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 = 0.486528, antiderivative size = 288, normalized size of antiderivative = 1., 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 2826
Rule 2766
Rule 2978
Rule 2748
Rule 2643
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*}
Mathematica [A] time = 2.86747, 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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Maple [F] time = 0.349, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c \left ( d\sin \left ( fx+e \right ) \right ) ^{p} \right ) ^{n}}{ \left ( a+a\sin \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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