Optimal. Leaf size=92 \[ \frac{\sinh ^5(a+b x) \cosh ^3(a+b x)}{8 b}-\frac{5 \sinh ^3(a+b x) \cosh ^3(a+b x)}{48 b}+\frac{5 \sinh (a+b x) \cosh ^3(a+b x)}{64 b}-\frac{5 \sinh (a+b x) \cosh (a+b x)}{128 b}-\frac{5 x}{128} \]
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Rubi [A] time = 0.103597, antiderivative size = 92, 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} \[ \frac{\sinh ^5(a+b x) \cosh ^3(a+b x)}{8 b}-\frac{5 \sinh ^3(a+b x) \cosh ^3(a+b x)}{48 b}+\frac{5 \sinh (a+b x) \cosh ^3(a+b x)}{64 b}-\frac{5 \sinh (a+b x) \cosh (a+b x)}{128 b}-\frac{5 x}{128} \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cosh ^2(a+b x) \sinh ^6(a+b x) \, dx &=\frac{\cosh ^3(a+b x) \sinh ^5(a+b x)}{8 b}-\frac{5}{8} \int \cosh ^2(a+b x) \sinh ^4(a+b x) \, dx\\ &=-\frac{5 \cosh ^3(a+b x) \sinh ^3(a+b x)}{48 b}+\frac{\cosh ^3(a+b x) \sinh ^5(a+b x)}{8 b}+\frac{5}{16} \int \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx\\ &=\frac{5 \cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac{5 \cosh ^3(a+b x) \sinh ^3(a+b x)}{48 b}+\frac{\cosh ^3(a+b x) \sinh ^5(a+b x)}{8 b}-\frac{5}{64} \int \cosh ^2(a+b x) \, dx\\ &=-\frac{5 \cosh (a+b x) \sinh (a+b x)}{128 b}+\frac{5 \cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac{5 \cosh ^3(a+b x) \sinh ^3(a+b x)}{48 b}+\frac{\cosh ^3(a+b x) \sinh ^5(a+b x)}{8 b}-\frac{5 \int 1 \, dx}{128}\\ &=-\frac{5 x}{128}-\frac{5 \cosh (a+b x) \sinh (a+b x)}{128 b}+\frac{5 \cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac{5 \cosh ^3(a+b x) \sinh ^3(a+b x)}{48 b}+\frac{\cosh ^3(a+b x) \sinh ^5(a+b x)}{8 b}\\ \end{align*}
Mathematica [A] time = 0.123181, size = 52, normalized size = 0.57 \[ \frac{48 \sinh (2 (a+b x))+24 \sinh (4 (a+b x))-16 \sinh (6 (a+b x))+3 \sinh (8 (a+b x))-120 b x}{3072 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 79, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{5} \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{8}}-{\frac{5\, \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}{48}}+{\frac{5\, \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }{64}}-{\frac{5\,\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{128}}-{\frac{5\,bx}{128}}-{\frac{5\,a}{128}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00577, size = 149, normalized size = 1.62 \begin{align*} -\frac{{\left (16 \, e^{\left (-2 \, b x - 2 \, a\right )} - 24 \, e^{\left (-4 \, b x - 4 \, a\right )} - 48 \, e^{\left (-6 \, b x - 6 \, a\right )} - 3\right )} e^{\left (8 \, b x + 8 \, a\right )}}{6144 \, b} - \frac{5 \,{\left (b x + a\right )}}{128 \, b} - \frac{48 \, e^{\left (-2 \, b x - 2 \, a\right )} + 24 \, e^{\left (-4 \, b x - 4 \, a\right )} - 16 \, e^{\left (-6 \, b x - 6 \, a\right )} + 3 \, e^{\left (-8 \, b x - 8 \, a\right )}}{6144 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04398, size = 382, normalized size = 4.15 \begin{align*} \frac{3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + 3 \,{\left (7 \, \cosh \left (b x + a\right )^{3} - 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} +{\left (21 \, \cosh \left (b x + a\right )^{5} - 40 \, \cosh \left (b x + a\right )^{3} + 12 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 15 \, b x + 3 \,{\left (\cosh \left (b x + a\right )^{7} - 4 \, \cosh \left (b x + a\right )^{5} + 4 \, \cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{384 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.8281, size = 189, normalized size = 2.05 \begin{align*} \begin{cases} - \frac{5 x \sinh ^{8}{\left (a + b x \right )}}{128} + \frac{5 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{32} - \frac{15 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{64} + \frac{5 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{32} - \frac{5 x \cosh ^{8}{\left (a + b x \right )}}{128} + \frac{5 \sinh ^{7}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{128 b} + \frac{73 \sinh ^{5}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{384 b} - \frac{55 \sinh ^{3}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{384 b} + \frac{5 \sinh{\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{128 b} & \text{for}\: b \neq 0 \\x \sinh ^{6}{\left (a \right )} \cosh ^{2}{\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 = 1.20593, size = 154, normalized size = 1.67 \begin{align*} -\frac{240 \, b x -{\left (250 \, e^{\left (8 \, b x + 8 \, a\right )} - 48 \, e^{\left (6 \, b x + 6 \, a\right )} - 24 \, e^{\left (4 \, b x + 4 \, a\right )} + 16 \, e^{\left (2 \, b x + 2 \, a\right )} - 3\right )} e^{\left (-8 \, b x - 8 \, a\right )} + 240 \, a - 3 \, e^{\left (8 \, b x + 8 \, a\right )} + 16 \, e^{\left (6 \, b x + 6 \, a\right )} - 24 \, e^{\left (4 \, b x + 4 \, a\right )} - 48 \, e^{\left (2 \, b x + 2 \, a\right )}}{6144 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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