Optimal. Leaf size=111 \[ \frac{\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}-\frac{3 \sinh (a+b x) \cosh ^7(a+b x)}{80 b}+\frac{\sinh (a+b x) \cosh ^5(a+b x)}{160 b}+\frac{\sinh (a+b x) \cosh ^3(a+b x)}{128 b}+\frac{3 \sinh (a+b x) \cosh (a+b x)}{256 b}+\frac{3 x}{256} \]
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Rubi [A] time = 0.0989776, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ \frac{\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}-\frac{3 \sinh (a+b x) \cosh ^7(a+b x)}{80 b}+\frac{\sinh (a+b x) \cosh ^5(a+b x)}{160 b}+\frac{\sinh (a+b x) \cosh ^3(a+b x)}{128 b}+\frac{3 \sinh (a+b x) \cosh (a+b x)}{256 b}+\frac{3 x}{256} \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cosh ^6(a+b x) \sinh ^4(a+b x) \, dx &=\frac{\cosh ^7(a+b x) \sinh ^3(a+b x)}{10 b}-\frac{3}{10} \int \cosh ^6(a+b x) \sinh ^2(a+b x) \, dx\\ &=-\frac{3 \cosh ^7(a+b x) \sinh (a+b x)}{80 b}+\frac{\cosh ^7(a+b x) \sinh ^3(a+b x)}{10 b}+\frac{3}{80} \int \cosh ^6(a+b x) \, dx\\ &=\frac{\cosh ^5(a+b x) \sinh (a+b x)}{160 b}-\frac{3 \cosh ^7(a+b x) \sinh (a+b x)}{80 b}+\frac{\cosh ^7(a+b x) \sinh ^3(a+b x)}{10 b}+\frac{1}{32} \int \cosh ^4(a+b x) \, dx\\ &=\frac{\cosh ^3(a+b x) \sinh (a+b x)}{128 b}+\frac{\cosh ^5(a+b x) \sinh (a+b x)}{160 b}-\frac{3 \cosh ^7(a+b x) \sinh (a+b x)}{80 b}+\frac{\cosh ^7(a+b x) \sinh ^3(a+b x)}{10 b}+\frac{3}{128} \int \cosh ^2(a+b x) \, dx\\ &=\frac{3 \cosh (a+b x) \sinh (a+b x)}{256 b}+\frac{\cosh ^3(a+b x) \sinh (a+b x)}{128 b}+\frac{\cosh ^5(a+b x) \sinh (a+b x)}{160 b}-\frac{3 \cosh ^7(a+b x) \sinh (a+b x)}{80 b}+\frac{\cosh ^7(a+b x) \sinh ^3(a+b x)}{10 b}+\frac{3 \int 1 \, dx}{256}\\ &=\frac{3 x}{256}+\frac{3 \cosh (a+b x) \sinh (a+b x)}{256 b}+\frac{\cosh ^3(a+b x) \sinh (a+b x)}{128 b}+\frac{\cosh ^5(a+b x) \sinh (a+b x)}{160 b}-\frac{3 \cosh ^7(a+b x) \sinh (a+b x)}{80 b}+\frac{\cosh ^7(a+b x) \sinh ^3(a+b x)}{10 b}\\ \end{align*}
Mathematica [A] time = 0.128019, size = 62, normalized size = 0.56 \[ \frac{20 \sinh (2 (a+b x))-40 \sinh (4 (a+b x))-10 \sinh (6 (a+b x))+5 \sinh (8 (a+b x))+2 \sinh (10 (a+b x))+120 b x}{10240 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 84, 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 ) ^{7}}{10}}-{\frac{3\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{7}}{80}}+{\frac{3\,\sinh \left ( bx+a \right ) }{80} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{6}}+{\frac{5\, \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{24}}+{\frac{5\,\cosh \left ( bx+a \right ) }{16}} \right ) }+{\frac{3\,bx}{256}}+{\frac{3\,a}{256}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02041, size = 178, normalized size = 1.6 \begin{align*} \frac{{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} - 10 \, e^{\left (-4 \, b x - 4 \, a\right )} - 40 \, e^{\left (-6 \, b x - 6 \, a\right )} + 20 \, e^{\left (-8 \, b x - 8 \, a\right )} + 2\right )} e^{\left (10 \, b x + 10 \, a\right )}}{20480 \, b} + \frac{3 \,{\left (b x + a\right )}}{256 \, b} - \frac{20 \, e^{\left (-2 \, b x - 2 \, a\right )} - 40 \, e^{\left (-4 \, b x - 4 \, a\right )} - 10 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + 2 \, e^{\left (-10 \, b x - 10 \, a\right )}}{20480 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06793, size = 544, normalized size = 4.9 \begin{align*} \frac{5 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{9} + 10 \,{\left (6 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{7} +{\left (126 \, \cosh \left (b x + a\right )^{5} + 70 \, \cosh \left (b x + a\right )^{3} - 15 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 10 \,{\left (6 \, \cosh \left (b x + a\right )^{7} + 7 \, \cosh \left (b x + a\right )^{5} - 5 \, \cosh \left (b x + a\right )^{3} - 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 30 \, b x + 5 \,{\left (\cosh \left (b x + a\right )^{9} + 2 \, \cosh \left (b x + a\right )^{7} - 3 \, \cosh \left (b x + a\right )^{5} - 8 \, \cosh \left (b x + a\right )^{3} + 2 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{2560 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 32.4585, size = 231, normalized size = 2.08 \begin{align*} \begin{cases} - \frac{3 x \sinh ^{10}{\left (a + b x \right )}}{256} + \frac{15 x \sinh ^{8}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{256} - \frac{15 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{128} + \frac{15 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{128} - \frac{15 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{8}{\left (a + b x \right )}}{256} + \frac{3 x \cosh ^{10}{\left (a + b x \right )}}{256} + \frac{3 \sinh ^{9}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{256 b} - \frac{7 \sinh ^{7}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{128 b} + \frac{\sinh ^{5}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{10 b} + \frac{7 \sinh ^{3}{\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{128 b} - \frac{3 \sinh{\left (a + b x \right )} \cosh ^{9}{\left (a + b x \right )}}{256 b} & \text{for}\: b \neq 0 \\x \sinh ^{4}{\left (a \right )} \cosh ^{6}{\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 = 1.25138, size = 184, normalized size = 1.66 \begin{align*} \frac{240 \, b x -{\left (274 \, e^{\left (10 \, b x + 10 \, a\right )} + 20 \, e^{\left (8 \, b x + 8 \, a\right )} - 40 \, e^{\left (6 \, b x + 6 \, a\right )} - 10 \, e^{\left (4 \, b x + 4 \, a\right )} + 5 \, e^{\left (2 \, b x + 2 \, a\right )} + 2\right )} e^{\left (-10 \, b x - 10 \, a\right )} + 240 \, a + 2 \, e^{\left (10 \, b x + 10 \, a\right )} + 5 \, e^{\left (8 \, b x + 8 \, a\right )} - 10 \, e^{\left (6 \, b x + 6 \, a\right )} - 40 \, e^{\left (4 \, b x + 4 \, a\right )} + 20 \, e^{\left (2 \, b x + 2 \, a\right )}}{20480 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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