Optimal. Leaf size=82 \[ \frac{2^{m+\frac{3}{4}} (\sin (c+d x)+1)^{\frac{1}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left (-\frac{1}{4},\frac{5}{4}-m;\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{d e \sqrt{e \cos (c+d x)}} \]
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Rubi [A] time = 0.0950441, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2689, 70, 69} \[ \frac{2^{m+\frac{3}{4}} (\sin (c+d x)+1)^{\frac{1}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left (-\frac{1}{4},\frac{5}{4}-m;\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{d e \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{3/2}} \, dx &=\frac{\left (a^2 \sqrt [4]{a-a \sin (c+d x)} \sqrt [4]{a+a \sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{-\frac{5}{4}+m}}{(a-a x)^{5/4}} \, dx,x,\sin (c+d x)\right )}{d e \sqrt{e \cos (c+d x)}}\\ &=\frac{\left (2^{-\frac{5}{4}+m} a \sqrt [4]{a-a \sin (c+d x)} (a+a \sin (c+d x))^m \left (\frac{a+a \sin (c+d x)}{a}\right )^{\frac{1}{4}-m}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{1}{2}+\frac{x}{2}\right )^{-\frac{5}{4}+m}}{(a-a x)^{5/4}} \, dx,x,\sin (c+d x)\right )}{d e \sqrt{e \cos (c+d x)}}\\ &=\frac{2^{\frac{3}{4}+m} \, _2F_1\left (-\frac{1}{4},\frac{5}{4}-m;\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac{1}{4}-m} (a+a \sin (c+d x))^m}{d e \sqrt{e \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.107823, size = 82, normalized size = 1. \[ \frac{2^{m+\frac{3}{4}} (\sin (c+d x)+1)^{\frac{1}{4}-m} (a (\sin (c+d x)+1))^m \, _2F_1\left (-\frac{1}{4},\frac{5}{4}-m;\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{d e \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.085, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e \cos \left (d x + c\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{e^{2} \cos \left (d x + c\right )^{2}}, x\right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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