Optimal. Leaf size=83 \[ -\frac {\left (\frac {7}{2}\right )^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},m+1;\frac {3}{2};\frac {a-a \sin (e+f x)}{7 (\sin (e+f x) a+a)}\right )}{f} \]
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Rubi [A] time = 0.11, antiderivative size = 118, normalized size of antiderivative = 1.42, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2788, 132} \[ -\frac {\sqrt {\frac {1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (4 \sin (e+f x)+3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},-m;1-m;\frac {2 (4 \sin (e+f x)+3)}{7 (\sin (e+f x)+1)}\right )}{\sqrt {7} f m (1-\sin (e+f x))} \]
Antiderivative was successfully verified.
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Rule 132
Rule 2788
Rubi steps
\begin {align*} \int (3+4 \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+4 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 {2 (3+4 \sin (e+f x))}{7 (1+\sin (e+f x))}\right ) \sqrt {\frac {1-\sin (e+f x)}{1+\sin (e+f x)}} (3+4 \sin (e+f x))^{-m} (a+a \sin (e+f x))^m}{\sqrt {7} f m (1-\sin (e+f x))}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 90, normalized size = 1.08 \[ -\frac {2\ 7^{-m-1} \cot \left (\frac {1}{4} (2 e+2 f x+\pi )\right ) \sin ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right )^{-m} (a (\sin (e+f x)+1))^m \, _2F_1\left (\frac {1}{2},m+1;\frac {3}{2};\frac {1}{7} \tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )}{f} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (4 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1}, x\right ) \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (4 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \left (3+4 \sin \left (f x +e \right )\right )^{-1-m} \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (4 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (4\,\sin \left (e+f\,x\right )+3\right )}^{m+1}} \,d x \]
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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