Optimal. Leaf size=127 \[ -\frac{b e^{c+d x} \sinh (a+b x)}{4 \left (b^2-d^2\right )}+\frac{3 b e^{c+d x} \sinh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}+\frac{d e^{c+d x} \cosh (a+b x)}{4 \left (b^2-d^2\right )}-\frac{d e^{c+d x} \cosh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )} \]
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Rubi [A] time = 0.0907898, antiderivative size = 127, 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, 5475} \[ -\frac{b e^{c+d x} \sinh (a+b x)}{4 \left (b^2-d^2\right )}+\frac{3 b e^{c+d x} \sinh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}+\frac{d e^{c+d x} \cosh (a+b x)}{4 \left (b^2-d^2\right )}-\frac{d e^{c+d x} \cosh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )} \]
Antiderivative was successfully verified.
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Rule 5509
Rule 5475
Rubi steps
\begin{align*} \int e^{c+d x} \cosh (a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac{1}{4} e^{c+d x} \cosh (a+b x)+\frac{1}{4} e^{c+d x} \cosh (3 a+3 b x)\right ) \, dx\\ &=-\left (\frac{1}{4} \int e^{c+d x} \cosh (a+b x) \, dx\right )+\frac{1}{4} \int e^{c+d x} \cosh (3 a+3 b x) \, dx\\ &=\frac{d e^{c+d x} \cosh (a+b x)}{4 \left (b^2-d^2\right )}-\frac{d e^{c+d x} \cosh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}-\frac{b e^{c+d x} \sinh (a+b x)}{4 \left (b^2-d^2\right )}+\frac{3 b e^{c+d x} \sinh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}\\ \end{align*}
Mathematica [A] time = 0.953156, size = 80, normalized size = 0.63 \[ \frac{1}{4} e^{c+d x} \left (\frac{3 b \sinh (3 (a+b x))-d \cosh (3 (a+b x))}{9 b^2-d^2}+\frac{d \cosh (a+b x)-b \sinh (a+b x)}{(b-d) (b+d)}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 178, normalized size = 1.4 \begin{align*} -{\frac{\sinh \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}-{\frac{\sinh \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}+{\frac{\sinh \left ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{24\,b-8\,d}}+{\frac{\sinh \left ( 3\,a+c+ \left ( 3\,b+d \right ) x \right ) }{24\,b+8\,d}}+{\frac{\cosh \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}-{\frac{\cosh \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}-{\frac{\cosh \left ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{24\,b-8\,d}}+{\frac{\cosh \left ( 3\,a+c+ \left ( 3\,b+d \right ) x \right ) }{24\,b+8\,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 = 1.82452, size = 887, normalized size = 6.98 \begin{align*} -\frac{3 \,{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} - 3 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} +{\left (9 \, b^{3} - b d^{2} - 9 \,{\left (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 ) +{\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} -{\left (9 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) +{\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} + 3 \,{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \,{\left (b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )^{3} -{\left (9 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) +{\left (9 \, b^{3} - b d^{2} - 9 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \,{\left ({\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \,{\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} +{\left (9 \, b^{4} - 10 \, 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 [A] time = 102.679, size = 1059, normalized size = 8.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20754, size = 116, normalized size = 0.91 \begin{align*} \frac{e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{8 \,{\left (3 \, b + d\right )}} - \frac{e^{\left (b x + d x + a + c\right )}}{8 \,{\left (b + d\right )}} + \frac{e^{\left (-b x + d x - a + c\right )}}{8 \,{\left (b - d\right )}} - \frac{e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{8 \,{\left (3 \, b - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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