Integrand size = 24, antiderivative size = 51 \[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (1+m,2-\frac {i n}{2},2+\frac {i n}{2},2+m,i a x,-i a x\right )}{c^2 (1+m)} \] Output:
x^(1+m)*AppellF1(1+m,2+1/2*I*n,2-1/2*I*n,2+m,-I*a*x,I*a*x)/c^2/(1+m)
\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^2} \, dx=\int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^2} \, dx \] Input:
Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^2,x]
Output:
Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^2, x]
Time = 0.44 (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 = {5605, 150}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^m e^{n \arctan (a x)}}{\left (a^2 c x^2+c\right )^2} \, dx\) |
\(\Big \downarrow \) 5605 |
\(\displaystyle \frac {\int x^m (1-i a x)^{\frac {i n}{2}-2} (i a x+1)^{-\frac {i n}{2}-2}dx}{c^2}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {x^{m+1} \operatorname {AppellF1}\left (m+1,2-\frac {i n}{2},\frac {i n}{2}+2,m+2,i a x,-i a x\right )}{c^2 (m+1)}\) |
Input:
Int[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^2,x]
Output:
(x^(1 + m)*AppellF1[1 + m, 2 - (I/2)*n, 2 + (I/2)*n, 2 + m, I*a*x, (-I)*a* x])/(c^2*(1 + m))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[c^p Int[x^m*(1 - I*a*x)^(p + I*(n/2))*(1 + I*a*x)^(p - I* (n/2)), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (Integer Q[p] || GtQ[c, 0])
\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{m}}{\left (a^{2} c \,x^{2}+c \right )^{2}}d x\]
Input:
int(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^2,x)
Output:
int(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^2,x)
\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \] Input:
integrate(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^2,x, algorithm="fricas")
Output:
integral(x^m*e^(n*arctan(a*x))/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2), x)
\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x^{m} e^{n \operatorname {atan}{\left (a x \right )}}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \] Input:
integrate(exp(n*atan(a*x))*x**m/(a**2*c*x**2+c)**2,x)
Output:
Integral(x**m*exp(n*atan(a*x))/(a**4*x**4 + 2*a**2*x**2 + 1), x)/c**2
\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \] Input:
integrate(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^2,x, algorithm="maxima")
Output:
integrate(x^m*e^(n*arctan(a*x))/(a^2*c*x^2 + c)^2, x)
Exception generated. \[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[0,1,1,0,0]%%%} / %%%{1,[0,0,0,1,2]%%%} Error: Bad Argum ent Value
Timed out. \[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^2} \, dx=\int \frac {x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \] Input:
int((x^m*exp(n*atan(a*x)))/(c + a^2*c*x^2)^2,x)
Output:
int((x^m*exp(n*atan(a*x)))/(c + a^2*c*x^2)^2, x)
\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {x^{m} e^{\mathit {atan} \left (a x \right ) n}-\left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n} x}{a^{4} x^{4}+2 a^{2} x^{2}+1}d x \right ) a^{4} m \,x^{2}+2 \left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n} x}{a^{4} x^{4}+2 a^{2} x^{2}+1}d x \right ) a^{4} x^{2}-\left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n} x}{a^{4} x^{4}+2 a^{2} x^{2}+1}d x \right ) a^{2} m +2 \left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n} x}{a^{4} x^{4}+2 a^{2} x^{2}+1}d x \right ) a^{2}-\left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n}}{a^{4} x^{5}+2 a^{2} x^{3}+x}d x \right ) a^{2} m \,x^{2}-\left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n}}{a^{4} x^{5}+2 a^{2} x^{3}+x}d x \right ) m}{a \,c^{2} n \left (a^{2} x^{2}+1\right )} \] Input:
int(exp(n*atan(a*x))*x^m/(a^2*c*x^2+c)^2,x)
Output:
(x**m*e**(atan(a*x)*n) - int((x**m*e**(atan(a*x)*n)*x)/(a**4*x**4 + 2*a**2 *x**2 + 1),x)*a**4*m*x**2 + 2*int((x**m*e**(atan(a*x)*n)*x)/(a**4*x**4 + 2 *a**2*x**2 + 1),x)*a**4*x**2 - int((x**m*e**(atan(a*x)*n)*x)/(a**4*x**4 + 2*a**2*x**2 + 1),x)*a**2*m + 2*int((x**m*e**(atan(a*x)*n)*x)/(a**4*x**4 + 2*a**2*x**2 + 1),x)*a**2 - int((x**m*e**(atan(a*x)*n))/(a**4*x**5 + 2*a**2 *x**3 + x),x)*a**2*m*x**2 - int((x**m*e**(atan(a*x)*n))/(a**4*x**5 + 2*a** 2*x**3 + x),x)*m)/(a*c**2*n*(a**2*x**2 + 1))