\(\int \frac {e^{n \arctan (a x)} x^m}{(c+a^2 c x^2)^3} \, dx\) [381]

Optimal result

Integrand size = 24, antiderivative size = 51 \[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (1+m,3-\frac {i n}{2},3+\frac {i n}{2},2+m,i a x,-i a x\right )}{c^3 (1+m)} \] Output:

x^(1+m)*AppellF1(1+m,3+1/2*I*n,3-1/2*I*n,2+m,-I*a*x,I*a*x)/c^3/(1+m)
 

Mathematica [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx \] Input:

Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^3,x]
 

Output:

Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^3, x]
 

Rubi [A] (verified)

Time = 0.45 (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.

\(\displaystyle \int \frac {x^m e^{n \arctan (a x)}}{\left (a^2 c x^2+c\right )^3} \, dx\)

\(\Big \downarrow \) 5605

\(\displaystyle \frac {\int x^m (1-i a x)^{\frac {i n}{2}-3} (i a x+1)^{-\frac {i n}{2}-3}dx}{c^3}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {x^{m+1} \operatorname {AppellF1}\left (m+1,3-\frac {i n}{2},\frac {i n}{2}+3,m+2,i a x,-i a x\right )}{c^3 (m+1)}\)

Input:

Int[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^3,x]
 

Output:

(x^(1 + m)*AppellF1[1 + m, 3 - (I/2)*n, 3 + (I/2)*n, 2 + m, I*a*x, (-I)*a* 
x])/(c^3*(1 + m))
 

Defintions of rubi rules used

rule 150

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])
 

rule 5605

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])
 

Maple [F]

\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{m}}{\left (a^{2} c \,x^{2}+c \right )^{3}}d x\]

Input:

int(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^3,x)
 

Output:

int(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^3,x)
 

Fricas [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \] Input:

integrate(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^3,x, algorithm="fricas")
 

Output:

integral(x^m*e^(n*arctan(a*x))/(a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^ 
2 + c^3), x)
 

Sympy [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{m} e^{n \operatorname {atan}{\left (a x \right )}}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \] Input:

integrate(exp(n*atan(a*x))*x**m/(a**2*c*x**2+c)**3,x)
 

Output:

Integral(x**m*exp(n*atan(a*x))/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1) 
, x)/c**3
 

Maxima [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \] Input:

integrate(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^3,x, algorithm="maxima")
 

Output:

integrate(x^m*e^(n*arctan(a*x))/(a^2*c*x^2 + c)^3, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^3,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,[0,0,0,1,3]%%%} Error: Bad Argum 
ent Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \] Input:

int((x^m*exp(n*atan(a*x)))/(c + a^2*c*x^2)^3,x)
 

Output:

int((x^m*exp(n*atan(a*x)))/(c + a^2*c*x^2)^3, x)
 

Reduce [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {x^{m} e^{\mathit {atan} \left (a x \right ) n}-\left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n} x}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a^{6} m \,x^{4}+4 \left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n} x}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a^{6} x^{4}-2 \left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n} x}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a^{4} m \,x^{2}+8 \left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n} x}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a^{4} x^{2}-\left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n} x}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a^{2} m +4 \left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n} x}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a^{2}-\left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n}}{a^{6} x^{7}+3 a^{4} x^{5}+3 a^{2} x^{3}+x}d x \right ) a^{4} m \,x^{4}-2 \left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n}}{a^{6} x^{7}+3 a^{4} x^{5}+3 a^{2} x^{3}+x}d x \right ) a^{2} m \,x^{2}-\left (\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n}}{a^{6} x^{7}+3 a^{4} x^{5}+3 a^{2} x^{3}+x}d x \right ) m}{a \,c^{3} n \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )} \] Input:

int(exp(n*atan(a*x))*x^m/(a^2*c*x^2+c)^3,x)
 

Output:

(x**m*e**(atan(a*x)*n) - int((x**m*e**(atan(a*x)*n)*x)/(a**6*x**6 + 3*a**4 
*x**4 + 3*a**2*x**2 + 1),x)*a**6*m*x**4 + 4*int((x**m*e**(atan(a*x)*n)*x)/ 
(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1),x)*a**6*x**4 - 2*int((x**m*e** 
(atan(a*x)*n)*x)/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1),x)*a**4*m*x** 
2 + 8*int((x**m*e**(atan(a*x)*n)*x)/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 
 + 1),x)*a**4*x**2 - int((x**m*e**(atan(a*x)*n)*x)/(a**6*x**6 + 3*a**4*x** 
4 + 3*a**2*x**2 + 1),x)*a**2*m + 4*int((x**m*e**(atan(a*x)*n)*x)/(a**6*x** 
6 + 3*a**4*x**4 + 3*a**2*x**2 + 1),x)*a**2 - int((x**m*e**(atan(a*x)*n))/( 
a**6*x**7 + 3*a**4*x**5 + 3*a**2*x**3 + x),x)*a**4*m*x**4 - 2*int((x**m*e* 
*(atan(a*x)*n))/(a**6*x**7 + 3*a**4*x**5 + 3*a**2*x**3 + x),x)*a**2*m*x**2 
 - int((x**m*e**(atan(a*x)*n))/(a**6*x**7 + 3*a**4*x**5 + 3*a**2*x**3 + x) 
,x)*m)/(a*c**3*n*(a**4*x**4 + 2*a**2*x**2 + 1))