Integrand size = 34, antiderivative size = 185 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=-\frac {(2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{2 d}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}-\frac {(A-i B) \operatorname {Hypergeometric2F1}\left (1,n,1+n,\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^n}{2 d n}-\frac {\left (2 i B n-A \left (2-n+n^2\right )\right ) \operatorname {Hypergeometric2F1}(1,n,1+n,1+i \tan (c+d x)) (a+i a \tan (c+d x))^n}{2 d n} \]
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Time = 0.67 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3679, 3681, 3562, 70, 3680, 67} \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=-\frac {\left (-A \left (n^2-n+2\right )+2 i B n\right ) (a+i a \tan (c+d x))^n \operatorname {Hypergeometric2F1}(1,n,n+1,i \tan (c+d x)+1)}{2 d n}-\frac {(A-i B) (a+i a \tan (c+d x))^n \operatorname {Hypergeometric2F1}\left (1,n,n+1,\frac {1}{2} (i \tan (c+d x)+1)\right )}{2 d n}-\frac {(2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{2 d}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d} \]
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Rule 67
Rule 70
Rule 3562
Rule 3679
Rule 3680
Rule 3681
Rubi steps \begin{align*} \text {integral}& = -\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}+\frac {\int \cot ^2(c+d x) (a+i a \tan (c+d x))^n (a (2 B+i A n)-a A (2-n) \tan (c+d x)) \, dx}{2 a} \\ & = -\frac {(2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{2 d}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}+\frac {\int \cot (c+d x) (a+i a \tan (c+d x))^n \left (a^2 \left (2 i B n-A \left (2-n+n^2\right )\right )-a^2 (1-n) (2 B+i A n) \tan (c+d x)\right ) \, dx}{2 a^2} \\ & = -\frac {(2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{2 d}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}+(-i A-B) \int (a+i a \tan (c+d x))^n \, dx+\frac {\left (2 i B n-A \left (2-n+n^2\right )\right ) \int \cot (c+d x) (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^n \, dx}{2 a} \\ & = -\frac {(2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{2 d}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}-\frac {(a (A-i B)) \text {Subst}\left (\int \frac {(a+x)^{-1+n}}{a-x} \, dx,x,i a \tan (c+d x)\right )}{d}+\frac {\left (a \left (2 i B n-A \left (2-n+n^2\right )\right )\right ) \text {Subst}\left (\int \frac {(a+i a x)^{-1+n}}{x} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = -\frac {(2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{2 d}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}-\frac {(A-i B) \operatorname {Hypergeometric2F1}\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 {\left (2 i B n-A \left (2-n+n^2\right )\right ) \operatorname {Hypergeometric2F1}(1,n,1+n,1+i \tan (c+d x)) (a+i a \tan (c+d x))^n}{2 d n} \\ \end{align*}
Time = 3.17 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.19 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\frac {(a+i a \tan (c+d x))^n \left (-i (A-i B) n \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {1}{2} (1+i \tan (c+d x))\right ) (-i+\tan (c+d x))+2 (A+i B+A n+i B n+2 A n \operatorname {Hypergeometric2F1}(3,1+n,2+n,1+i \tan (c+d x))+2 A n \operatorname {Hypergeometric2F1}(1,1+n,2+n,1+i \tan (c+d x)) (1+i \tan (c+d x))+2 i A n \operatorname {Hypergeometric2F1}(3,1+n,2+n,1+i \tan (c+d x)) \tan (c+d x)-2 B n \operatorname {Hypergeometric2F1}(2,1+n,2+n,1+i \tan (c+d x)) (-i+\tan (c+d x)))\right )}{4 d n (1+n)} \]
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\[\int \left (\cot ^{3}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )d x\]
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\[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3} \,d x } \]
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\[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n} \left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{3}{\left (c + d x \right )}\, dx \]
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\[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3} \,d x } \]
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\[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^3\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]
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