Optimal. Leaf size=26 \[ \frac{\sinh ^3(a+b x)}{3 b}+\frac{\sinh (a+b x)}{b} \]
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Rubi [A] time = 0.0117305, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2633} \[ \frac{\sinh ^3(a+b x)}{3 b}+\frac{\sinh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin{align*} \int \cosh ^3(a+b x) \, dx &=\frac{i \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=\frac{\sinh (a+b x)}{b}+\frac{\sinh ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0062334, size = 26, normalized size = 1. \[ \frac{\sinh ^3(a+b x)}{3 b}+\frac{\sinh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 23, normalized size = 0.9 \begin{align*}{\frac{\sinh \left ( bx+a \right ) }{b} \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04511, size = 73, normalized size = 2.81 \begin{align*} \frac{e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} + \frac{3 \, e^{\left (b x + a\right )}}{8 \, b} - \frac{3 \, e^{\left (-b x - a\right )}}{8 \, b} - \frac{e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70444, size = 89, normalized size = 3.42 \begin{align*} \frac{\sinh \left (b x + a\right )^{3} + 3 \,{\left (\cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.486892, size = 36, normalized size = 1.38 \begin{align*} \begin{cases} - \frac{2 \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac{\sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} & \text{for}\: b \neq 0 \\x \cosh ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2953, size = 65, normalized size = 2.5 \begin{align*} -\frac{{\left (9 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} - e^{\left (3 \, b x + 3 \, a\right )} - 9 \, e^{\left (b x + a\right )}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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