Optimal. Leaf size=88 \[ \frac {(a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1)} \]
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Rubi [A] time = 0.14, 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 = {2833, 70, 69} \[ \frac {(a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 2833
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 {\operatorname {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 ) \operatorname {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 = 0.14, size = 88, normalized size = 1.00 \[ \frac {(a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right ), x\right ) \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, 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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )\,{d x} \]
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 \[ \int \cos \left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \]
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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