Optimal. Leaf size=130 \[ \frac{(4 n+5) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},n+1;n+2;\sin (e+f x)\right )}{d f (n+1) (2 n+3) \sqrt{1-\sin (e+f x)} \sqrt{\sin (e+f x)+1}}-\frac{2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{\sin (e+f x)+1}} \]
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Rubi [A] time = 0.140899, antiderivative size = 130, 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, 64} \[ \frac{(4 n+5) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},n+1;n+2;\sin (e+f x)\right )}{d f (n+1) (2 n+3) \sqrt{1-\sin (e+f x)} \sqrt{\sin (e+f x)+1}}-\frac{2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{\sin (e+f x)+1}} \]
Antiderivative was successfully verified.
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Rule 2763
Rule 21
Rule 2776
Rule 64
Rubi steps
\begin{align*} \int (d \sin (e+f x))^n (1+\sin (e+f x))^{3/2} \, dx &=-\frac{2 \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{1+\sin (e+f x)}}+\frac{2 \int \frac{(d \sin (e+f x))^n \left (\frac{1}{2} d (5+4 n)+\frac{1}{2} d (5+4 n) \sin (e+f x)\right )}{\sqrt{1+\sin (e+f x)}} \, dx}{d (3+2 n)}\\ &=-\frac{2 \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{1+\sin (e+f x)}}+\frac{(5+4 n) \int (d \sin (e+f x))^n \sqrt{1+\sin (e+f x)} \, dx}{3+2 n}\\ &=-\frac{2 \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{1+\sin (e+f x)}}+\frac{((5+4 n) \cos (e+f x)) \operatorname{Subst}\left (\int \frac{(d x)^n}{\sqrt{1-x}} \, dx,x,\sin (e+f x)\right )}{f (3+2 n) \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ &=-\frac{2 \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{1+\sin (e+f x)}}+\frac{(5+4 n) \cos (e+f x) \, _2F_1\left (\frac{1}{2},1+n;2+n;\sin (e+f x)\right ) (d \sin (e+f x))^{1+n}}{d f (1+n) (3+2 n) \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 6.30222, size = 5129, normalized size = 39.45 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.116, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sin \left ( fx+e \right ) \right ) ^{n} \left ( 1+\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 (d \sin \left (f x + e\right )\right )^{n}{\left (\sin \left (f x + e\right ) + 1\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 (\left (d \sin \left (f x + e\right )\right )^{n}{\left (\sin \left (f x + e\right ) + 1\right )}^{\frac{3}{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 \left (d \sin \left (f x + e\right )\right )^{n}{\left (\sin \left (f x + e\right ) + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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