 b^5}-\frac{488 d^4 \cosh (a+b x)}{27 b^5}-\frac{2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac{(c+d x)^4 \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]
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Rubi [A] time = 0.359891, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3311, 3296, 2638, 2633} \[ -\frac{8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}+\frac{160 d^3 (c+d x) \sinh (a+b x)}{9 b^4}-\frac{80 d^2 (c+d x)^2 \cosh (a+b x)}{9 b^3}+\frac{4 d^2 (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{9 b^3}-\frac{4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}+\frac{8 d (c+d x)^3 \sinh (a+b x)}{3 b^2}+\frac{8 d^4 \cosh ^3(a+b x)}{81 b^5}-\frac{488 d^4 \cosh (a+b x)}{27 b^5}-\frac{2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac{(c+d x)^4 \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3311
Rule 3296
Rule 2638
Rule 2633
Rubi steps
\begin{align*} \int (c+d x)^4 \sinh ^3(a+b x) \, dx &=\frac{(c+d x)^4 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}-\frac{2}{3} \int (c+d x)^4 \sinh (a+b x) \, dx+\frac{\left (4 d^2\right ) \int (c+d x)^2 \sinh ^3(a+b x) \, dx}{3 b^2}\\ &=-\frac{2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac{4 d^2 (c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac{(c+d x)^4 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}-\frac{4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}+\frac{(8 d) \int (c+d x)^3 \cosh (a+b x) \, dx}{3 b}-\frac{\left (8 d^2\right ) \int (c+d x)^2 \sinh (a+b x) \, dx}{9 b^2}+\frac{\left (8 d^4\right ) \int \sinh ^3(a+b x) \, dx}{27 b^4}\\ &=-\frac{8 d^2 (c+d x)^2 \cosh (a+b x)}{9 b^3}-\frac{2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac{8 d (c+d x)^3 \sinh (a+b x)}{3 b^2}+\frac{4 d^2 (c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac{(c+d x)^4 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}-\frac{4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}-\frac{\left (8 d^2\right ) \int (c+d x)^2 \sinh (a+b x) \, dx}{b^2}+\frac{\left (16 d^3\right ) \int (c+d x) \cosh (a+b x) \, dx}{9 b^3}-\frac{\left (8 d^4\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (a+b x)\right )}{27 b^5}\\ &=-\frac{8 d^4 \cosh (a+b x)}{27 b^5}-\frac{80 d^2 (c+d x)^2 \cosh (a+b x)}{9 b^3}-\frac{2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac{8 d^4 \cosh ^3(a+b x)}{81 b^5}+\frac{16 d^3 (c+d x) \sinh (a+b x)}{9 b^4}+\frac{8 d (c+d x)^3 \sinh (a+b x)}{3 b^2}+\frac{4 d^2 (c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac{(c+d x)^4 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}-\frac{4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}+\frac{\left (16 d^3\right ) \int (c+d x) \cosh (a+b x) \, dx}{b^3}-\frac{\left (16 d^4\right ) \int \sinh (a+b x) \, dx}{9 b^4}\\ &=-\frac{56 d^4 \cosh (a+b x)}{27 b^5}-\frac{80 d^2 (c+d x)^2 \cosh (a+b x)}{9 b^3}-\frac{2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac{8 d^4 \cosh ^3(a+b x)}{81 b^5}+\frac{160 d^3 (c+d x) \sinh (a+b x)}{9 b^4}+\frac{8 d (c+d x)^3 \sinh (a+b x)}{3 b^2}+\frac{4 d^2 (c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac{(c+d x)^4 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}-\frac{4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}-\frac{\left (16 d^4\right ) \int \sinh (a+b x) \, dx}{b^4}\\ &=-\frac{488 d^4 \cosh (a+b x)}{27 b^5}-\frac{80 d^2 (c+d x)^2 \cosh (a+b x)}{9 b^3}-\frac{2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac{8 d^4 \cosh ^3(a+b x)}{81 b^5}+\frac{160 d^3 (c+d x) \sinh (a+b x)}{9 b^4}+\frac{8 d (c+d x)^3 \sinh (a+b x)}{3 b^2}+\frac{4 d^2 (c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac{(c+d x)^4 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}-\frac{4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}\\ \end{align*}
Mathematica [A] time = 1.01925, size = 150, normalized size = 0.67 \[ \frac{-243 \cosh (a+b x) \left (12 b^2 d^2 (c+d x)^2+b^4 (c+d x)^4+24 d^4\right )+\cosh (3 (a+b x)) \left (36 b^2 d^2 (c+d x)^2+27 b^4 (c+d x)^4+8 d^4\right )-24 b d (c+d x) \sinh (a+b x) \left (\cosh (2 (a+b x)) \left (3 b^2 (c+d x)^2+2 d^2\right )-39 b^2 (c+d x)^2-242 d^2\right )}{324 b^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.156, size = 1217, normalized size = 5.4 \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 [B] time = 1.35361, size = 863, normalized size = 3.84 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.82189, size = 1141, normalized size = 5.07 \begin{align*} \frac{{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} + 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \,{\left (9 \, b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 36 \,{\left (3 \, b^{4} c^{3} d + 2 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right )^{3} + 3 \,{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} + 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \,{\left (9 \, b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 36 \,{\left (3 \, b^{4} c^{3} d + 2 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 12 \,{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d + 2 \, b c d^{3} +{\left (9 \, b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \sinh \left (b x + a\right )^{3} - 243 \,{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + b^{4} c^{4} + 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4} + 6 \,{\left (b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 4 \,{\left (b^{4} c^{3} d + 6 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right ) + 36 \,{\left (27 \, b^{3} d^{4} x^{3} + 81 \, b^{3} c d^{3} x^{2} + 27 \, b^{3} c^{3} d + 162 \, b c d^{3} -{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d + 2 \, b c d^{3} +{\left (9 \, b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \cosh \left (b x + a\right )^{2} + 81 \,{\left (b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \sinh \left (b x + a\right )}{324 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.5076, size = 772, normalized size = 3.43 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2083, size = 883, normalized size = 3.92 \begin{align*} \frac{{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} - 36 \, b^{3} d^{4} x^{3} + 108 \, b^{4} c^{3} d x - 108 \, b^{3} c d^{3} x^{2} + 27 \, b^{4} c^{4} - 108 \, b^{3} c^{2} d^{2} x + 36 \, b^{2} d^{4} x^{2} - 36 \, b^{3} c^{3} d + 72 \, b^{2} c d^{3} x + 36 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 8 \, d^{4}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{648 \, b^{5}} - \frac{3 \,{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} - 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x - 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} - 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} - 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (b x + a\right )}}{8 \, b^{5}} - \frac{3 \,{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x + 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} + 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} + 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (-b x - a\right )}}{8 \, b^{5}} + \frac{{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 36 \, b^{3} d^{4} x^{3} + 108 \, b^{4} c^{3} d x + 108 \, b^{3} c d^{3} x^{2} + 27 \, b^{4} c^{4} + 108 \, b^{3} c^{2} d^{2} x + 36 \, b^{2} d^{4} x^{2} + 36 \, b^{3} c^{3} d + 72 \, b^{2} c d^{3} x + 36 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 8 \, d^{4}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{648 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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