Integrand size = 24, antiderivative size = 90 \[ \int \frac {e^{n \arctan (a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx=\frac {i a e^{n \arctan (a x)} (i+n)}{c n}-\frac {e^{n \arctan (a x)}}{c x}-\frac {2 i a e^{n \arctan (a x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {i n}{2},1-\frac {i n}{2},-1+\frac {2 i}{i+a x}\right )}{c} \]
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Time = 0.10 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.84, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5190, 105, 160, 12, 133} \[ \int \frac {e^{n \arctan (a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx=-\frac {2 i a (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1,-\frac {i n}{2},1-\frac {i n}{2},\frac {i a x+1}{1-i a x}\right )}{c}-\frac {a (1-i n) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c x} \]
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Rule 12
Rule 105
Rule 133
Rule 160
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^2} \, dx}{c} \\ & = -\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c x}-\frac {\int \frac {(1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} \left (-a n+a^2 x\right )}{x} \, dx}{c} \\ & = -\frac {a (1-i n) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c x}+\frac {\int \frac {a^2 n^2 (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}}}{x} \, dx}{a c n} \\ & = -\frac {a (1-i n) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c x}+\frac {(a n) \int \frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}}}{x} \, dx}{c} \\ & = -\frac {a (1-i n) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c x}-\frac {2 i a (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} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.58 \[ \int \frac {e^{n \arctan (a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx=\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left ((-2 i+n) (1+i a x) (i a x+n (i+a x))+2 a n^2 x (1-i a x) \operatorname {Hypergeometric2F1}\left (1,1+\frac {i n}{2},2+\frac {i n}{2},\frac {i+a x}{i-a x}\right )\right )}{c n (-2 i+n) x (-i+a x)} \]
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\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{x^{2} \left (a^{2} c \,x^{2}+c \right )}d x\]
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\[ \int \frac {e^{n \arctan (a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{2}} \,d x } \]
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\[ \int \frac {e^{n \arctan (a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{4} + x^{2}}\, dx}{c} \]
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\[ \int \frac {e^{n \arctan (a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{2}} \,d x } \]
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\[ \int \frac {e^{n \arctan (a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {e^{n \arctan (a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{x^2\,\left (c\,a^2\,x^2+c\right )} \,d x \]
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