Integrand size = 24, antiderivative size = 51 \[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (1+m,3-\frac {i n}{2},3+\frac {i n}{2},2+m,i a x,-i a x\right )}{c^3 (1+m)} \]
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Time = 0.06 (sec) , antiderivative size = 51, 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 = {5190, 138} \[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {x^{m+1} \operatorname {AppellF1}\left (m+1,3-\frac {i n}{2},\frac {i n}{2}+3,m+2,i a x,-i a x\right )}{c^3 (m+1)} \]
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Rule 138
Rule 5190
Rubi steps \begin{align*} \text {integral}& = \frac {\int x^m (1-i a x)^{-3+\frac {i n}{2}} (1+i a x)^{-3-\frac {i n}{2}} \, dx}{c^3} \\ & = \frac {x^{1+m} \operatorname {AppellF1}\left (1+m,3-\frac {i n}{2},3+\frac {i n}{2},2+m,i a x,-i a x\right )}{c^3 (1+m)} \\ \end{align*}
\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx \]
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\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{m}}{\left (a^{2} c \,x^{2}+c \right )^{3}}d x\]
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\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
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\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{m} e^{n \operatorname {atan}{\left (a x \right )}}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
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\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
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\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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