3.377 \(\int \frac{e^{2 i \tan ^{-1}(a x)} x^2}{(c+a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=38 \[ -\frac{2 a x+i}{6 a^3 c^3 (1-i a x)^3 (1+i a x)} \]

[Out]

-(I + 2*a*x)/(6*a^3*c^3*(1 - I*a*x)^3*(1 + I*a*x))

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Rubi [A]  time = 0.0767827, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {5082, 81} \[ -\frac{2 a x+i}{6 a^3 c^3 (1-i a x)^3 (1+i a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(I + 2*a*x)/(6*a^3*c^3*(1 - I*a*x)^3*(1 + I*a*x))

Rule 5082

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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] && (Int
egerQ[p] || GtQ[c, 0])

Rule 81

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*
x)^(n + 1)*(e + f*x)^(p + 1)*(2*a*d*f*(n + p + 3) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x))/(d^2
*f^2*(n + p + 2)*(n + p + 3)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && NeQ[n + p + 3,
0] && EqQ[d*f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1)
+ c*f*(p + 1))*(a*d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]

Rubi steps

\begin{align*} \int \frac{e^{2 i \tan ^{-1}(a x)} x^2}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x^2}{(1-i a x)^4 (1+i a x)^2} \, dx}{c^3}\\ &=-\frac{i+2 a x}{6 a^3 c^3 (1-i a x)^3 (1+i a x)}\\ \end{align*}

Mathematica [A]  time = 0.0309703, size = 36, normalized size = 0.95 \[ \frac{2 a x+i}{6 a^3 c^3 (a x-i) (a x+i)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I + 2*a*x)/(6*a^3*c^3*(-I + a*x)*(I + a*x)^3)

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Maple [A]  time = 0.051, size = 34, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{3} \left ( ax+i \right ) ^{3} \left ( ax-i \right ) } \left ({\frac{x}{3\,{a}^{2}}}+{\frac{{\frac{i}{6}}}{{a}^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+I*a*x)^2/(a^2*x^2+1)*x^2/(a^2*c*x^2+c)^3,x)

[Out]

1/c^3*(1/3*x/a^2+1/6*I/a^3)/(a*x+I)^3/(a*x-I)

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Maxima [B]  time = 1.50358, size = 84, normalized size = 2.21 \begin{align*} \frac{16 \, a^{3} x^{3} - 24 i \, a^{2} x^{2} - 8 i}{48 \,{\left (a^{9} c^{3} x^{6} + 3 \, a^{7} c^{3} x^{4} + 3 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^2/(a^2*x^2+1)*x^2/(a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

1/48*(16*a^3*x^3 - 24*I*a^2*x^2 - 8*I)/(a^9*c^3*x^6 + 3*a^7*c^3*x^4 + 3*a^5*c^3*x^2 + a^3*c^3)

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Fricas [A]  time = 2.02252, size = 104, normalized size = 2.74 \begin{align*} \frac{2 \, a x + i}{6 \, a^{7} c^{3} x^{4} + 12 i \, a^{6} c^{3} x^{3} + 12 i \, a^{4} c^{3} x - 6 \, a^{3} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^2/(a^2*x^2+1)*x^2/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

(2*a*x + I)/(6*a^7*c^3*x^4 + 12*I*a^6*c^3*x^3 + 12*I*a^4*c^3*x - 6*a^3*c^3)

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Sympy [A]  time = 0.859227, size = 60, normalized size = 1.58 \begin{align*} \frac{a^{2} \left (2 a^{9} x + i a^{8}\right )}{6 a^{17} c^{3} x^{4} + 12 i a^{16} c^{3} x^{3} + 12 i a^{14} c^{3} x - 6 a^{13} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)**2/(a**2*x**2+1)*x**2/(a**2*c*x**2+c)**3,x)

[Out]

a**2*(2*a**9*x + I*a**8)/(6*a**17*c**3*x**4 + 12*I*a**16*c**3*x**3 + 12*I*a**14*c**3*x - 6*a**13*c**3)

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Giac [A]  time = 1.15881, size = 65, normalized size = 1.71 \begin{align*} -\frac{1}{16 \,{\left (a x - i\right )} a^{3} c^{3}} + \frac{3 \, a^{2} x^{2} + 12 \, a i x - 5}{48 \,{\left (a x + i\right )}^{3} a^{3} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^2/(a^2*x^2+1)*x^2/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

-1/16/((a*x - i)*a^3*c^3) + 1/48*(3*a^2*x^2 + 12*a*i*x - 5)/((a*x + i)^3*a^3*c^3)