Optimal. Leaf size=122 \[ -\frac {2^{m+\frac {1}{2}} 5^{-m-\frac {1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left (\frac {\sin (e+f x)+1}{2 \sin (e+f x)+3}\right )^{\frac {1}{2}-m} (2 \sin (e+f x)+3)^{-m} \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1-\sin (e+f x)}{2 (2 \sin (e+f x)+3)}\right )}{f} \]
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Rubi [A] time = 0.11, antiderivative size = 122, 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} \[ -\frac {2^{m+\frac {1}{2}} 5^{-m-\frac {1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left (\frac {\sin (e+f x)+1}{2 \sin (e+f x)+3}\right )^{\frac {1}{2}-m} (2 \sin (e+f x)+3)^{-m} \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1-\sin (e+f x)}{2 (2 \sin (e+f x)+3)}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 132
Rule 2788
Rubi steps
\begin {align*} \int (1+\sin (e+f x))^m (3+2 \sin (e+f x))^{-1-m} \, dx &=\frac {\cos (e+f x) \operatorname {Subst}\left (\int \frac {(1+x)^{-\frac {1}{2}+m} (3+2 x)^{-1-m}}{\sqrt {1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}\\ &=-\frac {2^{\frac {1}{2}+m} 5^{-\frac {1}{2}-m} \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1-\sin (e+f x)}{2 (3+2 \sin (e+f x))}\right ) (1+\sin (e+f x))^{-1+m} \left (\frac {1+\sin (e+f x)}{3+2 \sin (e+f x)}\right )^{\frac {1}{2}-m} (3+2 \sin (e+f x))^{-m}}{f}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 131, normalized size = 1.07 \[ \frac {2\ 5^{-m-1} \tan \left (\frac {1}{4} (2 e+2 f x-\pi )\right ) (\sin (e+f x)+1)^m (2 \sin (e+f x)+3)^{-m} \left ((2 \sin (e+f x)+3) \sec ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )^m \, _2F_1\left (\frac {1}{2},m+1;\frac {3}{2};-\frac {1}{5} \cos ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right ) \sec ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )}{f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (2 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}, x\right ) \]
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 \[ \int {\left (2 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.68, size = 0, normalized size = 0.00 \[ \int \left (1+\sin \left (f x +e \right )\right )^{m} \left (3+2 \sin \left (f x +e \right )\right )^{-1-m}\, dx \]
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 \[ \int {\left (2 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}\,{d x} \]
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 \[ \int \frac {{\left (\sin \left (e+f\,x\right )+1\right )}^m}{{\left (2\,\sin \left (e+f\,x\right )+3\right )}^{m+1}} \,d x \]
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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