Optimal. Leaf size=116 \[ -\frac {\sqrt {-\frac {1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (\sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},-m;1-m;\frac {3-\sin (e+f x)}{\sin (e+f x)+1}\right )}{2 \sqrt {2} f m (1-\sin (e+f x))} \]
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Rubi [A] time = 0.10, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2788, 132} \[ -\frac {\sqrt {-\frac {1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (\sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},-m;1-m;\frac {3-\sin (e+f x)}{\sin (e+f x)+1}\right )}{2 \sqrt {2} f m (1-\sin (e+f x))} \]
Antiderivative was successfully verified.
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Rule 132
Rule 2788
Rubi steps
\begin {align*} \int (-3+\sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx &=\frac {\left (a^2 \cos (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(-3+x)^{-1-m} (a+a x)^{-\frac {1}{2}+m}}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2},-m;1-m;\frac {3-\sin (e+f x)}{1+\sin (e+f x)}\right ) (-3+\sin (e+f x))^{-m} \sqrt {-\frac {1-\sin (e+f x)}{1+\sin (e+f x)}} (a+a \sin (e+f x))^m}{2 \sqrt {2} f m (1-\sin (e+f x))}\\ \end {align*}
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Mathematica [A] time = 0.71, size = 155, normalized size = 1.34 \[ \frac {2^{-m} \cot \left (\frac {1}{4} (2 e+2 f x+\pi )\right ) (\sin (e+f x)-3)^{-m} \sin ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right )^{\frac {1}{2}-m} \cos ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )^{m-\frac {1}{2}} (a (\sin (e+f x)+1))^m \, _2F_1\left (\frac {1}{2},m+1;\frac {3}{2};-2 \tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right ) \left ((\sin (e+f x)-3) \left (-\sec ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )\right )^m}{f} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (\sin \left (f x + e\right ) - 3\right )}^{-m - 1}, x\right ) \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (\sin \left (f x + e\right ) - 3\right )}^{-m - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \left (-3+\sin \left (f x +e \right )\right )^{-1-m} \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (\sin \left (f x + e\right ) - 3\right )}^{-m - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (\sin \left (e+f\,x\right )-3\right )}^{m+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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