Optimal. Leaf size=72 \[ \frac{b \cos (e+f x) (b \csc (e+f x))^{n-1} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{1-n}{2},\frac{3-n}{2},\sin ^2(e+f x)\right )}{f (1-n) \sqrt{\cos ^2(e+f x)}} \]
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Rubi [A] time = 0.0796031, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2631, 2577} \[ \frac{b \cos (e+f x) (b \csc (e+f x))^{n-1} \, _2F_1\left (-\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n) \sqrt{\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2631
Rule 2577
Rubi steps
\begin{align*} \int \cos ^2(e+f x) (b \csc (e+f x))^n \, dx &=\left (b^2 (b \csc (e+f x))^{-1+n} (b \sin (e+f x))^{-1+n}\right ) \int \cos ^2(e+f x) (b \sin (e+f x))^{-n} \, dx\\ &=\frac{b \cos (e+f x) (b \csc (e+f x))^{-1+n} \, _2F_1\left (-\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n) \sqrt{\cos ^2(e+f x)}}\\ \end{align*}
Mathematica [B] time = 0.456792, size = 165, normalized size = 2.29 \[ -\frac{2 \tan \left (\frac{1}{2} (e+f x)\right ) \sec ^2\left (\frac{1}{2} (e+f x)\right )^{-n} (b \csc (e+f x))^n \left (\text{Hypergeometric2F1}\left (1-n,\frac{1}{2}-\frac{n}{2},\frac{3}{2}-\frac{n}{2},-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-4 \text{Hypergeometric2F1}\left (2-n,\frac{1}{2}-\frac{n}{2},\frac{3}{2}-\frac{n}{2},-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+4 \text{Hypergeometric2F1}\left (3-n,\frac{1}{2}-\frac{n}{2},\frac{3}{2}-\frac{n}{2},-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )}{f (n-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.946, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( b\csc \left ( fx+e \right ) \right ) ^{n}\, 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 \csc \left (f x + e\right )\right )^{n} \cos \left (f x + e\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 (\left (b \csc \left (f x + e\right )\right )^{n} \cos \left (f x + e\right )^{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*} \int \left (b \csc{\left (e + f x \right )}\right )^{n} \cos ^{2}{\left (e + f x \right )}\, 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 \csc \left (f x + e\right )\right )^{n} \cos \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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