Optimal. Leaf size=26 \[ \frac{(a \sinh (c+d x)+a \cosh (c+d x))^3}{3 d} \]
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Rubi [A] time = 0.0151791, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {3071} \[ \frac{(a \sinh (c+d x)+a \cosh (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 3071
Rubi steps
\begin{align*} \int (a \cosh (c+d x)+a \sinh (c+d x))^3 \, dx &=\frac{(a \cosh (c+d x)+a \sinh (c+d x))^3}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0763161, size = 25, normalized size = 0.96 \[ \frac{a^3 (\sinh (c+d x)+\cosh (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 24, normalized size = 0.9 \begin{align*}{\frac{{a}^{3} \left ( \cosh \left ( dx+c \right ) +\sinh \left ( dx+c \right ) \right ) ^{3}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06453, size = 197, normalized size = 7.58 \begin{align*} \frac{a^{3} \cosh \left (d x + c\right )^{3}}{d} + \frac{a^{3} \sinh \left (d x + c\right )^{3}}{d} + \frac{1}{24} \, a^{3}{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} - \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac{1}{24} \, a^{3}{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} + \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95203, size = 163, normalized size = 6.27 \begin{align*} \frac{a^{3} \cosh \left (d x + c\right )^{2} + 2 \, a^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{3} \sinh \left (d x + c\right )^{2}}{3 \,{\left (d \cosh \left (d x + c\right ) - d \sinh \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.877147, size = 83, normalized size = 3.19 \begin{align*} \begin{cases} \frac{a^{3} \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac{a^{3} \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} + \frac{a^{3} \sinh{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} + \frac{a^{3} \cosh ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a \sinh{\left (c \right )} + a \cosh{\left (c \right )}\right )^{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15384, size = 23, normalized size = 0.88 \begin{align*} \frac{a^{3} e^{\left (3 \, d x + 3 \, c\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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