Optimal. Leaf size=288 \[ \frac {2 \left (1-n^2 p^2\right ) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+2);\frac {1}{2} (n p+4);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (n p+2) \sqrt {\cos ^2(e+f x)}}-\frac {n p (1-2 n p) \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (n p+1) \sqrt {\cos ^2(e+f x)}}+\frac {2 (1-n p) \sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (\sin (e+f x)+1)}+\frac {\sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 f (a \sin (e+f x)+a)^2} \]
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Rubi [A] time = 0.49, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2826, 2766, 2978, 2748, 2643} \[ \frac {2 \left (1-n^2 p^2\right ) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+2);\frac {1}{2} (n p+4);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (n p+2) \sqrt {\cos ^2(e+f x)}}-\frac {n p (1-2 n p) \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (n p+1) \sqrt {\cos ^2(e+f x)}}+\frac {2 (1-n p) \sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (\sin (e+f x)+1)}+\frac {\sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 f (a \sin (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rule 2766
Rule 2826
Rule 2978
Rubi steps
\begin {align*} \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(a+a \sin (e+f x))^2} \, dx &=\left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \frac {(d \sin (e+f x))^{n p}}{(a+a \sin (e+f x))^2} \, dx\\ &=\frac {\cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 f (a+a \sin (e+f x))^2}+\frac {\left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \frac {(d \sin (e+f x))^{n p} (a d (2-n p)+a d n p \sin (e+f x))}{a+a \sin (e+f x)} \, dx}{3 a^2 d}\\ &=\frac {2 (1-n p) \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (1+\sin (e+f x))}+\frac {\cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 f (a+a \sin (e+f x))^2}+\frac {\left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{n p} \left (-a^2 d^2 n p (1-2 n p)+2 a^2 d^2 (1-n p) (1+n p) \sin (e+f x)\right ) \, dx}{3 a^4 d^2}\\ &=\frac {2 (1-n p) \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (1+\sin (e+f x))}+\frac {\cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 f (a+a \sin (e+f x))^2}-\frac {\left (n p (1-2 n p) (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{n p} \, dx}{3 a^2}+\frac {\left (2 (1-n p) (1+n p) (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{1+n p} \, dx}{3 a^2 d}\\ &=-\frac {n p (1-2 n p) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (1+n p) \sqrt {\cos ^2(e+f x)}}+\frac {2 \left (1-n^2 p^2\right ) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (2+n p);\frac {1}{2} (4+n p);\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (2+n p) \sqrt {\cos ^2(e+f x)}}+\frac {2 (1-n p) \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (1+\sin (e+f x))}+\frac {\cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 f (a+a \sin (e+f x))^2}\\ \end {align*}
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Mathematica [A] time = 2.90, size = 195, normalized size = 0.68 \[ \frac {\sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n \left (-\frac {2 \left (n^2 p^2-1\right ) \sqrt {\cos ^2(e+f x)} \tan (e+f x) \sec (e+f x) \, _2F_1\left (\frac {1}{2},\frac {n p}{2}+1;\frac {n p}{2}+2;\sin ^2(e+f x)\right )}{n p+2}+\frac {n p (2 n p-1) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right )}{(n p+1) \sqrt {\cos ^2(e+f x)}}+\frac {(2-2 n p) \sin (e+f x)-2 n p+3}{(\sin (e+f x)+1)^2}\right )}{3 a^2 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]
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 \[ \int \frac {\left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.43, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \left (d \sin \left (f x +e \right )\right )^{p}\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{2}}\, dx \]
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 \[ \int \frac {\left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (c\,{\left (d\,\sin \left (e+f\,x\right )\right )}^p\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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