Optimal. Leaf size=195 \[ \frac{3 \sinh (a+x (b-3 d)-3 c)}{32 (b-3 d)}-\frac{9 \sinh (a+x (b-d)-c)}{32 (b-d)}-\frac{\sinh (3 (a-c)+3 x (b-d))}{96 (b-d)}+\frac{3 \sinh (3 a+x (3 b-d)-c)}{32 (3 b-d)}+\frac{9 \sinh (a+x (b+d)+c)}{32 (b+d)}+\frac{\sinh (3 (a+c)+3 x (b+d))}{96 (b+d)}-\frac{3 \sinh (3 a+x (3 b+d)+c)}{32 (3 b+d)}-\frac{3 \sinh (a+x (b+3 d)+3 c)}{32 (b+3 d)} \]
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Rubi [A] time = 0.144429, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {5613, 2637} \[ \frac{3 \sinh (a+x (b-3 d)-3 c)}{32 (b-3 d)}-\frac{9 \sinh (a+x (b-d)-c)}{32 (b-d)}-\frac{\sinh (3 (a-c)+3 x (b-d))}{96 (b-d)}+\frac{3 \sinh (3 a+x (3 b-d)-c)}{32 (3 b-d)}+\frac{9 \sinh (a+x (b+d)+c)}{32 (b+d)}+\frac{\sinh (3 (a+c)+3 x (b+d))}{96 (b+d)}-\frac{3 \sinh (3 a+x (3 b+d)+c)}{32 (3 b+d)}-\frac{3 \sinh (a+x (b+3 d)+3 c)}{32 (b+3 d)} \]
Antiderivative was successfully verified.
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Rule 5613
Rule 2637
Rubi steps
\begin{align*} \int \sinh ^3(a+b x) \sinh ^3(c+d x) \, dx &=\int \left (\frac{3}{32} \cosh (a-3 c+(b-3 d) x)-\frac{9}{32} \cosh (a-c+(b-d) x)-\frac{1}{32} \cosh (3 (a-c)+3 (b-d) x)+\frac{3}{32} \cosh (3 a-c+(3 b-d) x)+\frac{9}{32} \cosh (a+c+(b+d) x)+\frac{1}{32} \cosh (3 (a+c)+3 (b+d) x)-\frac{3}{32} \cosh (3 a+c+(3 b+d) x)-\frac{3}{32} \cosh (a+3 c+(b+3 d) x)\right ) \, dx\\ &=-\left (\frac{1}{32} \int \cosh (3 (a-c)+3 (b-d) x) \, dx\right )+\frac{1}{32} \int \cosh (3 (a+c)+3 (b+d) x) \, dx+\frac{3}{32} \int \cosh (a-3 c+(b-3 d) x) \, dx+\frac{3}{32} \int \cosh (3 a-c+(3 b-d) x) \, dx-\frac{3}{32} \int \cosh (3 a+c+(3 b+d) x) \, dx-\frac{3}{32} \int \cosh (a+3 c+(b+3 d) x) \, dx-\frac{9}{32} \int \cosh (a-c+(b-d) x) \, dx+\frac{9}{32} \int \cosh (a+c+(b+d) x) \, dx\\ &=\frac{3 \sinh (a-3 c+(b-3 d) x)}{32 (b-3 d)}-\frac{9 \sinh (a-c+(b-d) x)}{32 (b-d)}-\frac{\sinh (3 (a-c)+3 (b-d) x)}{96 (b-d)}+\frac{3 \sinh (3 a-c+(3 b-d) x)}{32 (3 b-d)}+\frac{9 \sinh (a+c+(b+d) x)}{32 (b+d)}+\frac{\sinh (3 (a+c)+3 (b+d) x)}{96 (b+d)}-\frac{3 \sinh (3 a+c+(3 b+d) x)}{32 (3 b+d)}-\frac{3 \sinh (a+3 c+(b+3 d) x)}{32 (b+3 d)}\\ \end{align*}
Mathematica [A] time = 1.57155, size = 177, normalized size = 0.91 \[ \frac{1}{96} \left (\frac{9 \sinh (a+b x-3 c-3 d x)}{b-3 d}-\frac{27 \sinh (a+b x-c-d x)}{b-d}-\frac{\sinh (3 (a+b x-c-d x))}{b-d}+\frac{9 \sinh (3 a+3 b x-c-d x)}{3 b-d}-\frac{9 \sinh (3 a+3 b x+c+d x)}{3 b+d}-\frac{9 \sinh (a+b x+3 c+3 d x)}{b+3 d}+\frac{27 \sinh (a+x (b+d)+c)}{b+d}+\frac{\sinh (3 (a+x (b+d)+c))}{b+d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 