Integrand size = 28, antiderivative size = 54 \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x)) \, dx=\frac {a \operatorname {Hypergeometric2F1}(1,1+n p,2+n p,i \tan (e+f x)) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)} \]
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Time = 0.13 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {12, 1970, 66} \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x)) \, dx=\frac {a \tan (e+f x) \operatorname {Hypergeometric2F1}(1,n p+1,n p+2,i \tan (e+f x)) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+1)} \]
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Rule 12
Rule 66
Rule 1970
Rubi steps \begin{align*} \text {integral}& = \frac {i \text {Subst}\left (\int \frac {a \left (c (d x)^p\right )^n}{i+x} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(i a) \text {Subst}\left (\int \frac {\left (c (d x)^p\right )^n}{i+x} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\left (i a \tan ^{-n p}(e+f x) \left (c (d \tan (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \frac {x^{n p}}{i+x} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {a \operatorname {Hypergeometric2F1}(1,1+n p,2+n p,i \tan (e+f x)) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x)) \, dx=\frac {a \operatorname {Hypergeometric2F1}(1,1+n p,2+n p,i \tan (e+f x)) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)} \]
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\[\int \left (c \left (d \tan \left (f x +e \right )\right )^{p}\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )d x\]
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\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x)) \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )} \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n} \,d x } \]
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\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x)) \, dx=i a \left (\int \left (- i \left (c \left (d \tan {\left (e + f x \right )}\right )^{p}\right )^{n}\right )\, dx + \int \left (c \left (d \tan {\left (e + f x \right )}\right )^{p}\right )^{n} \tan {\left (e + f x \right )}\, dx\right ) \]
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\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x)) \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )} \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n} \,d x } \]
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\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x)) \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )} \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n} \,d x } \]
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Timed out. \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x)) \, dx=\int {\left (c\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^p\right )}^n\,\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right ) \,d x \]
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