Optimal. Leaf size=96 \[ -\frac {2 (4 n+5) \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right )}{f (2 n+3) \sqrt {\sin (e+f x)+1}}-\frac {2 \cos (e+f x) \sin ^{n+1}(e+f x)}{f (2 n+3) \sqrt {\sin (e+f x)+1}} \]
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Rubi [A] time = 0.11, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2763, 21, 2776, 65} \[ -\frac {2 (4 n+5) \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right )}{f (2 n+3) \sqrt {\sin (e+f x)+1}}-\frac {2 \cos (e+f x) \sin ^{n+1}(e+f x)}{f (2 n+3) \sqrt {\sin (e+f x)+1}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 65
Rule 2763
Rule 2776
Rubi steps
\begin {align*} \int \sin ^n(e+f x) (1+\sin (e+f x))^{3/2} \, dx &=-\frac {2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt {1+\sin (e+f x)}}+\frac {2 \int \frac {\sin ^n(e+f x) \left (\frac {1}{2} (5+4 n)+\frac {1}{2} (5+4 n) \sin (e+f x)\right )}{\sqrt {1+\sin (e+f x)}} \, dx}{3+2 n}\\ &=-\frac {2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt {1+\sin (e+f x)}}+\frac {(5+4 n) \int \sin ^n(e+f x) \sqrt {1+\sin (e+f x)} \, dx}{3+2 n}\\ &=-\frac {2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt {1+\sin (e+f x)}}+\frac {((5+4 n) \cos (e+f x)) \operatorname {Subst}\left (\int \frac {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 (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 {1+\sin (e+f x)}}-\frac {2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt {1+\sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 22.54, size = 5109, normalized size = 53.22 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sin \left (f x + e\right )^{n} {\left (\sin \left (f x + e\right ) + 1\right )}^{\frac {3}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \left (f x + e\right )^{n} {\left (\sin \left (f x + e\right ) + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{n}\left (f x +e \right )\right ) \left (1+\sin \left (f x +e \right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \left (f x + e\right )^{n} {\left (\sin \left (f x + e\right ) + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (e+f\,x\right )}^n\,{\left (\sin \left (e+f\,x\right )+1\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\sin {\left (e + f x \right )} + 1\right )^{\frac {3}{2}} \sin ^{n}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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