Optimal. Leaf size=59 \[ \frac{8 a^2 \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d} \]
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Rubi [A] time = 0.0293274, 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} \[ \frac{8 a^2 \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d} \]
Antiderivative was successfully verified.
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Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{3/2} \, dx &=\frac{2 a \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{3 d}+\frac{1}{3} (4 a) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{8 a^2 \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0640326, size = 55, normalized size = 0.93 \[ \frac{a \left (9 \sin \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{3}{2} (c+d x)\right )\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)}}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.83, size = 58, normalized size = 1. \begin{align*}{\frac{4\,{a}^{2}\sqrt{2}}{3\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+2 \right ){\frac{1}{\sqrt{ \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.92992, size = 51, normalized size = 0.86 \begin{align*} \frac{{\left (\sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 9 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} \sqrt{a}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54189, size = 117, normalized size = 1.98 \begin{align*} \frac{2 \,{\left (a \cos \left (d x + c\right ) + 5 \, a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right ) + d\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 \cos{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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