Integrand size = 26, antiderivative size = 209 \[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx=\frac {(1-n)^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{4 a^2 d f (1+n)}+\frac {(2-n) (d \tan (e+f x))^{1+n}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {i (2-n) n \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{2},\frac {4+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{2+n}}{4 a^2 d^2 f (2+n)}+\frac {(d \tan (e+f x))^{1+n}}{4 d f (a+i a \tan (e+f x))^2} \]
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Time = 0.44 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3640, 3677, 3619, 3557, 371} \[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx=\frac {i (2-n) n (d \tan (e+f x))^{n+2} \operatorname {Hypergeometric2F1}\left (1,\frac {n+2}{2},\frac {n+4}{2},-\tan ^2(e+f x)\right )}{4 a^2 d^2 f (n+2)}+\frac {(1-n)^2 (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(e+f x)\right )}{4 a^2 d f (n+1)}+\frac {(2-n) (d \tan (e+f x))^{n+1}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {(d \tan (e+f x))^{n+1}}{4 d f (a+i a \tan (e+f x))^2} \]
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Rule 371
Rule 3557
Rule 3619
Rule 3640
Rule 3677
Rubi steps \begin{align*} \text {integral}& = \frac {(d \tan (e+f x))^{1+n}}{4 d f (a+i a \tan (e+f x))^2}+\frac {\int \frac {(d \tan (e+f x))^n (a d (3-n)-i a d (1-n) \tan (e+f x))}{a+i a \tan (e+f x)} \, dx}{4 a^2 d} \\ & = \frac {(2-n) (d \tan (e+f x))^{1+n}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {(d \tan (e+f x))^{1+n}}{4 d f (a+i a \tan (e+f x))^2}+\frac {\int (d \tan (e+f x))^n \left (2 a^2 d^2 (1-n)^2+2 i a^2 d^2 (2-n) n \tan (e+f x)\right ) \, dx}{8 a^4 d^2} \\ & = \frac {(2-n) (d \tan (e+f x))^{1+n}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {(d \tan (e+f x))^{1+n}}{4 d f (a+i a \tan (e+f x))^2}+\frac {(1-n)^2 \int (d \tan (e+f x))^n \, dx}{4 a^2}+\frac {(i (2-n) n) \int (d \tan (e+f x))^{1+n} \, dx}{4 a^2 d} \\ & = \frac {(2-n) (d \tan (e+f x))^{1+n}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {(d \tan (e+f x))^{1+n}}{4 d f (a+i a \tan (e+f x))^2}+\frac {\left (d (1-n)^2\right ) \text {Subst}\left (\int \frac {x^n}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{4 a^2 f}+\frac {(i (2-n) n) \text {Subst}\left (\int \frac {x^{1+n}}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{4 a^2 f} \\ & = \frac {(1-n)^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{4 a^2 d f (1+n)}+\frac {(2-n) (d \tan (e+f x))^{1+n}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {i (2-n) n \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{2},\frac {4+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{2+n}}{4 a^2 d^2 f (2+n)}+\frac {(d \tan (e+f x))^{1+n}}{4 d f (a+i a \tan (e+f x))^2} \\ \end{align*}
Time = 3.10 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.06 \[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx=\frac {2 a^3 d (1+n) (2+n) (d \tan (e+f x))^{1+n}+(a+i a \tan (e+f x)) \left (-2 a^2 d (-2+n) (1+n) (2+n) (d \tan (e+f x))^{1+n}+2 a (a+i a \tan (e+f x)) \left (d (-1+n)^2 (2+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}+i (2-n) n (1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{2},\frac {4+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{2+n}\right )\right )}{8 a^3 d^2 f (1+n) (2+n) (a+i a \tan (e+f x))^2} \]
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\[\int \frac {\left (d \tan \left (f x +e \right )\right )^{n}}{\left (a +i a \tan \left (f x +e \right )\right )^{2}}d x\]
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\[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx=\int { \frac {\left (d \tan \left (f x + e\right )\right )^{n}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx=- \frac {\int \frac {\left (d \tan {\left (e + f x \right )}\right )^{n}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx}{a^{2}} \]
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Exception generated. \[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx=\int { \frac {\left (d \tan \left (f x + e\right )\right )^{n}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx=\int \frac {{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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