Optimal. Leaf size=73 \[ \frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d} \]
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Rubi [A] time = 0.0745764, 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} \[ \frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 \sqrt{a+x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^2 \sqrt{a+x}-4 a (a+x)^{3/2}+(a+x)^{5/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{8 (a+a \sin (c+d x))^{3/2}}{3 a^3 d}-\frac{8 (a+a \sin (c+d x))^{5/2}}{5 a^4 d}+\frac{2 (a+a \sin (c+d x))^{7/2}}{7 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.0964159, size = 44, normalized size = 0.6 \[ \frac{2 \left (15 \sin ^2(c+d x)-54 \sin (c+d x)+71\right ) (a (\sin (c+d x)+1))^{3/2}}{105 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 41, normalized size = 0.6 \begin{align*} -{\frac{30\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+108\,\sin \left ( dx+c \right ) -172}{105\,{a}^{3}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 = 0.963007, size = 74, normalized size = 1.01 \begin{align*} \frac{2 \,{\left (15 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} - 84 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a + 140 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{2}\right )}}{105 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2489, size = 142, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (39 \, \cos \left (d x + c\right )^{2} -{\left (15 \, \cos \left (d x + c\right )^{2} - 32\right )} \sin \left (d x + c\right ) + 32\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{105 \, 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 = 1.20317, size = 74, normalized size = 1.01 \begin{align*} \frac{2 \,{\left (15 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} - 84 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a + 140 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{2}\right )}}{105 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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