Optimal. Leaf size=47 \[ \frac{4 \sqrt{a \sin (c+d x)+a}}{a^2 d}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d} \]
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Rubi [A] time = 0.0665267, antiderivative size = 47, 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} \[ \frac{4 \sqrt{a \sin (c+d x)+a}}{a^2 d}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a-x}{\sqrt{a+x}} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{2 a}{\sqrt{a+x}}-\sqrt{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{4 \sqrt{a+a \sin (c+d x)}}{a^2 d}-\frac{2 (a+a \sin (c+d x))^{3/2}}{3 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.0500463, size = 32, normalized size = 0.68 \[ -\frac{2 (\sin (c+d x)-5) \sqrt{a (\sin (c+d x)+1)}}{3 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 29, normalized size = 0.6 \begin{align*} -{\frac{2\,\sin \left ( dx+c \right ) -10}{3\,{a}^{2}d}\sqrt{a+a\sin \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964622, size = 49, normalized size = 1.04 \begin{align*} -\frac{2 \,{\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} - 6 \, \sqrt{a \sin \left (d x + c\right ) + a} a\right )}}{3 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20186, size = 78, normalized size = 1.66 \begin{align*} -\frac{2 \, \sqrt{a \sin \left (d x + c\right ) + a}{\left (\sin \left (d x + c\right ) - 5\right )}}{3 \, a^{2} 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 = 1.15639, size = 49, normalized size = 1.04 \begin{align*} -\frac{2 \,{\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} - 6 \, \sqrt{a \sin \left (d x + c\right ) + a} a\right )}}{3 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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