Optimal. Leaf size=131 \[ \frac{(B+i A) (a+i a \tan (c+d x))^n \text{Hypergeometric2F1}\left (1,n,n+1,\frac{1}{2} (1+i \tan (c+d x))\right )}{2 d n}-\frac{(B+i A n) (a+i a \tan (c+d x))^n \text{Hypergeometric2F1}(1,n,n+1,1+i \tan (c+d x))}{d n}-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^n}{d} \]
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Rubi [A] time = 0.326724, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {3598, 3600, 3481, 68, 3599, 65} \[ \frac{(B+i A) (a+i a \tan (c+d x))^n \, _2F_1\left (1,n;n+1;\frac{1}{2} (i \tan (c+d x)+1)\right )}{2 d n}-\frac{(B+i A n) (a+i a \tan (c+d x))^n \, _2F_1(1,n;n+1;i \tan (c+d x)+1)}{d n}-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^n}{d} \]
Antiderivative was successfully verified.
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Rule 3598
Rule 3600
Rule 3481
Rule 68
Rule 3599
Rule 65
Rubi steps
\begin{align*} \int \cot ^2(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx &=-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^n}{d}+\frac{\int \cot (c+d x) (a+i a \tan (c+d x))^n (a (B+i A n)-a A (1-n) \tan (c+d x)) \, dx}{a}\\ &=-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^n}{d}+(-A+i B) \int (a+i a \tan (c+d x))^n \, dx+\frac{(B+i A n) \int \cot (c+d x) (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^n \, dx}{a}\\ &=-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^n}{d}+\frac{(a (i A+B)) \operatorname{Subst}\left (\int \frac{(a+x)^{-1+n}}{a-x} \, dx,x,i a \tan (c+d x)\right )}{d}+\frac{(a (B+i A n)) \operatorname{Subst}\left (\int \frac{(a+i a x)^{-1+n}}{x} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^n}{d}+\frac{(i A+B) \, _2F_1\left (1,n;1+n;\frac{1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^n}{2 d n}-\frac{(B+i A n) \, _2F_1(1,n;1+n;1+i \tan (c+d x)) (a+i a \tan (c+d x))^n}{d n}\\ \end{align*}
Mathematica [F] time = 44.2511, size = 0, normalized size = 0. \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.761, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( dx+c \right ) \right ) ^{2} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{n} \left ( A+B\tan \left ( dx+c \right ) \right ) \, 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 (d x + c\right ) + A\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\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 (-\frac{{\left ({\left (A - i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, A e^{\left (2 i \, d x + 2 i \, c\right )} + A + i \, B\right )} \left (\frac{2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n}}{e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (d x + c\right ) + A\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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