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 = 0.08, antiderivative size = 90, normalized size of antiderivative = 1.00, 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 8
Rule 2568
Rule 2635
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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 0.39, size = 97, normalized size = 1.08 \[ \frac {7 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{5} + \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + {\left (7 \, \cosh \left (b x + a\right )^{5} - 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, b x + {\left (\cosh \left (b x + a\right )^{7} - 4 \, \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )}{128 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 60, normalized size = 0.67 \[ \frac {3}{128} \, x + \frac {e^{\left (8 \, b x + 8 \, a\right )}}{2048 \, b} - \frac {e^{\left (4 \, b x + 4 \, a\right )}}{256 \, b} + \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b} - \frac {e^{\left (-8 \, b x - 8 \, a\right )}}{2048 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 74, normalized size = 0.82 \[ \frac {\frac {\left (\sinh ^{3}\left (b x +a \right )\right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{8}-\frac {\sinh \left (b x +a \right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{16}+\frac {\left (\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )}{16}+\frac {3 b x}{128}+\frac {3 a}{128}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 66, normalized size = 0.73 \[ -\frac {{\left (8 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (8 \, b x + 8 \, a\right )}}{2048 \, b} + \frac {3 \, {\left (b x + a\right )}}{128 \, b} + \frac {8 \, e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )}}{2048 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 32, normalized size = 0.36 \[ \frac {3\,x}{128}-\frac {\frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{128}-\frac {\mathrm {sinh}\left (8\,a+8\,b\,x\right )}{1024}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.29, size = 189, normalized size = 2.10 \[ \begin {cases} \frac {3 x \sinh ^{8}{\left (a + b x \right )}}{128} - \frac {3 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{32} + \frac {9 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{64} - \frac {3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{32} + \frac {3 x \cosh ^{8}{\left (a + b x \right )}}{128} - \frac {3 \sinh ^{7}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{128 b} + \frac {11 \sinh ^{5}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{128 b} + \frac {11 \sinh ^{3}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{128 b} - \frac {3 \sinh {\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \sinh ^{4}{\relax (a )} \cosh ^{4}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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