Integrand size = 22, antiderivative size = 122 \[ \int \frac {e^{n \arctan (a x)} x}{c+a^2 c x^2} \, dx=\frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 c n}-\frac {i 2^{1-\frac {i n}{2}} (1-i a x)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2},\frac {i n}{2},1+\frac {i n}{2},\frac {1}{2} (1-i a x)\right )}{a^2 c n} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5190, 80, 71} \[ \int \frac {e^{n \arctan (a x)} x}{c+a^2 c x^2} \, dx=\frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 c n}-\frac {i 2^{1-\frac {i n}{2}} (1-i a x)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2},\frac {i n}{2},\frac {i n}{2}+1,\frac {1}{2} (1-i a x)\right )}{a^2 c n} \]
[In]
[Out]
Rule 71
Rule 80
Rule 5190
Rubi steps \begin{align*} \text {integral}& = \frac {\int x (1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} \, dx}{c} \\ & = \frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 c n}-\frac {i \int (1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \, dx}{a c} \\ & = \frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 c n}-\frac {i 2^{1-\frac {i n}{2}} (1-i a x)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2},\frac {i n}{2},1+\frac {i n}{2},\frac {1}{2} (1-i a x)\right )}{a^2 c n} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.89 \[ \int \frac {e^{n \arctan (a x)} x}{c+a^2 c x^2} \, dx=\frac {i (1-i a x)^{\frac {i n}{2}} (2+2 i a x)^{-\frac {i n}{2}} \left (2^{\frac {i n}{2}}-2 (1+i a x)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2},\frac {i n}{2},1+\frac {i n}{2},\frac {1}{2} (1-i a x)\right )\right )}{a^2 c n} \]
[In]
[Out]
\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x}{a^{2} c \,x^{2}+c}d x\]
[In]
[Out]
\[ \int \frac {e^{n \arctan (a x)} x}{c+a^2 c x^2} \, dx=\int { \frac {x e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c} \,d x } \]
[In]
[Out]
\[ \int \frac {e^{n \arctan (a x)} x}{c+a^2 c x^2} \, dx=\frac {\int \frac {x e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{2} + 1}\, dx}{c} \]
[In]
[Out]
\[ \int \frac {e^{n \arctan (a x)} x}{c+a^2 c x^2} \, dx=\int { \frac {x e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c} \,d x } \]
[In]
[Out]
\[ \int \frac {e^{n \arctan (a x)} x}{c+a^2 c x^2} \, dx=\int { \frac {x e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {e^{n \arctan (a x)} x}{c+a^2 c x^2} \, dx=\int \frac {x\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{c\,a^2\,x^2+c} \,d x \]
[In]
[Out]