\(\int \frac {e^{n \arctan (a x)}}{x (c+a^2 c x^2)} \, dx\) [358]

Optimal result

Integrand size = 24, antiderivative size = 65 \[ \int \frac {e^{n \arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\frac {i e^{n \arctan (a x)}}{c n}-\frac {2 i e^{n \arctan (a x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {i n}{2},1-\frac {i n}{2},e^{2 i \arctan (a x)}\right )}{c n} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.85 \[ \int \frac {e^{n \arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left ((2+i n) (-i+a x)+2 (n-i a n x) \operatorname {Hypergeometric2F1}\left (1,1+\frac {i n}{2},2+\frac {i n}{2},\frac {i+a x}{i-a x}\right )\right )}{c n (-2 i+n) (-i+a x)} \] Input:

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

Output:

((1 - I*a*x)^((I/2)*n)*((2 + I*n)*(-I + a*x) + 2*(n - I*a*n*x)*Hypergeomet 
ric2F1[1, 1 + (I/2)*n, 2 + (I/2)*n, (I + a*x)/(I - a*x)]))/(c*n*(-2*I + n) 
*(1 + I*a*x)^((I/2)*n)*(-I + a*x))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.85, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5605, 107, 141}

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.

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

\(\Big \downarrow \) 5605

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

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 
 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

rule 141

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( 
n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f 
))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, 
p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !Su 
mSimplerQ[p, 1]) &&  !ILtQ[m, 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]

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

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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(e**(atan(a*x)*n)/(a**2*x**3 + x),x)/c