3.374 \(\int e^{i n \tan ^{-1}(a x)} x^2 (c+a^2 c x^2)^{-1-\frac{n^2}{2}} \, dx\)

Optimal. Leaf size=60 \[ \frac{i (1-i a n x) \left (a^2 c x^2+c\right )^{-\frac{n^2}{2}} e^{i n \tan ^{-1}(a x)}}{a^3 c n \left (1-n^2\right )} \]

[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))

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Rubi [A]  time = 0.111918, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {5079} \[ \frac{i (1-i a n x) \left (a^2 c x^2+c\right )^{-\frac{n^2}{2}} e^{i n \tan ^{-1}(a x)}}{a^3 c n \left (1-n^2\right )} \]

Antiderivative was successfully verified.

[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 5079

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*} \int e^{i n \tan ^{-1}(a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac{n^2}{2}} \, dx &=\frac{i e^{i n \tan ^{-1}(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]  time = 0.024067, size = 55, normalized size = 0.92 \[ -\frac{(a n x+i) \left (a^2 c x^2+c\right )^{-\frac{n^2}{2}} e^{i n \tan ^{-1}(a x)}}{a^3 c n \left (n^2-1\right )} \]

Antiderivative was successfully verified.

[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)))

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Maple [A]  time = 0.06, size = 62, normalized size = 1. \begin{align*}{\frac{ \left ( -ax+i \right ) \left ( ax+i \right ) \left ( nax+i \right ){{\rm e}^{in\arctan \left ( ax \right ) }}}{n{a}^{3} \left ({n}^{2}-1 \right ) } \left ({a}^{2}c{x}^{2}+c \right ) ^{-1-{\frac{{n}^{2}}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 2.24294, size = 162, normalized size = 2.7 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [B]  time = 1.14228, size = 487, normalized size = 8.12 \begin{align*} -\frac{a^{3} n x^{3} e^{\left (\frac{1}{2} \, \pi i n - \frac{1}{2} \, n^{2} \log \left (a x + i\right ) - \frac{1}{2} \, n^{2} \log \left (a x - i\right ) - \frac{1}{2} \, n^{2} \log \left (c\right ) - \frac{1}{2} \, n \log \left (a x + i\right ) + \frac{1}{2} \, n \log \left (a x - i\right ) - \log \left (a x + i\right ) - \log \left (a x - i\right ) - \log \left (c\right )\right )} + a^{2} i x^{2} e^{\left (\frac{1}{2} \, \pi i n - \frac{1}{2} \, n^{2} \log \left (a x + i\right ) - \frac{1}{2} \, n^{2} \log \left (a x - i\right ) - \frac{1}{2} \, n^{2} \log \left (c\right ) - \frac{1}{2} \, n \log \left (a x + i\right ) + \frac{1}{2} \, n \log \left (a x - i\right ) - \log \left (a x + i\right ) - \log \left (a x - i\right ) - \log \left (c\right )\right )} + a n x e^{\left (\frac{1}{2} \, \pi i n - \frac{1}{2} \, n^{2} \log \left (a x + i\right ) - \frac{1}{2} \, n^{2} \log \left (a x - i\right ) - \frac{1}{2} \, n^{2} \log \left (c\right ) - \frac{1}{2} \, n \log \left (a x + i\right ) + \frac{1}{2} \, n \log \left (a x - i\right ) - \log \left (a x + i\right ) - \log \left (a x - i\right ) - \log \left (c\right )\right )} + i e^{\left (\frac{1}{2} \, \pi i n - \frac{1}{2} \, n^{2} \log \left (a x + i\right ) - \frac{1}{2} \, n^{2} \log \left (a x - i\right ) - \frac{1}{2} \, n^{2} \log \left (c\right ) - \frac{1}{2} \, n \log \left (a x + i\right ) + \frac{1}{2} \, n \log \left (a x - i\right ) - \log \left (a x + i\right ) - \log \left (a x - i\right ) - \log \left (c\right )\right )}}{a^{3} n^{3} - a^{3} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

-(a^3*n*x^3*e^(1/2*pi*i*n - 1/2*n^2*log(a*x + i) - 1/2*n^2*log(a*x - i) - 1/2*n^2*log(c) - 1/2*n*log(a*x + i)
+ 1/2*n*log(a*x - i) - log(a*x + i) - log(a*x - i) - log(c)) + a^2*i*x^2*e^(1/2*pi*i*n - 1/2*n^2*log(a*x + i)
- 1/2*n^2*log(a*x - i) - 1/2*n^2*log(c) - 1/2*n*log(a*x + i) + 1/2*n*log(a*x - i) - log(a*x + i) - log(a*x - i
) - log(c)) + a*n*x*e^(1/2*pi*i*n - 1/2*n^2*log(a*x + i) - 1/2*n^2*log(a*x - i) - 1/2*n^2*log(c) - 1/2*n*log(a
*x + i) + 1/2*n*log(a*x - i) - log(a*x + i) - log(a*x - i) - log(c)) + i*e^(1/2*pi*i*n - 1/2*n^2*log(a*x + i)
- 1/2*n^2*log(a*x - i) - 1/2*n^2*log(c) - 1/2*n*log(a*x + i) + 1/2*n*log(a*x - i) - log(a*x + i) - log(a*x - i
) - log(c)))/(a^3*n^3 - a^3*n)