Optimal. Leaf size=46 \[ \frac{\sinh ^3(a+b x) \cosh (a+b x)}{4 b}-\frac{3 \sinh (a+b x) \cosh (a+b x)}{8 b}+\frac{3 x}{8} \]
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Rubi [A] time = 0.0207817, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 8} \[ \frac{\sinh ^3(a+b x) \cosh (a+b x)}{4 b}-\frac{3 \sinh (a+b x) \cosh (a+b x)}{8 b}+\frac{3 x}{8} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sinh ^4(a+b x) \, dx &=\frac{\cosh (a+b x) \sinh ^3(a+b x)}{4 b}-\frac{3}{4} \int \sinh ^2(a+b x) \, dx\\ &=-\frac{3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac{\cosh (a+b x) \sinh ^3(a+b x)}{4 b}+\frac{3 \int 1 \, dx}{8}\\ &=\frac{3 x}{8}-\frac{3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac{\cosh (a+b x) \sinh ^3(a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0401589, size = 33, normalized size = 0.72 \[ \frac{12 (a+b x)-8 \sinh (2 (a+b x))+\sinh (4 (a+b x))}{32 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 39, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \left ({\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}{4}}-{\frac{3\,\sinh \left ( bx+a \right ) }{8}} \right ) \cosh \left ( bx+a \right ) +{\frac{3\,bx}{8}}+{\frac{3\,a}{8}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11297, size = 81, normalized size = 1.76 \begin{align*} \frac{3}{8} \, x + \frac{e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} - \frac{e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} + \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, 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 = 2.22838, size = 134, normalized size = 2.91 \begin{align*} \frac{\cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 3 \, b x +{\left (\cosh \left (b x + a\right )^{3} - 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.21722, size = 95, normalized size = 2.07 \begin{align*} \begin{cases} \frac{3 x \sinh ^{4}{\left (a + b x \right )}}{8} - \frac{3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4} + \frac{3 x \cosh ^{4}{\left (a + b x \right )}}{8} + \frac{5 \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{8 b} - \frac{3 \sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} & \text{for}\: b \neq 0 \\x \sinh ^{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 = 1.41966, size = 92, normalized size = 2. \begin{align*} \frac{24 \, b x -{\left (18 \, e^{\left (4 \, b x + 4 \, a\right )} - 8 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 24 \, a + e^{\left (4 \, b x + 4 \, a\right )} - 8 \, e^{\left (2 \, b x + 2 \, a\right )}}{64 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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