Optimal. Leaf size=60 \[ -\frac {\cos (e+f x) F_1\left (\frac {1}{2};-n,2;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right )}{2 f \sqrt {\sin (e+f x)+1}} \]
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Rubi [A] time = 0.07, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2785, 130, 429} \[ -\frac {\cos (e+f x) F_1\left (\frac {1}{2};-n,2;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right )}{2 f \sqrt {\sin (e+f x)+1}} \]
Antiderivative was successfully verified.
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Rule 130
Rule 429
Rule 2785
Rubi steps
\begin {align*} \int \frac {\sin ^n(e+f x)}{(1+\sin (e+f x))^{3/2}} \, dx &=-\frac {\cos (e+f x) \operatorname {Subst}\left (\int \frac {(1-x)^n}{(2-x)^2 \sqrt {x}} \, dx,x,1-\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}\\ &=-\frac {(2 \cos (e+f x)) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^n}{\left (2-x^2\right )^2} \, dx,x,\sqrt {1-\sin (e+f x)}\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}\\ &=-\frac {F_1\left (\frac {1}{2};-n,2;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x)}{2 f \sqrt {1+\sin (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 3.79, size = 263, normalized size = 4.38 \[ \frac {\sec (e+f x) \sin ^n(e+f x) \left (\sqrt {2-2 \sin (e+f x)} (\sin (e+f x)+1)^2 (-\sin (e+f x))^{-n} F_1\left (1;\frac {1}{2},-n;2;\frac {1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )-\frac {4 (\sin (e+f x)+1) \sqrt {1-\frac {2}{\sin (e+f x)+1}} \left (1-\frac {1}{\sin (e+f x)+1}\right )^{-n} \left (2 (2 n+1) F_1\left (\frac {1}{2}-n;-\frac {1}{2},-n;\frac {3}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {1}{\sin (e+f x)+1}\right )+(2 n-1) (\sin (e+f x)+1) F_1\left (-n-\frac {1}{2};-\frac {1}{2},-n;\frac {1}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {1}{\sin (e+f x)+1}\right )\right )}{4 n^2-1}\right )}{8 f \sqrt {\sin (e+f x)+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sin \left (f x + e\right )^{n} \sqrt {\sin \left (f x + e\right ) + 1}}{\cos \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) - 2}, x\right ) \]
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 \[ \int \frac {\sin \left (f x + e\right )^{n}}{{\left (\sin \left (f x + e\right ) + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{n}\left (f x +e \right )}{\left (1+\sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\sin \left (f x + e\right )^{n}}{{\left (\sin \left (f x + e\right ) + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\sin \left (e+f\,x\right )}^n}{{\left (\sin \left (e+f\,x\right )+1\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sin ^{n}{\left (e + f x \right )}}{\left (\sin {\left (e + f x \right )} + 1\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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