Optimal. Leaf size=59 \[ \frac {(a \sin (e+f x)+a)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1) (c-d)^2} \]
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Rubi [A] time = 0.10, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2833, 68} \[ \frac {(a \sin (e+f x)+a)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1) (c-d)^2} \]
Antiderivative was successfully verified.
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Rule 68
Rule 2833
Rubi steps
\begin {align*} \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+x)^m}{\left (c+\frac {d x}{a}\right )^2} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac {\, _2F_1\left (2,1+m;2+m;-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m}}{a (c-d)^2 f (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 59, normalized size = 1.00 \[ \frac {(a \sin (e+f x)+a)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1) (c-d)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.58, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{m}}{\left (c +d \sin \left (f x +e \right )\right )^{2}}\, dx \]
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 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\cos \left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
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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