Optimal. Leaf size=137 \[ \frac{d e^{c+d x} \sinh (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac{d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}-\frac{b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac{b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )} \]
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Rubi [A] time = 0.101286, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {5509, 5474} \[ \frac{d e^{c+d x} \sinh (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac{d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}-\frac{b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac{b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )} \]
Antiderivative was successfully verified.
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Rule 5509
Rule 5474
Rubi steps
\begin{align*} \int e^{c+d x} \cosh (a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{1}{4} e^{c+d x} \sinh (2 a+2 b x)+\frac{1}{8} e^{c+d x} \sinh (4 a+4 b x)\right ) \, dx\\ &=\frac{1}{8} \int e^{c+d x} \sinh (4 a+4 b x) \, dx-\frac{1}{4} \int e^{c+d x} \sinh (2 a+2 b x) \, dx\\ &=-\frac{b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac{b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}+\frac{d e^{c+d x} \sinh (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac{d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}\\ \end{align*}
Mathematica [A] time = 1.11985, size = 86, normalized size = 0.63 \[ \frac{1}{8} e^{c+d x} \left (\frac{2 d \sinh (2 (a+b x))-4 b \cosh (2 (a+b x))}{4 b^2-d^2}+\frac{4 b \cosh (4 (a+b x))-d \sinh (4 (a+b x))}{16 b^2-d^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 202, normalized size = 1.5 \begin{align*}{\frac{\sinh \left ( 2\,a-c+ \left ( 2\,b-d \right ) x \right ) }{16\,b-8\,d}}-{\frac{\sinh \left ( 2\,a+c+ \left ( 2\,b+d \right ) x \right ) }{16\,b+8\,d}}-{\frac{\sinh \left ( \left ( 4\,b-d \right ) x+4\,a-c \right ) }{64\,b-16\,d}}+{\frac{\sinh \left ( \left ( 4\,b+d \right ) x+4\,a+c \right ) }{64\,b+16\,d}}-{\frac{\cosh \left ( 2\,a-c+ \left ( 2\,b-d \right ) x \right ) }{16\,b-8\,d}}-{\frac{\cosh \left ( 2\,a+c+ \left ( 2\,b+d \right ) x \right ) }{16\,b+8\,d}}+{\frac{\cosh \left ( \left ( 4\,b-d \right ) x+4\,a-c \right ) }{64\,b-16\,d}}+{\frac{\cosh \left ( \left ( 4\,b+d \right ) x+4\,a+c \right ) }{64\,b+16\,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 = 1.95394, size = 1161, normalized size = 8.47 \begin{align*} -\frac{{\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} -{\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{4} +{\left (16 \, b^{3} - b d^{2} - 6 \,{\left (4 \, 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 )^{2} +{\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} -{\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) -{\left ({\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{4} -{\left (16 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) -{\left ({\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{4} -{\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} +{\left (4 \, b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )^{4} -{\left (16 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2} -{\left (16 \, b^{3} - b d^{2} - 6 \,{\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} -{\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} -{\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{2 \,{\left ({\left (64 \, b^{4} - 20 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \,{\left (64 \, b^{4} - 20 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} +{\left (64 \, b^{4} - 20 \, 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 [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 = 1.1639, size = 126, normalized size = 0.92 \begin{align*} \frac{e^{\left (4 \, b x + d x + 4 \, a + c\right )}}{16 \,{\left (4 \, b + d\right )}} - \frac{e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{8 \,{\left (2 \, b + d\right )}} - \frac{e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{8 \,{\left (2 \, b - d\right )}} + \frac{e^{\left (-4 \, b x + d x - 4 \, a + c\right )}}{16 \,{\left (4 \, b - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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