Optimal. Leaf size=73 \[ \frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d} \]
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Rubi [A] time = 0.0672223, antiderivative size = 73, 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} \[ \frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 (a+x)^{3/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^2 (a+x)^{3/2}-4 a (a+x)^{5/2}+(a+x)^{7/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{8 (a+a \sin (c+d x))^{5/2}}{5 a^3 d}-\frac{8 (a+a \sin (c+d x))^{7/2}}{7 a^4 d}+\frac{2 (a+a \sin (c+d x))^{9/2}}{9 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.13584, size = 51, normalized size = 0.7 \[ \frac{2 (\sin (c+d x)+1)^3 \left (35 \sin ^2(c+d x)-110 \sin (c+d x)+107\right )}{315 d \sqrt{a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 41, normalized size = 0.6 \begin{align*} -{\frac{70\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+220\,\sin \left ( dx+c \right ) -284}{315\,{a}^{3}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 [B] time = 0.953644, size = 216, normalized size = 2.96 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{a \sin \left (d x + c\right ) + a} - \frac{42 \,{\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 + 15 \, \sqrt{a \sin \left (d x + c\right ) + a} a^{2}\right )}}{a^{2}} + \frac{35 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 180 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a + 378 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{2} - 420 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{3} + 315 \, \sqrt{a \sin \left (d x + c\right ) + a} a^{4}}{a^{4}}\right )}}{315 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84219, size = 165, normalized size = 2.26 \begin{align*} \frac{2 \,{\left (35 \, \cos \left (d x + c\right )^{4} + 8 \, \cos \left (d x + c\right )^{2} + 8 \,{\left (5 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 64\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{315 \, 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 = 1.23034, size = 74, normalized size = 1.01 \begin{align*} \frac{2 \,{\left (35 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 180 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a + 252 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{2}\right )}}{315 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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