Optimal. Leaf size=137 \[ \frac{3 d e^{c+d x} \sinh (2 a+2 b x)}{32 \left (4 b^2-d^2\right )}-\frac{d e^{c+d x} \sinh (6 a+6 b x)}{32 \left (36 b^2-d^2\right )}-\frac{3 b e^{c+d x} \cosh (2 a+2 b x)}{16 \left (4 b^2-d^2\right )}+\frac{3 b e^{c+d x} \cosh (6 a+6 b x)}{16 \left (36 b^2-d^2\right )} \]
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Rubi [A] time = 0.113393, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5509, 5474} \[ \frac{3 d e^{c+d x} \sinh (2 a+2 b x)}{32 \left (4 b^2-d^2\right )}-\frac{d e^{c+d x} \sinh (6 a+6 b x)}{32 \left (36 b^2-d^2\right )}-\frac{3 b e^{c+d x} \cosh (2 a+2 b x)}{16 \left (4 b^2-d^2\right )}+\frac{3 b e^{c+d x} \cosh (6 a+6 b x)}{16 \left (36 b^2-d^2\right )} \]
Antiderivative was successfully verified.
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Rule 5509
Rule 5474
Rubi steps
\begin{align*} \int e^{c+d x} \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{3}{32} e^{c+d x} \sinh (2 a+2 b x)+\frac{1}{32} e^{c+d x} \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac{1}{32} \int e^{c+d x} \sinh (6 a+6 b x) \, dx-\frac{3}{32} \int e^{c+d x} \sinh (2 a+2 b x) \, dx\\ &=-\frac{3 b e^{c+d x} \cosh (2 a+2 b x)}{16 \left (4 b^2-d^2\right )}+\frac{3 b e^{c+d x} \cosh (6 a+6 b x)}{16 \left (36 b^2-d^2\right )}+\frac{3 d e^{c+d x} \sinh (2 a+2 b x)}{32 \left (4 b^2-d^2\right )}-\frac{d e^{c+d x} \sinh (6 a+6 b x)}{32 \left (36 b^2-d^2\right )}\\ \end{align*}
Mathematica [A] time = 0.985342, size = 113, normalized size = 0.82 \[ \frac{e^{c+d x} \left (6 b \left (d^2-36 b^2\right ) \cosh (2 (a+b x))+6 \left (4 b^3-b d^2\right ) \cosh (6 (a+b x))+2 d \sinh (2 (a+b x)) \left (\left (d^2-4 b^2\right ) \cosh (4 (a+b x))+52 b^2-d^2\right )\right )}{32 \left (-40 b^2 d^2+144 b^4+d^4\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 202, normalized size = 1.5 \begin{align*}{\frac{3\,\sinh \left ( 2\,a-c+ \left ( 2\,b-d \right ) x \right ) }{128\,b-64\,d}}-{\frac{3\,\sinh \left ( 2\,a+c+ \left ( 2\,b+d \right ) x \right ) }{128\,b+64\,d}}-{\frac{\sinh \left ( \left ( 6\,b-d \right ) x+6\,a-c \right ) }{384\,b-64\,d}}+{\frac{\sinh \left ( \left ( 6\,b+d \right ) x+6\,a+c \right ) }{384\,b+64\,d}}-{\frac{3\,\cosh \left ( 2\,a-c+ \left ( 2\,b-d \right ) x \right ) }{128\,b-64\,d}}-{\frac{3\,\cosh \left ( 2\,a+c+ \left ( 2\,b+d \right ) x \right ) }{128\,b+64\,d}}+{\frac{\cosh \left ( \left ( 6\,b-d \right ) x+6\,a-c \right ) }{384\,b-64\,d}}+{\frac{\cosh \left ( \left ( 6\,b+d \right ) x+6\,a+c \right ) }{384\,b+64\,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 = 2.02437, size = 1611, normalized size = 11.76 \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 [A] time = 1.2048, size = 126, normalized size = 0.92 \begin{align*} \frac{e^{\left (6 \, b x + d x + 6 \, a + c\right )}}{64 \,{\left (6 \, b + d\right )}} - \frac{3 \, e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{64 \,{\left (2 \, b + d\right )}} - \frac{3 \, e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{64 \,{\left (2 \, b - d\right )}} + \frac{e^{\left (-6 \, b x + d x - 6 \, a + c\right )}}{64 \,{\left (6 \, b - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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