Optimal. Leaf size=213 \[ \frac{d^3 \left (a^2 (2-n)+b^2 (1-n)\right ) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}+\frac{2 a b d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right )}{f (2-n) \sqrt{\cos ^2(e+f x)}} \]
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Rubi [A] time = 0.269088, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3238, 3788, 3772, 2643, 4046} \[ \frac{d^3 \left (a^2 (2-n)+b^2 (1-n)\right ) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}+\frac{2 a b d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right )}{f (2-n) \sqrt{\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 3788
Rule 3772
Rule 2643
Rule 4046
Rubi steps
\begin{align*} \int (d \csc (e+f x))^n (a+b \sin (e+f x))^2 \, dx &=d^2 \int (d \csc (e+f x))^{-2+n} (b+a \csc (e+f x))^2 \, dx\\ &=(2 a b d) \int (d \csc (e+f x))^{-1+n} \, dx+d^2 \int (d \csc (e+f x))^{-2+n} \left (b^2+a^2 \csc ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{-2+n}}{f (1-n)}+\left (d^2 \left (b^2+\frac{a^2 (2-n)}{1-n}\right )\right ) \int (d \csc (e+f x))^{-2+n} \, dx+\left (2 a b d (d \csc (e+f x))^n \left (\frac{\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac{\sin (e+f x)}{d}\right )^{1-n} \, dx\\ &=\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{-2+n}}{f (1-n)}+\frac{2 a b \cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right ) \sin ^2(e+f x)}{f (2-n) \sqrt{\cos ^2(e+f x)}}+\left (d^2 \left (b^2+\frac{a^2 (2-n)}{1-n}\right ) (d \csc (e+f x))^n \left (\frac{\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac{\sin (e+f x)}{d}\right )^{2-n} \, dx\\ &=\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{-2+n}}{f (1-n)}+\frac{2 a b \cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right ) \sin ^2(e+f x)}{f (2-n) \sqrt{\cos ^2(e+f x)}}+\frac{\left (b^2+\frac{a^2 (2-n)}{1-n}\right ) \cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right ) \sin ^3(e+f x)}{f (3-n) \sqrt{\cos ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.374521, size = 135, normalized size = 0.63 \[ -\frac{d \cos (e+f x) \sin ^2(e+f x)^{\frac{n-1}{2}} (d \csc (e+f x))^{n-1} \left (a \left (a \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{3}{2};\cos ^2(e+f x)\right )+2 b \sqrt{\sin ^2(e+f x)} \csc (e+f x) \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{3}{2};\cos ^2(e+f x)\right )\right )+b^2 \, _2F_1\left (\frac{1}{2},\frac{n-1}{2};\frac{3}{2};\cos ^2(e+f x)\right )\right )}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 3.652, size = 0, normalized size = 0. \begin{align*} \int \left ( d\csc \left ( fx+e \right ) \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 (d \csc \left (f x + e\right )\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 (d \csc \left (f x + e\right )\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 (d \csc{\left (e + f x \right )}\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 (d \csc \left (f x + e\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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