Optimal. Leaf size=279 \[ \frac {(n+1) (2 A (1-n)+2 B n+B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left (\frac {1}{2},\frac {n+2}{2};\frac {n+4}{2};\sin ^2(e+f x)\right )}{3 a^2 d^2 f (n+2) \sqrt {\cos ^2(e+f x)}}-\frac {n (-2 A n+A+2 B (n+1)) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(e+f x)\right )}{3 a^2 d f (n+1) \sqrt {\cos ^2(e+f x)}}+\frac {(2 A (1-n)+2 B n+B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 a^2 d f (\sin (e+f x)+1)}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 d f (a \sin (e+f x)+a)^2} \]
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Rubi [A] time = 0.49, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2978, 2748, 2643} \[ \frac {(n+1) (2 A (1-n)+2 B n+B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left (\frac {1}{2},\frac {n+2}{2};\frac {n+4}{2};\sin ^2(e+f x)\right )}{3 a^2 d^2 f (n+2) \sqrt {\cos ^2(e+f x)}}-\frac {n (-2 A n+A+2 B (n+1)) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(e+f x)\right )}{3 a^2 d f (n+1) \sqrt {\cos ^2(e+f x)}}+\frac {(2 A (1-n)+2 B n+B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 a^2 d f (\sin (e+f x)+1)}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 d f (a \sin (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rule 2978
Rubi steps
\begin {align*} \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx &=\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{3 d f (a+a \sin (e+f x))^2}+\frac {\int \frac {(d \sin (e+f x))^n (a d (2 A+B-A n+B n)+a (A-B) d n \sin (e+f x))}{a+a \sin (e+f x)} \, dx}{3 a^2 d}\\ &=\frac {(B+2 A (1-n)+2 B n) \cos (e+f x) (d \sin (e+f x))^{1+n}}{3 a^2 d f (1+\sin (e+f x))}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{3 d f (a+a \sin (e+f x))^2}+\frac {\int (d \sin (e+f x))^n \left (-a^2 d^2 n (A-2 A n+2 B (1+n))+a^2 d^2 (1+n) (2 A (1-n)+B (1+2 n)) \sin (e+f x)\right ) \, dx}{3 a^4 d^2}\\ &=\frac {(B+2 A (1-n)+2 B n) \cos (e+f x) (d \sin (e+f x))^{1+n}}{3 a^2 d f (1+\sin (e+f x))}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{3 d f (a+a \sin (e+f x))^2}+\frac {((1+n) (B+2 A (1-n)+2 B n)) \int (d \sin (e+f x))^{1+n} \, dx}{3 a^2 d}-\frac {(n (A-2 A n+2 B (1+n))) \int (d \sin (e+f x))^n \, dx}{3 a^2}\\ &=-\frac {n (A-2 A n+2 B (1+n)) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+n}}{3 a^2 d f (1+n) \sqrt {\cos ^2(e+f x)}}+\frac {(1+n) (B+2 A (1-n)+2 B n) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {2+n}{2};\frac {4+n}{2};\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+n}}{3 a^2 d^2 f (2+n) \sqrt {\cos ^2(e+f x)}}+\frac {(B+2 A (1-n)+2 B n) \cos (e+f x) (d \sin (e+f x))^{1+n}}{3 a^2 d f (1+\sin (e+f x))}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{3 d f (a+a \sin (e+f x))^2}\\ \end {align*}
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Mathematica [A] time = 1.33, size = 212, normalized size = 0.76 \[ \frac {\sin (e+f x) \cos (e+f x) (d \sin (e+f x))^n \left (-\frac {n (-2 A n+A+2 B (n+1)) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(e+f x)\right )}{(n+1) \sqrt {\cos ^2(e+f x)}}+\frac {(n+1) (-2 A (n-1)+2 B n+B) \sin (e+f x) \, _2F_1\left (\frac {1}{2},\frac {n+2}{2};\frac {n+4}{2};\sin ^2(e+f x)\right )}{(n+2) \sqrt {\cos ^2(e+f x)}}+\frac {n (B-A)}{\sin (e+f x)+1}+\frac {A (-n)+2 A+B n+B}{\sin (e+f x)+1}+\frac {A-B}{(\sin (e+f x)+1)^2}\right )}{3 a^2 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 11.46, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sin \left (f x +e \right )\right )^{n} \left (A +B \sin \left (f x +e \right )\right )}{\left (a +a \sin \left (f x +e \right )\right )^{2}}\, dx \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d\,\sin \left (e+f\,x\right )\right )}^n\,\left (A+B\,\sin \left (e+f\,x\right )\right )}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
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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