Integrand size = 24, antiderivative size = 43 \[ \int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx=\frac {a \operatorname {Hypergeometric2F1}(1,1+n,2+n,-i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)} \]
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Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3618, 66} \[ \int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx=\frac {a (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}(1,n+1,n+2,-i \tan (e+f x))}{d f (n+1)} \]
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Rule 66
Rule 3618
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {\left (\frac {i d x}{a}\right )^n}{-a^2+a x} \, dx,x,-i a \tan (e+f x)\right )}{f} \\ & = \frac {a \operatorname {Hypergeometric2F1}(1,1+n,2+n,-i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.02 \[ \int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx=\frac {a \operatorname {Hypergeometric2F1}(1,1+n,2+n,-i \tan (e+f x)) \tan (e+f x) (d \tan (e+f x))^n}{f (1+n)} \]
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\[\int \left (d \tan \left (f x +e \right )\right )^{n} \left (a -i a \tan \left (f x +e \right )\right )d x\]
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\[ \int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx=\int { {\left (-i \, a \tan \left (f x + e\right ) + a\right )} \left (d \tan \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx=- i a \left (\int i \left (d \tan {\left (e + f x \right )}\right )^{n}\, dx + \int \left (d \tan {\left (e + f x \right )}\right )^{n} \tan {\left (e + f x \right )}\, dx\right ) \]
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\[ \int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx=\int { {\left (-i \, a \tan \left (f x + e\right ) + a\right )} \left (d \tan \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx=\int { {\left (-i \, a \tan \left (f x + e\right ) + a\right )} \left (d \tan \left (f x + e\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx=\int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,\left (a-a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right ) \,d x \]
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