Optimal. Leaf size=116 \[ -\frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2}-m,-m;1-m;-\frac {2 \sin (e+f x)}{1-\sin (e+f x)}\right ) (d \sin (e+f x))^{-m} \left (\frac {1+\sin (e+f x)}{1-\sin (e+f x)}\right )^{\frac {1}{2}-m} (a+a \sin (e+f x))^m}{d f m (1+\sin (e+f x))} \]
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Rubi [A]
time = 0.14, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2866, 2865,
2864, 134} \begin {gather*} -\frac {\cos (e+f x) \left (\frac {\sin (e+f x)+1}{1-\sin (e+f x)}\right )^{\frac {1}{2}-m} (a \sin (e+f x)+a)^m (d \sin (e+f x))^{-m} \, _2F_1\left (\frac {1}{2}-m,-m;1-m;-\frac {2 \sin (e+f x)}{1-\sin (e+f x)}\right )}{d f m (\sin (e+f x)+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 134
Rule 2864
Rule 2865
Rule 2866
Rubi steps
\begin {align*} \int (d \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx &=\left ((1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int (d \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx\\ &=\frac {\left (\sin ^m(e+f x) (d \sin (e+f x))^{-m} (1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int \sin ^{-1-m}(e+f x) (1+\sin (e+f x))^m \, dx}{d}\\ &=-\frac {\left (\cos (e+f x) \sin ^m(e+f x) (d \sin (e+f x))^{-m} (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^m\right ) \text {Subst}\left (\int \frac {(1-x)^{-1-m} (2-x)^{-\frac {1}{2}+m}}{\sqrt {x}} \, dx,x,1-\sin (e+f x)\right )}{d f \sqrt {1-\sin (e+f x)}}\\ &=-\frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2}-m,-m;1-m;-\frac {2 \sin (e+f x)}{1-\sin (e+f x)}\right ) (d \sin (e+f x))^{-m} \left (\frac {1+\sin (e+f x)}{1-\sin (e+f x)}\right )^{\frac {1}{2}-m} (a+a \sin (e+f x))^m}{d f m (1+\sin (e+f x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.61, size = 194, normalized size = 1.67 \begin {gather*} \frac {(1-i) 2^m \, _2F_1\left (1+m,1+2 m;2 (1+m);\sqrt {2} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right ) \csc \left (\frac {1}{2} (e+f x)\right )\right ) ((1-i) (1+\cos (e+f x)-i \sin (e+f x)))^m ((1+i) (1-\cos (e+f x)+i \sin (e+f x)))^{-m} (d \sin (e+f x))^{-m} (a (1+\sin (e+f x)))^m (\cos (e+f x)-i (1+\sin (e+f x))) (\cosh (m \log (2))-\sinh (m \log (2)))}{d f (1+2 m) (-1+\cos (e+f x)-i \sin (e+f x))} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (d \sin \left (f x +e \right )\right )^{-1-m} \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx\]
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 \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (d \sin {\left (e + f x \right )}\right )^{- m - 1}\, dx \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 (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (d\,\sin \left (e+f\,x\right )\right )}^{m+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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