Optimal. Leaf size=90 \[ \frac{\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}-\frac{\sinh (a+b x) \cosh ^5(a+b x)}{16 b}+\frac{\sinh (a+b x) \cosh ^3(a+b x)}{64 b}+\frac{3 \sinh (a+b x) \cosh (a+b x)}{128 b}+\frac{3 x}{128} \]
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Rubi [A] time = 0.0839527, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ \frac{\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}-\frac{\sinh (a+b x) \cosh ^5(a+b x)}{16 b}+\frac{\sinh (a+b x) \cosh ^3(a+b x)}{64 b}+\frac{3 \sinh (a+b x) \cosh (a+b x)}{128 b}+\frac{3 x}{128} \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cosh ^4(a+b x) \sinh ^4(a+b x) \, dx &=\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}-\frac{3}{8} \int \cosh ^4(a+b x) \sinh ^2(a+b x) \, dx\\ &=-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{16 b}+\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}+\frac{1}{16} \int \cosh ^4(a+b x) \, dx\\ &=\frac{\cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{16 b}+\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}+\frac{3}{64} \int \cosh ^2(a+b x) \, dx\\ &=\frac{3 \cosh (a+b x) \sinh (a+b x)}{128 b}+\frac{\cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{16 b}+\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}+\frac{3 \int 1 \, dx}{128}\\ &=\frac{3 x}{128}+\frac{3 \cosh (a+b x) \sinh (a+b x)}{128 b}+\frac{\cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{16 b}+\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}\\ \end{align*}
Mathematica [A] time = 0.0428134, size = 33, normalized size = 0.37 \[ \frac{24 (a+b x)-8 \sinh (4 (a+b x))+\sinh (8 (a+b x))}{1024 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 74, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{3} \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{8}}-{\frac{\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{16}}+{\frac{\sinh \left ( bx+a \right ) }{16} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( bx+a \right ) }{8}} \right ) }+{\frac{3\,bx}{128}}+{\frac{3\,a}{128}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21238, size = 89, normalized size = 0.99 \begin{align*} -\frac{{\left (8 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (8 \, b x + 8 \, a\right )}}{2048 \, b} + \frac{3 \,{\left (b x + a\right )}}{128 \, b} + \frac{8 \, e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )}}{2048 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06411, size = 263, normalized size = 2.92 \begin{align*} \frac{7 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{5} + \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} +{\left (7 \, \cosh \left (b x + a\right )^{5} - 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, b x +{\left (\cosh \left (b x + a\right )^{7} - 4 \, \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )}{128 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.0885, size = 189, normalized size = 2.1 \begin{align*} \begin{cases} \frac{3 x \sinh ^{8}{\left (a + b x \right )}}{128} - \frac{3 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{32} + \frac{9 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{64} - \frac{3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{32} + \frac{3 x \cosh ^{8}{\left (a + b x \right )}}{128} - \frac{3 \sinh ^{7}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{128 b} + \frac{11 \sinh ^{5}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{128 b} + \frac{11 \sinh ^{3}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{128 b} - \frac{3 \sinh{\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{128 b} & \text{for}\: b \neq 0 \\x \sinh ^{4}{\left (a \right )} \cosh ^{4}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2121, size = 92, normalized size = 1.02 \begin{align*} \frac{48 \, b x -{\left (18 \, e^{\left (8 \, b x + 8 \, a\right )} - 8 \, e^{\left (4 \, b x + 4 \, a\right )} + 1\right )} e^{\left (-8 \, b x - 8 \, a\right )} + 48 \, a + e^{\left (8 \, b x + 8 \, a\right )} - 8 \, e^{\left (4 \, b x + 4 \, a\right )}}{2048 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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