\(\int e^{i n \arctan (a x)} x^2 (c+a^2 c x^2)^{-1-\frac {n^2}{2}} \, dx\) [374]

Optimal result

Integrand size = 35, antiderivative size = 60 \[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=\frac {i e^{i n \arctan (a x)} (1-i a n x) \left (c+a^2 c x^2\right )^{-\frac {n^2}{2}}}{a^3 c n \left (1-n^2\right )} \]

[Out]

I*exp(I*n*arctan(a*x))*(1-I*a*n*x)/a^3/c/n/(-n^2+1)/((a^2*c*x^2+c)^(1/2*n^2))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {5187} \[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=\frac {i (1-i a n x) e^{i n \arctan (a x)} \left (a^2 c x^2+c\right )^{-\frac {n^2}{2}}}{a^3 c n \left (1-n^2\right )} \]

[In]

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

[Out]

(I*E^(I*n*ArcTan[a*x])*(1 - I*a*n*x))/(a^3*c*n*(1 - n^2)*(c + a^2*c*x^2)^(n^2/2))

Rule 5187

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

Rubi steps \begin{align*} \text {integral}& = \frac {i e^{i n \arctan (a x)} (1-i a n x) \left (c+a^2 c x^2\right )^{-\frac {n^2}{2}}}{a^3 c n \left (1-n^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.92 \[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=-\frac {e^{i n \arctan (a x)} (i+a n x) \left (c+a^2 c x^2\right )^{-\frac {n^2}{2}}}{a^3 c n \left (-1+n^2\right )} \]

[In]

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

[Out]

-((E^(I*n*ArcTan[a*x])*(I + a*n*x))/(a^3*c*n*(-1 + n^2)*(c + a^2*c*x^2)^(n^2/2)))

Maple [A] (verified)

Time = 5.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.03

method	result	size
gosper	\(\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (n a x +i\right ) {\mathrm e}^{i n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}}}{a^{3} n \left (n^{2}-1\right )}\)	\(62\)
parallelrisch	\(-\frac {{\mathrm e}^{i n \arctan \left (a x \right )} x^{3} \left (a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}} a^{3} n +i {\mathrm e}^{i n \arctan \left (a x \right )} x^{2} \left (a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}} a^{2}+{\mathrm e}^{i n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}} x a n +i {\mathrm e}^{i n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}}}{a^{3} n \left (n^{2}-1\right )}\)	\(149\)

[In]

int(exp(I*n*arctan(a*x))*x^2*(a^2*c*x^2+c)^(-1-1/2*n^2),x,method=_RETURNVERBOSE)

[Out]

(I-a*x)*(I+a*x)*(n*a*x+I)*exp(I*n*arctan(a*x))*(a^2*c*x^2+c)^(-1-1/2*n^2)/a^3/n/(n^2-1)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.30 \[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=-\frac {{\left (a^{3} n x^{3} + i \, a^{2} x^{2} + a n x + i\right )} {\left (a^{2} c x^{2} + c\right )}^{-\frac {1}{2} \, n^{2} - 1}}{{\left (a^{3} n^{3} - a^{3} n\right )} \left (-\frac {a x + i}{a x - i}\right )^{\frac {1}{2} \, n}} \]

[In]

integrate(exp(I*n*arctan(a*x))*x^2*(a^2*c*x^2+c)^(-1-1/2*n^2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=\text {Timed out} \]

[In]

integrate(exp(I*n*atan(a*x))*x**2*(a**2*c*x**2+c)**(-1-1/2*n**2),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate(exp(I*n*arctan(a*x))*x^2*(a^2*c*x^2+c)^(-1-1/2*n^2),x, algorithm="maxima")

[Out]

integrate((a^2*c*x^2 + c)^(-1/2*n^2 - 1)*x^2*e^(I*n*arctan(a*x)), x)

Giac [F]

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

[In]

integrate(exp(I*n*arctan(a*x))*x^2*(a^2*c*x^2+c)^(-1-1/2*n^2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=\int \frac {x^2\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}}}{{\left (c\,a^2\,x^2+c\right )}^{\frac {n^2}{2}+1}} \,d x \]

[In]

int((x^2*exp(n*atan(a*x)*1i))/(c + a^2*c*x^2)^(n^2/2 + 1),x)

[Out]

int((x^2*exp(n*atan(a*x)*1i))/(c + a^2*c*x^2)^(n^2/2 + 1), x)