Optimal. Leaf size=60 \[ \frac {d (c+d \sin (e+f x))^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {c+d \sin (e+f x)}{c-d}\right )}{a^2 f (n+1) (c-d)^2} \]
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Rubi [A] time = 0.11, antiderivative size = 60, 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} \[ \frac {d (c+d \sin (e+f x))^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {c+d \sin (e+f x)}{c-d}\right )}{a^2 f (n+1) (c-d)^2} \]
Antiderivative was successfully verified.
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Rule 68
Rule 2833
Rubi steps
\begin {align*} \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (c+\frac {d x}{a}\right )^n}{(a+x)^2} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac {d \, _2F_1\left (2,1+n;2+n;\frac {c+d \sin (e+f x)}{c-d}\right ) (c+d \sin (e+f x))^{1+n}}{a^2 (c-d)^2 f (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 61, normalized size = 1.02 \[ \frac {d (c+d \sin (e+f x))^{n+1} \, _2F_1\left (2,n+1;n+2;-\frac {c+d \sin (e+f x)}{d-c}\right )}{a^2 f (n+1) (d-c)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.90, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{2}}\, dx \]
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 \[ \int \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {\cos \left (e+f\,x\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
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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