Integrand size = 24, antiderivative size = 65 \[ \int \frac {e^{n \arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\frac {i e^{n \arctan (a x)}}{c n}-\frac {2 i e^{n \arctan (a x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {i n}{2},1-\frac {i n}{2},e^{2 i \arctan (a x)}\right )}{c n} \]
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Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.88, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5190, 98, 133} \[ \int \frac {e^{n \arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {2 i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1,-\frac {i n}{2},1-\frac {i n}{2},\frac {i a x+1}{1-i a x}\right )}{c n} \]
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Rule 98
Rule 133
Rule 5190
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}}}{x} \, dx}{c} \\ & = \frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}+\frac {\int \frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}}}{x} \, dx}{c} \\ & = \frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {2 i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1,-\frac {i n}{2},1-\frac {i n}{2},\frac {1+i a x}{1-i a x}\right )}{c n} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.85 \[ \int \frac {e^{n \arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left ((2+i n) (-i+a x)+2 (n-i a n x) \operatorname {Hypergeometric2F1}\left (1,1+\frac {i n}{2},2+\frac {i n}{2},\frac {i+a x}{i-a x}\right )\right )}{c n (-2 i+n) (-i+a x)} \]
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\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{x \left (a^{2} c \,x^{2}+c \right )}d x\]
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\[ \int \frac {e^{n \arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x} \,d x } \]
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\[ \int \frac {e^{n \arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{3} + x}\, dx}{c} \]
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\[ \int \frac {e^{n \arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x} \,d x } \]
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\[ \int \frac {e^{n \arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x} \,d x } \]
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Timed out. \[ \int \frac {e^{n \arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{x\,\left (c\,a^2\,x^2+c\right )} \,d x \]
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