\(\int \frac {e^{n \arctan (a x)} x}{c+a^2 c x^2} \, dx\) [356]

Optimal result

Integrand size = 22, antiderivative size = 122 \[ \int \frac {e^{n \arctan (a x)} x}{c+a^2 c x^2} \, dx=\frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 c n}-\frac {i 2^{1-\frac {i n}{2}} (1-i a x)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2},\frac {i n}{2},1+\frac {i n}{2},\frac {1}{2} (1-i a x)\right )}{a^2 c n} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.89 \[ \int \frac {e^{n \arctan (a x)} x}{c+a^2 c x^2} \, dx=\frac {i (1-i a x)^{\frac {i n}{2}} (2+2 i a x)^{-\frac {i n}{2}} \left (2^{\frac {i n}{2}}-2 (1+i a x)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2},\frac {i n}{2},1+\frac {i n}{2},\frac {1}{2} (1-i a x)\right )\right )}{a^2 c n} \] Input:

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

Output:

(I*(1 - I*a*x)^((I/2)*n)*(2^((I/2)*n) - 2*(1 + I*a*x)^((I/2)*n)*Hypergeome 
tric2F1[(I/2)*n, (I/2)*n, 1 + (I/2)*n, (1 - I*a*x)/2]))/(a^2*c*n*(2 + (2*I 
)*a*x)^((I/2)*n))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5605, 88, 79}

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.

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

\(\Big \downarrow \) 5605

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {\frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 n}-\frac {i 2^{1-\frac {i n}{2}} (1-i a x)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2},\frac {i n}{2},\frac {i n}{2}+1,\frac {1}{2} (1-i a x)\right )}{a^2 n}}{c}\)

Input:

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

Output:

((I*(1 - I*a*x)^((I/2)*n))/(a^2*n*(1 + I*a*x)^((I/2)*n)) - (I*2^(1 - (I/2) 
*n)*(1 - I*a*x)^((I/2)*n)*Hypergeometric2F1[(I/2)*n, (I/2)*n, 1 + (I/2)*n, 
 (1 - I*a*x)/2])/(a^2*n))/c
 

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 1]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(x*e^(n*arctan(a*x))/(a^2*c*x^2 + c), x)
 

Sympy [F]

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

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

Output:

Integral(x*exp(n*atan(a*x))/(a**2*x**2 + 1), x)/c
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {e^{n \arctan (a x)} x}{c+a^2 c x^2} \, dx=\frac {\int \frac {e^{\mathit {atan} \left (a x \right ) n} x}{a^{2} x^{2}+1}d x}{c} \] Input:

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

Output:

int((e**(atan(a*x)*n)*x)/(a**2*x**2 + 1),x)/c