Optimal. Leaf size=67 \[ \frac{\cot ^m(e+f x) (b \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{b f (-m+n+1)} \]
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Rubi [A] time = 0.067043, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2604, 3476, 364} \[ \frac{\cot ^m(e+f x) (b \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{b f (-m+n+1)} \]
Antiderivative was successfully verified.
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Rule 2604
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \cot ^m(e+f x) (b \tan (e+f x))^n \, dx &=\left (\cot ^m(e+f x) (b \tan (e+f x))^m\right ) \int (b \tan (e+f x))^{-m+n} \, dx\\ &=\frac{\left (b \cot ^m(e+f x) (b \tan (e+f x))^m\right ) \operatorname{Subst}\left (\int \frac{x^{-m+n}}{b^2+x^2} \, dx,x,b \tan (e+f x)\right )}{f}\\ &=\frac{\cot ^m(e+f x) \, _2F_1\left (1,\frac{1}{2} (1-m+n);\frac{1}{2} (3-m+n);-\tan ^2(e+f x)\right ) (b \tan (e+f x))^{1+n}}{b f (1-m+n)}\\ \end{align*}
Mathematica [A] time = 0.0663491, size = 64, normalized size = 0.96 \[ \frac{\cot ^{m-1}(e+f x) (b \tan (e+f x))^n \, _2F_1\left (1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{f (-m+n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.307, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( fx+e \right ) \right ) ^{m} \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} \cot \left (f x + e\right )^{m}\,{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} \cot \left (f x + e\right )^{m}, 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} \cot ^{m}{\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} \cot \left (f x + e\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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