Optimal. Leaf size=88 \[ \frac {\, _2F_1\left (1+m,-n;2+m;-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{a f (1+m)} \]
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Rubi [A]
time = 0.10, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2912, 72, 71}
\begin {gather*} \frac {(a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 2912
Rubi steps
\begin {align*} \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx &=\frac {\text {Subst}\left (\int (a+x)^m \left (c+\frac {d x}{a}\right )^n \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac {\left ((c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}\right ) \text {Subst}\left (\int (a+x)^m \left (\frac {c}{c-d}+\frac {d x}{a (c-d)}\right )^n \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac {\, _2F_1\left (1+m,-n;2+m;-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{a f (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 88, normalized size = 1.00 \begin {gather*} \frac {\, _2F_1\left (1+m,-n;2+m;-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{a f (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{n}\, dx\]
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 (c + d \sin {\left (e + f x \right )}\right )^{n} \cos {\left (e + f x \right )}\, 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 \cos \left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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