Optimal. Leaf size=49 \[ \frac{4 (a \sin (c+d x)+a)^{3/2}}{3 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d} \]
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Rubi [A] time = 0.0625084, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2667, 43} \[ \frac{4 (a \sin (c+d x)+a)^{3/2}}{3 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x) \sqrt{a+x} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a \sqrt{a+x}-(a+x)^{3/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{4 (a+a \sin (c+d x))^{3/2}}{3 a^2 d}-\frac{2 (a+a \sin (c+d x))^{5/2}}{5 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.0637287, size = 34, normalized size = 0.69 \[ -\frac{2 (3 \sin (c+d x)-7) (a (\sin (c+d x)+1))^{3/2}}{15 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.088, size = 31, normalized size = 0.6 \begin{align*} -{\frac{6\,\sin \left ( dx+c \right ) -14}{15\,{a}^{2}d} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953746, size = 101, normalized size = 2.06 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{a \sin \left (d x + c\right ) + a} - \frac{3 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 10 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{a \sin \left (d x + c\right ) + a} a^{2}}{a^{2}}\right )}}{15 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86179, size = 104, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) + 4\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{15 \, a 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 [A] time = 1.15475, size = 51, normalized size = 1.04 \begin{align*} -\frac{2 \,{\left (3 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 10 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a\right )}}{15 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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