\(\int \frac {e^{n \arctan (a x)}}{(c+a^2 c x^2)^{2/3}} \, dx\) [376]

Optimal result

Integrand size = 23, antiderivative size = 120 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=-\frac {3\ 2^{\frac {1}{3}-\frac {i n}{2}} (1-i a x)^{\frac {1}{6} (2+3 i n)} \left (1+a^2 x^2\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{6} (2+3 i n),\frac {1}{6} (4+3 i n),\frac {1}{6} (8+3 i n),\frac {1}{2} (1-i a x)\right )}{a (2 i-3 n) \left (c+a^2 c x^2\right )^{2/3}} \] Output:

-3*2^(1/3-1/2*I*n)*(1-I*a*x)^(1/3+1/2*I*n)*(a^2*x^2+1)^(2/3)*hypergeom([2/ 
3+1/2*I*n, 1/3+1/2*I*n],[4/3+1/2*I*n],1/2-1/2*I*a*x)/a/(2*I-3*n)/(a^2*c*x^ 
2+c)^(2/3)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx=\frac {3\ 2^{\frac {1}{3}-\frac {i n}{2}} (1-i a x)^{\frac {1}{3}+\frac {i n}{2}} \left (1+a^2 x^2\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3}+\frac {i n}{2},\frac {2}{3}+\frac {i n}{2},\frac {4}{3}+\frac {i n}{2},\frac {1}{2}-\frac {i a x}{2}\right )}{a (-2 i+3 n) \left (c+a^2 c x^2\right )^{2/3}} \] Input:

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

Output:

(3*2^(1/3 - (I/2)*n)*(1 - I*a*x)^(1/3 + (I/2)*n)*(1 + a^2*x^2)^(2/3)*Hyper 
geometric2F1[1/3 + (I/2)*n, 2/3 + (I/2)*n, 4/3 + (I/2)*n, 1/2 - (I/2)*a*x] 
)/(a*(-2*I + 3*n)*(c + a^2*c*x^2)^(2/3))
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5599, 5596, 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.

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

\(\Big \downarrow \) 5599

\(\displaystyle \frac {\left (a^2 x^2+1\right )^{2/3} \int \frac {e^{n \arctan (a x)}}{\left (a^2 x^2+1\right )^{2/3}}dx}{\left (a^2 c x^2+c\right )^{2/3}}\)

\(\Big \downarrow \) 5596

\(\displaystyle \frac {\left (a^2 x^2+1\right )^{2/3} \int (1-i a x)^{\frac {1}{6} (3 i n-4)} (i a x+1)^{\frac {1}{6} (-3 i n-4)}dx}{\left (a^2 c x^2+c\right )^{2/3}}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {3\ 2^{\frac {1}{3}-\frac {i n}{2}} \left (a^2 x^2+1\right )^{2/3} (1-i a x)^{\frac {1}{6} (2+3 i n)} \operatorname {Hypergeometric2F1}\left (\frac {1}{6} (3 i n+2),\frac {1}{6} (3 i n+4),\frac {1}{6} (3 i n+8),\frac {1}{2} (1-i a x)\right )}{a (-3 n+2 i) \left (a^2 c x^2+c\right )^{2/3}}\)

Input:

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

Output:

(-3*2^(1/3 - (I/2)*n)*(1 - I*a*x)^((2 + (3*I)*n)/6)*(1 + a^2*x^2)^(2/3)*Hy 
pergeometric2F1[(2 + (3*I)*n)/6, (4 + (3*I)*n)/6, (8 + (3*I)*n)/6, (1 - I* 
a*x)/2])/(a*(2*I - 3*n)*(c + a^2*c*x^2)^(2/3))
 

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 5596

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> 
Simp[c^p   Int[(1 - I*a*x)^(p + I*(n/2))*(1 + I*a*x)^(p - I*(n/2)), x], x] 
/; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0])
 

rule 5599

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> S 
imp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 + a^2*x^2)^FracPart[p])   Int[ 
(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && E 
qQ[d, a^2*c] &&  !(IntegerQ[p] || GtQ[c, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(e**(atan(a*x)*n)/(a**2*x**2 + 1)**(2/3),x)/c**(2/3)