Optimal. Leaf size=59 \[ \frac{8 a^2 \sinh (c+d x)}{3 d \sqrt{a \cosh (c+d x)+a}}+\frac{2 a \sinh (c+d x) \sqrt{a \cosh (c+d x)+a}}{3 d} \]
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Rubi [A] time = 0.0287345, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ \frac{8 a^2 \sinh (c+d x)}{3 d \sqrt{a \cosh (c+d x)+a}}+\frac{2 a \sinh (c+d x) \sqrt{a \cosh (c+d x)+a}}{3 d} \]
Antiderivative was successfully verified.
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Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \cosh (c+d x))^{3/2} \, dx &=\frac{2 a \sqrt{a+a \cosh (c+d x)} \sinh (c+d x)}{3 d}+\frac{1}{3} (4 a) \int \sqrt{a+a \cosh (c+d x)} \, dx\\ &=\frac{8 a^2 \sinh (c+d x)}{3 d \sqrt{a+a \cosh (c+d x)}}+\frac{2 a \sqrt{a+a \cosh (c+d x)} \sinh (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0633739, size = 55, normalized size = 0.93 \[ \frac{a \left (9 \sinh \left (\frac{1}{2} (c+d x)\right )+\sinh \left (\frac{3}{2} (c+d x)\right )\right ) \text{sech}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cosh (c+d x)+1)}}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 58, normalized size = 1. \begin{align*}{\frac{4\,{a}^{2}\sqrt{2}}{3\,d}\cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sinh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+2 \right ){\frac{1}{\sqrt{a \left ( \cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.55873, size = 109, normalized size = 1.85 \begin{align*} \frac{\sqrt{2} a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )}}{6 \, d} + \frac{3 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{2 \, d} - \frac{3 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{2 \, d} - \frac{\sqrt{2} a^{\frac{3}{2}} e^{\left (-\frac{3}{2} \, d x - \frac{3}{2} \, c\right )}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85382, size = 385, normalized size = 6.53 \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left (a \cosh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{3} + 9 \, a \cosh \left (d x + c\right )^{2} + 3 \,{\left (a \cosh \left (d x + c\right ) + 3 \, a\right )} \sinh \left (d x + c\right )^{2} - 9 \, a \cosh \left (d x + c\right ) + 3 \,{\left (a \cosh \left (d x + c\right )^{2} + 6 \, a \cosh \left (d x + c\right ) - 3 \, a\right )} \sinh \left (d x + c\right ) - a\right )} \sqrt{\frac{a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \cosh{\left (c + d x \right )} + a\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11523, size = 101, normalized size = 1.71 \begin{align*} -\frac{\sqrt{2}{\left ({\left (9 \, a^{\frac{3}{2}} e^{\left (d x + \frac{3}{2} \, c\right )} + a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, c\right )}\right )} e^{\left (-\frac{3}{2} \, d x - 2 \, c\right )} -{\left (a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, d x + \frac{15}{2} \, c\right )} + 9 \, a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, d x + \frac{13}{2} \, c\right )}\right )} e^{\left (-6 \, c\right )}\right )}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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