Optimal. Leaf size=139 \[ \frac{b e^{a+b x} \sinh ^3(c+d x)}{b^2-9 d^2}+\frac{6 b d^2 e^{a+b x} \sinh (c+d x)}{-10 b^2 d^2+b^4+9 d^4}-\frac{6 d^3 e^{a+b x} \cosh (c+d x)}{-10 b^2 d^2+b^4+9 d^4}-\frac{3 d e^{a+b x} \sinh ^2(c+d x) \cosh (c+d x)}{b^2-9 d^2} \]
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Rubi [A] time = 0.0630272, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5476, 5474} \[ \frac{b e^{a+b x} \sinh ^3(c+d x)}{b^2-9 d^2}+\frac{6 b d^2 e^{a+b x} \sinh (c+d x)}{-10 b^2 d^2+b^4+9 d^4}-\frac{6 d^3 e^{a+b x} \cosh (c+d x)}{-10 b^2 d^2+b^4+9 d^4}-\frac{3 d e^{a+b x} \sinh ^2(c+d x) \cosh (c+d x)}{b^2-9 d^2} \]
Antiderivative was successfully verified.
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Rule 5476
Rule 5474
Rubi steps
\begin{align*} \int e^{a+b x} \sinh ^3(c+d x) \, dx &=-\frac{3 d e^{a+b x} \cosh (c+d x) \sinh ^2(c+d x)}{b^2-9 d^2}+\frac{b e^{a+b x} \sinh ^3(c+d x)}{b^2-9 d^2}+\frac{\left (6 d^2\right ) \int e^{a+b x} \sinh (c+d x) \, dx}{b^2-9 d^2}\\ &=-\frac{6 d^3 e^{a+b x} \cosh (c+d x)}{b^4-10 b^2 d^2+9 d^4}+\frac{6 b d^2 e^{a+b x} \sinh (c+d x)}{b^4-10 b^2 d^2+9 d^4}-\frac{3 d e^{a+b x} \cosh (c+d x) \sinh ^2(c+d x)}{b^2-9 d^2}+\frac{b e^{a+b x} \sinh ^3(c+d x)}{b^2-9 d^2}\\ \end{align*}
Mathematica [A] time = 0.488643, size = 108, normalized size = 0.78 \[ \frac{e^{a+b x} \left (3 d \left (b^2-9 d^2\right ) \cosh (c+d x)+\left (3 d^3-3 b^2 d\right ) \cosh (3 (c+d x))+2 b \sinh (c+d x) \left (\left (b^2-d^2\right ) \cosh (2 (c+d x))-b^2+13 d^2\right )\right )}{4 \left (-10 b^2 d^2+b^4+9 d^4\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 166, normalized size = 1.2 \begin{align*} -{\frac{\sinh \left ( a-3\,c+ \left ( b-3\,d \right ) x \right ) }{8\,b-24\,d}}+{\frac{3\,\sinh \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}-{\frac{3\,\sinh \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}+{\frac{\sinh \left ( a+3\,c+ \left ( b+3\,d \right ) x \right ) }{8\,b+24\,d}}-{\frac{\cosh \left ( a-3\,c+ \left ( b-3\,d \right ) x \right ) }{8\,b-24\,d}}+{\frac{3\,\cosh \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}-{\frac{3\,\cosh \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}+{\frac{\cosh \left ( a+3\,c+ \left ( b+3\,d \right ) x \right ) }{8\,b+24\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.79787, size = 752, normalized size = 5.41 \begin{align*} -\frac{3 \,{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{3} -{\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) +{\left (b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \,{\left (b^{2} d - 9 \, d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) + 9 \,{\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) +{\left (b^{2} d - d^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )^{2} + 3 \,{\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (d x + c\right )^{3} -{\left (b^{2} d - 9 \, d^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right ) - 3 \,{\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} -{\left (b^{3} - 9 \, b d^{2}\right )} \cosh \left (b x + a\right ) -{\left (b^{3} - 9 \, b d^{2} -{\left (b^{3} - b d^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \,{\left (b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13631, size = 113, normalized size = 0.81 \begin{align*} \frac{e^{\left (b x + 3 \, d x + a + 3 \, c\right )}}{8 \,{\left (b + 3 \, d\right )}} - \frac{3 \, e^{\left (b x + d x + a + c\right )}}{8 \,{\left (b + d\right )}} + \frac{3 \, e^{\left (b x - d x + a - c\right )}}{8 \,{\left (b - d\right )}} - \frac{e^{\left (b x - 3 \, d x + a - 3 \, c\right )}}{8 \,{\left (b - 3 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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