Optimal. Leaf size=15 \[ \frac{\sinh ^4(a+b x)}{4 b} \]
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Rubi [A] time = 0.0207488, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2564, 30} \[ \frac{\sinh ^4(a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 2564
Rule 30
Rubi steps
\begin{align*} \int \cosh (a+b x) \sinh ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,i \sinh (a+b x)\right )}{b}\\ &=\frac{\sinh ^4(a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.002444, size = 15, normalized size = 1. \[ \frac{\sinh ^4(a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 14, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{4}}{4\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01369, size = 18, normalized size = 1.2 \begin{align*} \frac{\sinh \left (b x + a\right )^{4}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.83843, size = 146, normalized size = 9.73 \begin{align*} \frac{\cosh \left (b x + a\right )^{4} + \sinh \left (b x + a\right )^{4} + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right )^{2} - 4 \, \cosh \left (b x + a\right )^{2}}{32 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.12259, size = 20, normalized size = 1.33 \begin{align*} \begin{cases} \frac{\sinh ^{4}{\left (a + b x \right )}}{4 b} & \text{for}\: b \neq 0 \\x \sinh ^{3}{\left (a \right )} \cosh{\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 = 1.20192, size = 66, normalized size = 4.4 \begin{align*} \frac{{\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}\right )}^{2} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} - 4 \, 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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