Optimal. Leaf size=144 \[ \frac{6 b^3 e^{c+d x} \sinh (a+b x)}{-10 b^2 d^2+9 b^4+d^4}-\frac{d e^{c+d x} \cosh ^3(a+b x)}{9 b^2-d^2}-\frac{6 b^2 d e^{c+d x} \cosh (a+b x)}{-10 b^2 d^2+9 b^4+d^4}+\frac{3 b e^{c+d x} \sinh (a+b x) \cosh ^2(a+b x)}{9 b^2-d^2} \]
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Rubi [A] time = 0.0566672, antiderivative size = 144, 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 = {5477, 5475} \[ \frac{6 b^3 e^{c+d x} \sinh (a+b x)}{-10 b^2 d^2+9 b^4+d^4}-\frac{d e^{c+d x} \cosh ^3(a+b x)}{9 b^2-d^2}-\frac{6 b^2 d e^{c+d x} \cosh (a+b x)}{-10 b^2 d^2+9 b^4+d^4}+\frac{3 b e^{c+d x} \sinh (a+b x) \cosh ^2(a+b x)}{9 b^2-d^2} \]
Antiderivative was successfully verified.
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Rule 5477
Rule 5475
Rubi steps
\begin{align*} \int e^{c+d x} \cosh ^3(a+b x) \, dx &=-\frac{d e^{c+d x} \cosh ^3(a+b x)}{9 b^2-d^2}+\frac{3 b e^{c+d x} \cosh ^2(a+b x) \sinh (a+b x)}{9 b^2-d^2}+\frac{\left (6 b^2\right ) \int e^{c+d x} \cosh (a+b x) \, dx}{9 b^2-d^2}\\ &=-\frac{6 b^2 d e^{c+d x} \cosh (a+b x)}{9 b^4-10 b^2 d^2+d^4}-\frac{d e^{c+d x} \cosh ^3(a+b x)}{9 b^2-d^2}+\frac{6 b^3 e^{c+d x} \sinh (a+b x)}{9 b^4-10 b^2 d^2+d^4}+\frac{3 b e^{c+d x} \cosh ^2(a+b x) \sinh (a+b x)}{9 b^2-d^2}\\ \end{align*}
Mathematica [A] time = 0.444232, size = 106, normalized size = 0.74 \[ \frac{e^{c+d x} \left (3 d \left (d^2-9 b^2\right ) \cosh (a+b x)+\left (d^3-b^2 d\right ) \cosh (3 (a+b x))+6 b \sinh (a+b x) \left (\left (b^2-d^2\right ) \cosh (2 (a+b x))+5 b^2-d^2\right )\right )}{4 \left (-10 b^2 d^2+9 b^4+d^4\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 178, normalized size = 1.2 \begin{align*}{\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 ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{24\,b-8\,d}}+{\frac{\sinh \left ( 3\,a+c+ \left ( 3\,b+d \right ) x \right ) }{24\,b+8\,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 ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{24\,b-8\,d}}+{\frac{\cosh \left ( 3\,a+c+ \left ( 3\,b+d \right ) x \right ) }{24\,b+8\,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 = 2.12696, size = 898, normalized size = 6.24 \begin{align*} -\frac{3 \,{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} - 3 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - 3 \,{\left (9 \, b^{3} - b d^{2} + 3 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) +{\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} + 3 \,{\left (9 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) +{\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} + 3 \,{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \,{\left (b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )^{3} + 3 \,{\left (9 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) - 3 \,{\left (9 \, b^{3} - b d^{2} + 3 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \,{\left ({\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \,{\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} +{\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 82.6233, size = 1114, normalized size = 7.74 \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 [A] time = 1.17066, size = 116, normalized size = 0.81 \begin{align*} \frac{e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{8 \,{\left (3 \, b + 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 (-3 \, b x + d x - 3 \, a + c\right )}}{8 \,{\left (3 \, b - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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