Optimal. Leaf size=106 \[ -\frac{2 a^2 (4 n+5) \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) \sin ^{n+1}(e+f x)}{f (2 n+3) \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.140269, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2763, 21, 2776, 65} \[ -\frac{2 a^2 (4 n+5) \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) \sin ^{n+1}(e+f x)}{f (2 n+3) \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2763
Rule 21
Rule 2776
Rule 65
Rubi steps
\begin{align*} \int \sin ^n(e+f x) (a+a \sin (e+f x))^{3/2} \, dx &=-\frac{2 a^2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\frac{2 \int \frac{\sin ^n(e+f x) \left (\frac{1}{2} a^2 (5+4 n)+\frac{1}{2} a^2 (5+4 n) \sin (e+f x)\right )}{\sqrt{a+a \sin (e+f x)}} \, dx}{3+2 n}\\ &=-\frac{2 a^2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\frac{(a (5+4 n)) \int \sin ^n(e+f x) \sqrt{a+a \sin (e+f x)} \, dx}{3+2 n}\\ &=-\frac{2 a^2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\frac{\left (a^3 (5+4 n) \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{x^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{f (3+2 n) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{2 a^2 (5+4 n) \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f (3+2 n) \sqrt{a+a \sin (e+f x)}}-\frac{2 a^2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 6.33124, size = 5111, normalized size = 48.22 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.125, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( fx+e \right ) \right ) ^{n} \left ( a+a\sin \left ( fx+e \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{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \sin \left (f x + e\right )^{n}\,{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 ({\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \sin \left (f x + e\right )^{n}, 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{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \sin \left (f x + e\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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