Optimal. Leaf size=106 \[ -\frac {2^{-\frac {1}{2}-m} \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1-\sin (e+f x)}{3+\sin (e+f x)}\right ) (1+\sin (e+f x))^{-1+m} \left (\frac {1+\sin (e+f x)}{3+\sin (e+f x)}\right )^{\frac {1}{2}-m} (3+\sin (e+f x))^{-m}}{f} \]
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Rubi [A]
time = 0.07, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2867, 134}
\begin {gather*} -\frac {2^{-m-\frac {1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left (\frac {\sin (e+f x)+1}{\sin (e+f x)+3}\right )^{\frac {1}{2}-m} (\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)}{\sin (e+f x)+3}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 134
Rule 2867
Rubi steps
\begin {align*} \int (1+\sin (e+f x))^m (3+\sin (e+f x))^{-1-m} \, dx &=\frac {\cos (e+f x) \text {Subst}\left (\int \frac {(1+x)^{-\frac {1}{2}+m} (3+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} \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1-\sin (e+f x)}{3+\sin (e+f x)}\right ) (1+\sin (e+f x))^{-1+m} \left (\frac {1+\sin (e+f x)}{3+\sin (e+f x)}\right )^{\frac {1}{2}-m} (3+\sin (e+f x))^{-m}}{f}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 167, normalized size = 1.58 \begin {gather*} \frac {2^{-1-2 m} \cos ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )^{-m} \, _2F_1\left (\frac {1}{2},1+m;\frac {3}{2};-\frac {1}{2} \cos ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ) \sec ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right ) (1+\sin (e+f x))^m \left (\frac {\cos ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{3+\sin (e+f x)}\right )^m \left (\sec ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right ) (3+\sin (e+f x))\right )^m \tan \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (1+\sin \left (f x +e \right )\right )^{m} \left (3+\sin \left (f x +e \right )\right )^{-1-m}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (\sin \left (e+f\,x\right )+1\right )}^m}{{\left (\sin \left (e+f\,x\right )+3\right )}^{m+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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