184, normalized size = 0.9 \begin{align*}{\frac{3\,\sinh \left ( a-3\,c+ \left ( b-3\,d \right ) x \right ) }{32\,b-96\,d}}-{\frac{9\,\sinh \left ( a-c+ \left ( b-d \right ) x \right ) }{32\,b-32\,d}}+{\frac{9\,\sinh \left ( a+c+ \left ( b+d \right ) x \right ) }{32\,b+32\,d}}-{\frac{3\,\sinh \left ( a+3\,c+ \left ( b+3\,d \right ) x \right ) }{32\,b+96\,d}}-{\frac{\sinh \left ( \left ( 3\,b-3\,d \right ) x+3\,a-3\,c \right ) }{96\,b-96\,d}}+{\frac{3\,\sinh \left ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{96\,b-32\,d}}-{\frac{3\,\sinh \left ( 3\,a+c+ \left ( 3\,b+d \right ) x \right ) }{96\,b+32\,d}}+{\frac{\sinh \left ( \left ( 3\,b+3\,d \right ) x+3\,a+3\,c \right ) }{96\,b+96\,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 = 2.08868, size = 1710, normalized size = 8.77 \begin{align*} \text{result too large to display} \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 [B] time = 1.24565, size = 504, normalized size = 2.58 \begin{align*} \frac{e^{\left (3 \, b x + 3 \, d x + 3 \, a + 3 \, c\right )}}{192 \,{\left (b + d\right )}} - \frac{3 \, e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{64 \,{\left (3 \, b + d\right )}} + \frac{3 \, e^{\left (3 \, b x - d x + 3 \, a - c\right )}}{64 \,{\left (3 \, b - d\right )}} - \frac{e^{\left (3 \, b x - 3 \, d x + 3 \, a - 3 \, c\right )}}{192 \,{\left (b - d\right )}} - \frac{3 \, e^{\left (b x + 3 \, d x + a + 3 \, c\right )}}{64 \,{\left (b + 3 \, d\right )}} + \frac{9 \, e^{\left (b x + d x + a + c\right )}}{64 \,{\left (b + d\right )}} - \frac{9 \, e^{\left (b x - d x + a - c\right )}}{64 \,{\left (b - d\right )}} + \frac{3 \, e^{\left (b x - 3 \, d x + a - 3 \, c\right )}}{64 \,{\left (b - 3 \, d\right )}} - \frac{3 \, e^{\left (-b x + 3 \, d x - a + 3 \, c\right )}}{64 \,{\left (b - 3 \, d\right )}} + \frac{9 \, e^{\left (-b x + d x - a + c\right )}}{64 \,{\left (b - d\right )}} - \frac{9 \, e^{\left (-b x - d x - a - c\right )}}{64 \,{\left (b + d\right )}} + \frac{3 \, e^{\left (-b x - 3 \, d x - a - 3 \, c\right )}}{64 \,{\left (b + 3 \, d\right )}} + \frac{e^{\left (-3 \, b x + 3 \, d x - 3 \, a + 3 \, c\right )}}{192 \,{\left (b - d\right )}} - \frac{3 \, e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{64 \,{\left (3 \, b - d\right )}} + \frac{3 \, e^{\left (-3 \, b x - d x - 3 \, a - c\right )}}{64 \,{\left (3 \, b + d\right )}} - \frac{e^{\left (-3 \, b x - 3 \, d x - 3 \, a - 3 \, c\right )}}{192 \,{\left (b + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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