Optimal. Leaf size=132 \[ -\frac{d \left (a^2-b^2\right ) (d \cot (e+f x))^{n-1} \, _2F_1\left (1,\frac{n-1}{2};\frac{n+1}{2};-\cot ^2(e+f x)\right )}{f (1-n)}+\frac{a^2 d (d \cot (e+f x))^{n-1}}{f (1-n)}-\frac{2 a b (d \cot (e+f x))^n \, _2F_1\left (1,\frac{n}{2};\frac{n+2}{2};-\cot ^2(e+f x)\right )}{f n} \]
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Rubi [A] time = 0.217563, antiderivative size = 132, 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 = {3673, 3543, 3538, 3476, 364} \[ -\frac{d \left (a^2-b^2\right ) (d \cot (e+f x))^{n-1} \, _2F_1\left (1,\frac{n-1}{2};\frac{n+1}{2};-\cot ^2(e+f x)\right )}{f (1-n)}+\frac{a^2 d (d \cot (e+f x))^{n-1}}{f (1-n)}-\frac{2 a b (d \cot (e+f x))^n \, _2F_1\left (1,\frac{n}{2};\frac{n+2}{2};-\cot ^2(e+f x)\right )}{f n} \]
Antiderivative was successfully verified.
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Rule 3673
Rule 3543
Rule 3538
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int (d \cot (e+f x))^n (a+b \tan (e+f x))^2 \, dx &=d^2 \int (d \cot (e+f x))^{-2+n} (b+a \cot (e+f x))^2 \, dx\\ &=\frac{a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}+d^2 \int (d \cot (e+f x))^{-2+n} \left (-a^2+b^2+2 a b \cot (e+f x)\right ) \, dx\\ &=\frac{a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}+(2 a b d) \int (d \cot (e+f x))^{-1+n} \, dx-\left (\left (a^2-b^2\right ) d^2\right ) \int (d \cot (e+f x))^{-2+n} \, dx\\ &=\frac{a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}-\frac{\left (2 a b d^2\right ) \operatorname{Subst}\left (\int \frac{x^{-1+n}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{f}+\frac{\left (\left (a^2-b^2\right ) d^3\right ) \operatorname{Subst}\left (\int \frac{x^{-2+n}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{f}\\ &=\frac{a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}-\frac{\left (a^2-b^2\right ) d (d \cot (e+f x))^{-1+n} \, _2F_1\left (1,\frac{1}{2} (-1+n);\frac{1+n}{2};-\cot ^2(e+f x)\right )}{f (1-n)}-\frac{2 a b (d \cot (e+f x))^n \, _2F_1\left (1,\frac{n}{2};\frac{2+n}{2};-\cot ^2(e+f x)\right )}{f n}\\ \end{align*}
Mathematica [A] time = 0.447957, size = 107, normalized size = 0.81 \[ -\frac{d (d \cot (e+f x))^{n-1} \left (a \left (a n+2 b (n-1) \cot (e+f x) \, _2F_1\left (1,\frac{n}{2};\frac{n+2}{2};-\cot ^2(e+f x)\right )\right )-n \left (a^2-b^2\right ) \, _2F_1\left (1,\frac{n-1}{2};\frac{n+1}{2};-\cot ^2(e+f x)\right )\right )}{f (n-1) n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.322, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{n} \left ( a+b\tan \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 \tan \left (f x + e\right ) + a\right )}^{2} \left (d \cot \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} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}\right )} \left (d \cot \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 \cot{\left (e + f x \right )}\right )^{n} \left (a + b \tan{\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 \tan \left (f x + e\right ) + a\right )}^{2} \left (d \cot \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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