Optimal. Leaf size=58 \[ -\frac {\cos (e+f x) F_1\left (\frac {1}{2};-n,1;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right )}{f \sqrt {\sin (e+f x)+1}} \]
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Rubi [A] time = 0.07, antiderivative size = 58, 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} \[ -\frac {\cos (e+f x) F_1\left (\frac {1}{2};-n,1;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right )}{f \sqrt {\sin (e+f x)+1}} \]
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)}{\sqrt {1+\sin (e+f x)}} \, dx &=-\frac {\cos (e+f x) \operatorname {Subst}\left (\int \frac {(1-x)^n}{(2-x) \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}{2-x^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,1;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x)}{f \sqrt {1+\sin (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 1.58, size = 225, normalized size = 3.88 \[ \frac {\sqrt {\sin (e+f x)+1} \cos (e+f x) (-\sin (e+f x))^{-n} \sin ^n(e+f x) \left (1-\frac {1}{\sin (e+f x)+1}\right )^{-n} \left (4 \sqrt {\frac {\sin (e+f x)-1}{\sin (e+f x)+1}} (-\sin (e+f x))^n F_1\left (-n-\frac {1}{2};-\frac {1}{2},-n;\frac {1}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {1}{\sin (e+f x)+1}\right )-(2 n+1) \sqrt {2-2 \sin (e+f x)} \left (1-\frac {1}{\sin (e+f x)+1}\right )^n F_1\left (1;\frac {1}{2},-n;2;\frac {1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )\right )}{4 f (2 n+1) (\sin (e+f x)-1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sin \left (f x + e\right )^{n}}{\sqrt {\sin \left (f x + e\right ) + 1}}, x\right ) \]
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 \[ \int \frac {\sin \left (f x + e\right )^{n}}{\sqrt {\sin \left (f x + e\right ) + 1}}\,{d x} \]
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 \[ \int \frac {\sin ^{n}\left (f x +e \right )}{\sqrt {1+\sin \left (f x +e \right )}}\, dx \]
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 \[ \int \frac {\sin \left (f x + e\right )^{n}}{\sqrt {\sin \left (f x + e\right ) + 1}}\,{d x} \]
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 \[ \int \frac {{\sin \left (e+f\,x\right )}^n}{\sqrt {\sin \left (e+f\,x\right )+1}} \,d x \]
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 \[ \int \frac {\sin ^{n}{\left (e + f x \right )}}{\sqrt {\sin {\left (e + f x \right )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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