Optimal. Leaf size=71 \[ \frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a^4 d}+\frac{8 \sqrt{a \sin (c+d x)+a}}{a^3 d} \]
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Rubi [A] time = 0.074642, antiderivative size = 71, 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} \[ \frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a^4 d}+\frac{8 \sqrt{a \sin (c+d x)+a}}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2}{\sqrt{a+x}} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{4 a^2}{\sqrt{a+x}}-4 a \sqrt{a+x}+(a+x)^{3/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{8 \sqrt{a+a \sin (c+d x)}}{a^3 d}-\frac{8 (a+a \sin (c+d x))^{3/2}}{3 a^4 d}+\frac{2 (a+a \sin (c+d x))^{5/2}}{5 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.0712361, size = 44, normalized size = 0.62 \[ \frac{2 \left (3 \sin ^2(c+d x)-14 \sin (c+d x)+43\right ) \sqrt{a (\sin (c+d x)+1)}}{15 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.081, size = 41, normalized size = 0.6 \begin{align*} -{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+28\,\sin \left ( dx+c \right ) -92}{15\,{a}^{3}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 = 0.951528, size = 74, normalized size = 1.04 \begin{align*} \frac{2 \,{\left (3 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 20 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 60 \, \sqrt{a \sin \left (d x + c\right ) + a} a^{2}\right )}}{15 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1591, size = 111, normalized size = 1.56 \begin{align*} -\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{2} + 14 \, \sin \left (d x + c\right ) - 46\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{15 \, a^{3} 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 = 1.20441, size = 74, normalized size = 1.04 \begin{align*} \frac{2 \,{\left (3 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 20 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 60 \, \sqrt{a \sin \left (d x + c\right ) + a} a^{2}\right )}}{15 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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