Integrand size = 22, antiderivative size = 49 \[ \int e^{n \arctan (a x)} x^m \left (c+a^2 c x^2\right ) \, dx=\frac {c x^{1+m} \operatorname {AppellF1}\left (1+m,-1-\frac {i n}{2},-1+\frac {i n}{2},2+m,i a x,-i a x\right )}{1+m} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5190, 138} \[ \int e^{n \arctan (a x)} x^m \left (c+a^2 c x^2\right ) \, dx=\frac {c x^{m+1} \operatorname {AppellF1}\left (m+1,-\frac {i n}{2}-1,\frac {i n}{2}-1,m+2,i a x,-i a x\right )}{m+1} \]
[In]
[Out]
Rule 138
Rule 5190
Rubi steps \begin{align*} \text {integral}& = c \int x^m (1-i a x)^{1+\frac {i n}{2}} (1+i a x)^{1-\frac {i n}{2}} \, dx \\ & = \frac {c x^{1+m} \operatorname {AppellF1}\left (1+m,-1-\frac {i n}{2},-1+\frac {i n}{2},2+m,i a x,-i a x\right )}{1+m} \\ \end{align*}
\[ \int e^{n \arctan (a x)} x^m \left (c+a^2 c x^2\right ) \, dx=\int e^{n \arctan (a x)} x^m \left (c+a^2 c x^2\right ) \, dx \]
[In]
[Out]
\[\int {\mathrm e}^{n \arctan \left (a x \right )} x^{m} \left (a^{2} c \,x^{2}+c \right )d x\]
[In]
[Out]
\[ \int e^{n \arctan (a x)} x^m \left (c+a^2 c x^2\right ) \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x^{m} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \]
[In]
[Out]
\[ \int e^{n \arctan (a x)} x^m \left (c+a^2 c x^2\right ) \, dx=c \left (\int x^{m} e^{n \operatorname {atan}{\left (a x \right )}}\, dx + \int a^{2} x^{2} x^{m} e^{n \operatorname {atan}{\left (a x \right )}}\, dx\right ) \]
[In]
[Out]
\[ \int e^{n \arctan (a x)} x^m \left (c+a^2 c x^2\right ) \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x^{m} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \]
[In]
[Out]
\[ \int e^{n \arctan (a x)} x^m \left (c+a^2 c x^2\right ) \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x^{m} e^{\left (n \arctan \left (a x\right )\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{n \arctan (a x)} x^m \left (c+a^2 c x^2\right ) \, dx=\int x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,\left (c\,a^2\,x^2+c\right ) \,d x \]
[In]
[Out]