Optimal. Leaf size=83 \[ \frac{b e^{c+d x} \sinh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}-\frac{d e^{c+d x} \cosh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}-\frac{e^{c+d x}}{8 d} \]
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Rubi [A] time = 0.0792998, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5509, 2194, 5475} \[ \frac{b e^{c+d x} \sinh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}-\frac{d e^{c+d x} \cosh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}-\frac{e^{c+d x}}{8 d} \]
Antiderivative was successfully verified.
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Rule 5509
Rule 2194
Rule 5475
Rubi steps
\begin{align*} \int e^{c+d x} \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac{1}{8} e^{c+d x}+\frac{1}{8} e^{c+d x} \cosh (4 a+4 b x)\right ) \, dx\\ &=-\left (\frac{1}{8} \int e^{c+d x} \, dx\right )+\frac{1}{8} \int e^{c+d x} \cosh (4 a+4 b x) \, dx\\ &=-\frac{e^{c+d x}}{8 d}-\frac{d e^{c+d x} \cosh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}+\frac{b e^{c+d x} \sinh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}\\ \end{align*}
Mathematica [A] time = 0.355294, size = 58, normalized size = 0.7 \[ \frac{e^{c+d x} \left (d^2 \cosh (4 (a+b x))-4 b d \sinh (4 (a+b x))+16 b^2-d^2\right )}{8 \left (d^3-16 b^2 d\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 124, normalized size = 1.5 \begin{align*} -{\frac{\sinh \left ( dx+c \right ) }{8\,d}}+{\frac{\sinh \left ( \left ( 4\,b-d \right ) x+4\,a-c \right ) }{64\,b-16\,d}}+{\frac{\sinh \left ( \left ( 4\,b+d \right ) x+4\,a+c \right ) }{64\,b+16\,d}}-{\frac{\cosh \left ( dx+c \right ) }{8\,d}}-{\frac{\cosh \left ( \left ( 4\,b-d \right ) x+4\,a-c \right ) }{64\,b-16\,d}}+{\frac{\cosh \left ( \left ( 4\,b+d \right ) x+4\,a+c \right ) }{64\,b+16\,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 = 1.98238, size = 770, normalized size = 9.28 \begin{align*} \frac{16 \, b d \cosh \left (b x + a\right )^{3} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - 6 \, d^{2} \cosh \left (b x + a\right )^{2} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + 16 \, b d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - d^{2} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{4} -{\left (d^{2} \cosh \left (b x + a\right )^{4} + 16 \, b^{2} - d^{2}\right )} \cosh \left (d x + c\right ) -{\left (d^{2} \cosh \left (b x + a\right )^{4} - 16 \, b d \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, d^{2} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} - 16 \, b d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + d^{2} \sinh \left (b x + a\right )^{4} + 16 \, b^{2} - d^{2}\right )} \sinh \left (d x + c\right )}{8 \,{\left ({\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{4} - 2 \,{\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} +{\left (16 \, b^{2} d - d^{3}\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 = 150.693, size = 831, normalized size = 10.01 \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 = 1.18579, size = 78, normalized size = 0.94 \begin{align*} \frac{e^{\left (4 \, b x + d x + 4 \, a + c\right )}}{16 \,{\left (4 \, b + d\right )}} - \frac{e^{\left (-4 \, b x + d x - 4 \, a + c\right )}}{16 \,{\left (4 \, b - d\right )}} - \frac{e^{\left (d x + c\right )}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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