Optimal. Leaf size=28 \[ \frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
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Rubi [A] time = 0.0424421, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {2749} \[ \frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 2749
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{3/2} \left (-\frac{3 B}{5}+B \cos (c+d x)\right ) \, dx &=\frac{2 B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.103002, size = 45, normalized size = 1.61 \[ \frac{8 a B \sin \left (\frac{1}{2} (c+d x)\right ) \cos ^3\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)}}{5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.658, size = 48, normalized size = 1.7 \begin{align*}{\frac{8\,{a}^{2}B\sqrt{2}}{5\,d} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}\sin \left ({\frac{dx}{2}}+{\frac{c}{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 [B] time = 1.90831, size = 124, normalized size = 4.43 \begin{align*} \frac{{\left (\sqrt{2} a \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 \, \sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 20 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} B \sqrt{a} - 2 \,{\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 )} B \sqrt{a}}{10 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37911, size = 95, normalized size = 3.39 \begin{align*} \frac{2 \,{\left (B a \cos \left (d x + c\right ) + B a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{5 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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