Optimal. Leaf size=49 \[ \frac{4 (a \sin (c+d x)+a)^{5/2}}{5 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d} \]
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Rubi [A] time = 0.0649647, 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} \[ \frac{4 (a \sin (c+d x)+a)^{5/2}}{5 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \cos ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x) (a+x)^{3/2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a (a+x)^{3/2}-(a+x)^{5/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{4 (a+a \sin (c+d x))^{5/2}}{5 a^2 d}-\frac{2 (a+a \sin (c+d x))^{7/2}}{7 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.238839, size = 54, normalized size = 1.1 \[ -\frac{2 (5 \sin (c+d x)-9) \sqrt{a (\sin (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4}{35 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 31, normalized size = 0.6 \begin{align*} -{\frac{10\,\sin \left ( dx+c \right ) -18}{35\,{a}^{2}d} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960687, size = 51, normalized size = 1.04 \begin{align*} -\frac{2 \,{\left (5 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} - 14 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a\right )}}{35 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59104, size = 124, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (\cos \left (d x + c\right )^{2} +{\left (5 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 8\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{35 \, 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 = 2.02199, size = 58, normalized size = 1.18 \begin{align*} -\frac{2 \,{\left (\frac{5 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}{a^{2}} - \frac{14 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{a}\right )}}{35 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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