Optimal. Leaf size=46 \[ \frac{\sinh (a+b x) \cosh ^3(a+b x)}{4 b}-\frac{\sinh (a+b x) \cosh (a+b x)}{8 b}-\frac{x}{8} \]
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Rubi [A] time = 0.0435783, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ \frac{\sinh (a+b x) \cosh ^3(a+b x)}{4 b}-\frac{\sinh (a+b x) \cosh (a+b x)}{8 b}-\frac{x}{8} \]
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 ^2(a+b x) \, dx &=\frac{\cosh ^3(a+b x) \sinh (a+b x)}{4 b}-\frac{1}{4} \int \cosh ^2(a+b x) \, dx\\ &=-\frac{\cosh (a+b x) \sinh (a+b x)}{8 b}+\frac{\cosh ^3(a+b x) \sinh (a+b x)}{4 b}-\frac{\int 1 \, dx}{8}\\ &=-\frac{x}{8}-\frac{\cosh (a+b x) \sinh (a+b x)}{8 b}+\frac{\cosh ^3(a+b x) \sinh (a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0295206, size = 23, normalized size = 0.5 \[ \frac{\sinh (4 (a+b x))-4 (a+b x)}{32 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 43, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }{4}}-{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{8}}-{\frac{bx}{8}}-{\frac{a}{8}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01234, size = 53, normalized size = 1.15 \begin{align*} -\frac{b x + a}{8 \, b} + \frac{e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} - \frac{e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05852, size = 104, normalized size = 2.26 \begin{align*} \frac{\cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - b x}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.118, size = 92, normalized size = 2. \begin{align*} \begin{cases} - \frac{x \sinh ^{4}{\left (a + b x \right )}}{8} + \frac{x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4} - \frac{x \cosh ^{4}{\left (a + b x \right )}}{8} + \frac{\sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{8 b} + \frac{\sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} & \text{for}\: b \neq 0 \\x \sinh ^{2}{\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 = 1.16746, size = 65, normalized size = 1.41 \begin{align*} -\frac{8 \, b x -{\left (2 \, e^{\left (4 \, b x + 4 \, a\right )} - 1\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 8 \, a - e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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