Optimal. Leaf size=78 \[ \frac{\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \cot (e+f x))^m \text{Hypergeometric2F1}\left (1,\frac{1}{2} (-m+2 p+1),\frac{1}{2} (-m+2 p+3),-\tan ^2(e+f x)\right )}{f (-m+2 p+1)} \]
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Rubi [A] time = 0.122044, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3658, 2604, 3476, 364} \[ \frac{\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \cot (e+f x))^m \, _2F_1\left (1,\frac{1}{2} (-m+2 p+1);\frac{1}{2} (-m+2 p+3);-\tan ^2(e+f x)\right )}{f (-m+2 p+1)} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 2604
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx &=\left (\tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \int (d \cot (e+f x))^m \tan ^{2 p}(e+f x) \, dx\\ &=\left ((d \cot (e+f x))^m \tan ^{m-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \int \tan ^{-m+2 p}(e+f x) \, dx\\ &=\frac{\left ((d \cot (e+f x))^m \tan ^{m-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \operatorname{Subst}\left (\int \frac{x^{-m+2 p}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(d \cot (e+f x))^m \, _2F_1\left (1,\frac{1}{2} (1-m+2 p);\frac{1}{2} (3-m+2 p);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1-m+2 p)}\\ \end{align*}
Mathematica [A] time = 0.136519, size = 70, normalized size = 0.9 \[ -\frac{d \left (b \tan ^2(e+f x)\right )^p (d \cot (e+f x))^{m-1} \text{Hypergeometric2F1}\left (1,-\frac{m}{2}+p+\frac{1}{2},-\frac{m}{2}+p+\frac{3}{2},-\tan ^2(e+f x)\right )}{f (m-2 p-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.646, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{m} \left ( b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{p}\, 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 )^{2}\right )^{p} \left (d \cot \left (f x + e\right )\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 )^{2}\right )^{p} \left (d \cot \left (f x + e\right )\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 ^{2}{\left (e + f x \right )}\right )^{p} \left (d \cot{\left (e + f x \right )}\right )^{m}\, 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 )^{2}\right )^{p} \left (d \cot \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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