Optimal. Leaf size=231 \[ \frac{\left (a^2 (n p+2)+b^2 (n p+1)\right ) \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+1) (n p+2) \sqrt{\cos ^2(e+f x)}}+\frac{2 a b \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{b^2 \sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2)} \]
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Rubi [A] time = 0.232943, antiderivative size = 221, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2826, 2789, 2643, 3014} \[ \frac{\left (\frac{a^2}{n p+1}+\frac{b^2}{n p+2}\right ) \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f \sqrt{\cos ^2(e+f x)}}+\frac{2 a b \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{b^2 \sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2)} \]
Antiderivative was successfully verified.
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Rule 2826
Rule 2789
Rule 2643
Rule 3014
Rubi steps
\begin{align*} \int \left (c (d \sin (e+f x))^p\right )^n (a+b \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 (d \sin (e+f x))^{n p} (a+b \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 (d \sin (e+f x))^{n p} \left (a^2+b^2 \sin ^2(e+f x)\right ) \, dx+\frac{\left (2 a b (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}{d}\\ &=-\frac{b^2 \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p)}+\frac{2 a b \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}{f (2+n p) \sqrt{\cos ^2(e+f x)}}+\left (\left (a^2+\frac{b^2 (1+n p)}{2+n p}\right ) (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\\ &=-\frac{b^2 \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p)}+\frac{\left (a^2+\frac{b^2 (1+n p)}{2+n p}\right ) \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}{f (1+n p) \sqrt{\cos ^2(e+f x)}}+\frac{2 a b \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}{f (2+n p) \sqrt{\cos ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.331043, size = 152, normalized size = 0.66 \[ -\frac{\cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-n p-1)} \left (c (d \sin (e+f x))^p\right )^n \left (a \left (a \sin (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1-n p);\frac{3}{2};\cos ^2(e+f x)\right )+2 b \sqrt{\sin ^2(e+f x)} \, _2F_1\left (\frac{1}{2},-\frac{n p}{2};\frac{3}{2};\cos ^2(e+f x)\right )\right )+b^2 \sin (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-n p-1);\frac{3}{2};\cos ^2(e+f x)\right )\right )}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.382, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( d\sin \left ( fx+e \right ) \right ) ^{p} \right ) ^{n} \left ( a+b\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{\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}\,{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 (-{\left (b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}\right )} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}, 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*} \int \left (c \left (d \sin{\left (e + f x \right )}\right )^{p}\right )^{n} \left (a + b \sin{\left (e + f x \right )}\right )^{2}\, dx \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{\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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