Optimal. Leaf size=78 \[ -\frac{\cos (e+f x) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n \, _2F_1\left (\frac{1-n}{2},\frac{2-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right )}{f (1-n)} \]
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Rubi [A] time = 0.0782896, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2601, 2576} \[ -\frac{\cos (e+f x) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n \, _2F_1\left (\frac{1-n}{2},\frac{2-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right )}{f (1-n)} \]
Antiderivative was successfully verified.
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Rule 2601
Rule 2576
Rubi steps
\begin{align*} \int \csc (e+f x) (b \tan (e+f x))^n \, dx &=\left (\cos ^n(e+f x) \sin ^{-n}(e+f x) (b \tan (e+f x))^n\right ) \int \cos ^{-n}(e+f x) \sin ^{-1+n}(e+f x) \, dx\\ &=-\frac{\cos (e+f x) \, _2F_1\left (\frac{1-n}{2},\frac{2-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n}{f (1-n)}\\ \end{align*}
Mathematica [A] time = 0.201303, size = 64, normalized size = 0.82 \[ \frac{(b \tan (e+f x))^n \, _2F_1\left (\frac{n}{2},n;\frac{n}{2}+1;\tan ^2\left (\frac{1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right )\right )^n}{f n} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.232, size = 0, normalized size = 0. \begin{align*} \int \csc \left ( fx+e \right ) \left ( b\tan \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 \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )\,{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 \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right ), 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 \tan{\left (e + f x \right )}\right )^{n} \csc{\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 \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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