Optimal. Leaf size=64 \[ -\frac {\left (\frac {7}{2}\right )^{-1-m} \cos (e+f x) \, _2F_1\left (\frac {1}{2},1+m;\frac {3}{2};\frac {1-\sin (e+f x)}{7 (1+\sin (e+f x))}\right )}{f (1+\sin (e+f x))} \]
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Rubi [A]
time = 0.08, antiderivative size = 122, normalized size of antiderivative = 1.91, 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 = {2867, 134}
\begin {gather*} -\frac {2^{m+\frac {1}{2}} 7^{-m-\frac {1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left (\frac {\sin (e+f x)+1}{4 \sin (e+f x)+3}\right )^{\frac {1}{2}-m} (4 \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 (4 \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+4 \sin (e+f x))^{-1-m} \, dx &=\frac {\cos (e+f x) \text {Subst}\left (\int \frac {(1+x)^{-\frac {1}{2}+m} (3+4 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} 7^{-\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+4 \sin (e+f x))}\right ) (1+\sin (e+f x))^{-1+m} \left (\frac {1+\sin (e+f x)}{3+4 \sin (e+f x)}\right )^{\frac {1}{2}-m} (3+4 \sin (e+f x))^{-m}}{f}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 88, normalized size = 1.38 \begin {gather*} -\frac {2\ 7^{-1-m} \cot \left (\frac {1}{4} (2 e+\pi +2 f x)\right ) \, _2F_1\left (\frac {1}{2},1+m;\frac {3}{2};\frac {1}{7} \tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right ) (1+\sin (e+f x))^m \sin ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )^{-m}}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \left (1+\sin \left (f x +e \right )\right )^{m} \left (3+4 \sin \left (f x +e \right )\right )^{-1-m}\, dx\]
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 = -0.02 \begin {gather*} \int \frac {{\left (\sin \left (e+f\,x\right )+1\right )}^m}{{\left (4\,\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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