Optimal. Leaf size=28 \[ \frac{a \sinh (c+d x)}{d}+\frac{b \sinh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0216874, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {3190} \[ \frac{a \sinh (c+d x)}{d}+\frac{b \sinh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3190
Rubi steps
\begin{align*} \int \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b x^2\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{a \sinh (c+d x)}{d}+\frac{b \sinh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0117645, size = 39, normalized size = 1.39 \[ \frac{a \sinh (c) \cosh (d x)}{d}+\frac{a \cosh (c) \sinh (d x)}{d}+\frac{b \sinh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 25, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{b \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{3}}+a\sinh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0335, size = 35, normalized size = 1.25 \begin{align*} \frac{b \sinh \left (d x + c\right )^{3}}{3 \, d} + \frac{a \sinh \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4489, size = 103, normalized size = 3.68 \begin{align*} \frac{b \sinh \left (d x + c\right )^{3} + 3 \,{\left (b \cosh \left (d x + c\right )^{2} + 4 \, a - b\right )} \sinh \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.487066, size = 36, normalized size = 1.29 \begin{align*} \begin{cases} \frac{a \sinh{\left (c + d x \right )}}{d} + \frac{b \sinh ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \cosh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13263, size = 97, normalized size = 3.46 \begin{align*} \frac{b e^{\left (3 \, d x + 3 \, c\right )} + 12 \, a e^{\left (d x + c\right )} - 3 \, b e^{\left (d x + c\right )} -{\left (12 \, a e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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