Optimal. Leaf size=195 \[ -\frac{b e^{c+d x} \sinh (a+b x)}{8 \left (b^2-d^2\right )}+\frac{3 b e^{c+d x} \sinh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}+\frac{5 b e^{c+d x} \sinh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )}+\frac{d e^{c+d x} \cosh (a+b x)}{8 \left (b^2-d^2\right )}-\frac{d e^{c+d x} \cosh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}-\frac{d e^{c+d x} \cosh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )} \]
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Rubi [A] time = 0.132763, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5509, 5475} \[ -\frac{b e^{c+d x} \sinh (a+b x)}{8 \left (b^2-d^2\right )}+\frac{3 b e^{c+d x} \sinh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}+\frac{5 b e^{c+d x} \sinh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )}+\frac{d e^{c+d x} \cosh (a+b x)}{8 \left (b^2-d^2\right )}-\frac{d e^{c+d x} \cosh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}-\frac{d e^{c+d x} \cosh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )} \]
Antiderivative was successfully verified.
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Rule 5509
Rule 5475
Rubi steps
\begin{align*} \int e^{c+d x} \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac{1}{8} e^{c+d x} \cosh (a+b x)+\frac{1}{16} e^{c+d x} \cosh (3 a+3 b x)+\frac{1}{16} e^{c+d x} \cosh (5 a+5 b x)\right ) \, dx\\ &=\frac{1}{16} \int e^{c+d x} \cosh (3 a+3 b x) \, dx+\frac{1}{16} \int e^{c+d x} \cosh (5 a+5 b x) \, dx-\frac{1}{8} \int e^{c+d x} \cosh (a+b x) \, dx\\ &=\frac{d e^{c+d x} \cosh (a+b x)}{8 \left (b^2-d^2\right )}-\frac{d e^{c+d x} \cosh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}-\frac{d e^{c+d x} \cosh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )}-\frac{b e^{c+d x} \sinh (a+b x)}{8 \left (b^2-d^2\right )}+\frac{3 b e^{c+d x} \sinh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}+\frac{5 b e^{c+d x} \sinh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )}\\ \end{align*}
Mathematica [A] time = 1.20979, size = 118, normalized size = 0.61 \[ \frac{1}{16} e^{c+d x} \left (\frac{3 b \sinh (3 (a+b x))-d \cosh (3 (a+b x))}{9 b^2-d^2}+\frac{5 b \sinh (5 (a+b x))-d \cosh (5 (a+b x))}{25 b^2-d^2}+\frac{2 d \cosh (a+b x)-2 b \sinh (a+b x)}{(b-d) (b+d)}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 278, normalized size = 1.4 \begin{align*} -{\frac{\sinh \left ( a-c+ \left ( b-d \right ) x \right ) }{16\,b-16\,d}}-{\frac{\sinh \left ( a+c+ \left ( b+d \right ) x \right ) }{16\,b+16\,d}}+{\frac{\sinh \left ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{96\,b-32\,d}}+{\frac{\sinh \left ( 3\,a+c+ \left ( 3\,b+d \right ) x \right ) }{96\,b+32\,d}}+{\frac{\sinh \left ( \left ( 5\,b-d \right ) x+5\,a-c \right ) }{160\,b-32\,d}}+{\frac{\sinh \left ( \left ( 5\,b+d \right ) x+5\,a+c \right ) }{160\,b+32\,d}}+{\frac{\cosh \left ( a-c+ \left ( b-d \right ) x \right ) }{16\,b-16\,d}}-{\frac{\cosh \left ( a+c+ \left ( b+d \right ) x \right ) }{16\,b+16\,d}}-{\frac{\cosh \left ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{96\,b-32\,d}}+{\frac{\cosh \left ( 3\,a+c+ \left ( 3\,b+d \right ) x \right ) }{96\,b+32\,d}}-{\frac{\cosh \left ( \left ( 5\,b-d \right ) x+5\,a-c \right ) }{160\,b-32\,d}}+{\frac{\cosh \left ( \left ( 5\,b+d \right ) x+5\,a+c \right ) }{160\,b+32\,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 = 2.11295, size = 2202, normalized size = 11.29 \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 = 1.17169, size = 178, normalized size = 0.91 \begin{align*} \frac{e^{\left (5 \, b x + d x + 5 \, a + c\right )}}{32 \,{\left (5 \, b + d\right )}} + \frac{e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{32 \,{\left (3 \, b + d\right )}} - \frac{e^{\left (b x + d x + a + c\right )}}{16 \,{\left (b + d\right )}} + \frac{e^{\left (-b x + d x - a + c\right )}}{16 \,{\left (b - d\right )}} - \frac{e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{32 \,{\left (3 \, b - d\right )}} - \frac{e^{\left (-5 \, b x + d x - 5 \, a + c\right )}}{32 \,{\left (5 \, b - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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