Optimal. Leaf size=144 \[ \frac{\sinh (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}+\frac{3 \sinh (2 a+x (2 b-d)-c)}{16 (2 b-d)}+\frac{3 \sinh (2 a+x (2 b+d)+c)}{16 (2 b+d)}+\frac{\sinh (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}-\frac{3 \sinh (c+d x)}{8 d}-\frac{\sinh (3 c+3 d x)}{24 d} \]
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Rubi [A] time = 0.0925019, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {5618, 2637} \[ \frac{\sinh (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}+\frac{3 \sinh (2 a+x (2 b-d)-c)}{16 (2 b-d)}+\frac{3 \sinh (2 a+x (2 b+d)+c)}{16 (2 b+d)}+\frac{\sinh (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}-\frac{3 \sinh (c+d x)}{8 d}-\frac{\sinh (3 c+3 d x)}{24 d} \]
Antiderivative was successfully verified.
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Rule 5618
Rule 2637
Rubi steps
\begin{align*} \int \cosh ^3(c+d x) \sinh ^2(a+b x) \, dx &=\int \left (\frac{1}{16} \cosh (2 a-3 c+(2 b-3 d) x)+\frac{3}{16} \cosh (2 a-c+(2 b-d) x)-\frac{3}{8} \cosh (c+d x)-\frac{1}{8} \cosh (3 c+3 d x)+\frac{3}{16} \cosh (2 a+c+(2 b+d) x)+\frac{1}{16} \cosh (2 a+3 c+(2 b+3 d) x)\right ) \, dx\\ &=\frac{1}{16} \int \cosh (2 a-3 c+(2 b-3 d) x) \, dx+\frac{1}{16} \int \cosh (2 a+3 c+(2 b+3 d) x) \, dx-\frac{1}{8} \int \cosh (3 c+3 d x) \, dx+\frac{3}{16} \int \cosh (2 a-c+(2 b-d) x) \, dx+\frac{3}{16} \int \cosh (2 a+c+(2 b+d) x) \, dx-\frac{3}{8} \int \cosh (c+d x) \, dx\\ &=\frac{\sinh (2 a-3 c+(2 b-3 d) x)}{16 (2 b-3 d)}+\frac{3 \sinh (2 a-c+(2 b-d) x)}{16 (2 b-d)}-\frac{3 \sinh (c+d x)}{8 d}-\frac{\sinh (3 c+3 d x)}{24 d}+\frac{3 \sinh (2 a+c+(2 b+d) x)}{16 (2 b+d)}+\frac{\sinh (2 a+3 c+(2 b+3 d) x)}{16 (2 b+3 d)}\\ \end{align*}
Mathematica [A] time = 1.56935, size = 158, normalized size = 1.1 \[ \frac{1}{48} \left (\frac{3 \sinh (2 a+2 b x-3 c-3 d x)}{2 b-3 d}+\frac{9 \sinh (2 a+2 b x-c-d x)}{2 b-d}+\frac{9 \sinh (2 a+2 b x+c+d x)}{2 b+d}+\frac{3 \sinh (2 a+2 b x+3 c+3 d x)}{2 b+3 d}-\frac{18 \sinh (c) \cosh (d x)}{d}-\frac{2 \sinh (3 c) \cosh (3 d x)}{d}-\frac{18 \cosh (c) \sinh (d x)}{d}-\frac{2 \cosh (3 c) \sinh (3 d x)}{d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 133, normalized size = 0.9 \begin{align*}{\frac{\sinh \left ( 2\,a-3\,c+ \left ( 2\,b-3\,d \right ) x \right ) }{32\,b-48\,d}}+{\frac{3\,\sinh \left ( 2\,a-c+ \left ( 2\,b-d \right ) x \right ) }{32\,b-16\,d}}-{\frac{3\,\sinh \left ( dx+c \right ) }{8\,d}}-{\frac{\sinh \left ( 3\,dx+3\,c \right ) }{24\,d}}+{\frac{3\,\sinh \left ( 2\,a+c+ \left ( 2\,b+d \right ) x \right ) }{32\,b+16\,d}}+{\frac{\sinh \left ( 2\,a+3\,c+ \left ( 2\,b+3\,d \right ) x \right ) }{32\,b+48\,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 = 1.92453, size = 926, normalized size = 6.43 \begin{align*} \frac{36 \,{\left (4 \, b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} -{\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} + 9 \,{\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (b x + a\right )^{2} + 9 \,{\left (4 \, b^{2} d^{2} - d^{4}\right )} \sinh \left (b x + a\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 12 \,{\left ({\left (4 \, b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{3} + 3 \,{\left (4 \, b^{3} d - 9 \, b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right ) - 3 \,{\left (48 \, b^{4} - 120 \, b^{2} d^{2} + 27 \, d^{4} + 3 \,{\left (4 \, b^{2} d^{2} - 9 \, d^{4}\right )} \cosh \left (b x + a\right )^{2} +{\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} + 9 \,{\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \,{\left (4 \, b^{2} d^{2} - 9 \, d^{4} + 3 \,{\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \sinh \left (d x + c\right )}{24 \,{\left ({\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (b x + a\right )^{2} -{\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )} \sinh \left (b x + a\right )^{2}\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 = 1.21426, size = 351, normalized size = 2.44 \begin{align*} \frac{e^{\left (2 \, b x + 3 \, d x + 2 \, a + 3 \, c\right )}}{32 \,{\left (2 \, b + 3 \, d\right )}} + \frac{3 \, e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{32 \,{\left (2 \, b + d\right )}} + \frac{3 \, e^{\left (2 \, b x - d x + 2 \, a - c\right )}}{32 \,{\left (2 \, b - d\right )}} + \frac{e^{\left (2 \, b x - 3 \, d x + 2 \, a - 3 \, c\right )}}{32 \,{\left (2 \, b - 3 \, d\right )}} - \frac{e^{\left (-2 \, b x + 3 \, d x - 2 \, a + 3 \, c\right )}}{32 \,{\left (2 \, b - 3 \, d\right )}} - \frac{3 \, e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{32 \,{\left (2 \, b - d\right )}} - \frac{3 \, e^{\left (-2 \, b x - d x - 2 \, a - c\right )}}{32 \,{\left (2 \, b + d\right )}} - \frac{e^{\left (-2 \, b x - 3 \, d x - 2 \, a - 3 \, c\right )}}{32 \,{\left (2 \, b + 3 \, d\right )}} - \frac{e^{\left (3 \, d x + 3 \, c\right )}}{48 \, d} - \frac{3 \, e^{\left (d x + c\right )}}{16 \, d} + \frac{3 \, e^{\left (-d x - c\right )}}{16 \, d} + \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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