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

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

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 38, normalized size of antiderivative = 1.00, 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} \[ \frac {-2 a x+i}{6 a^3 c^3 (1-i a x) (1+i a x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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])

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

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Mathematica [A]  time = 0.04, size = 36, normalized size = 0.95 \[ \frac {2 a x-i}{6 a^3 c^3 (a x-i)^3 (a x+i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 49, normalized size = 1.29 \[ \frac {2 \, a x - i}{6 \, a^{7} c^{3} x^{4} - 12 i \, a^{6} c^{3} x^{3} - 12 i \, a^{4} c^{3} x - 6 \, a^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+I*a*x)^2*(a^2*x^2+1)/(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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giac [B]  time = 0.13, size = 93, normalized size = 2.45 \[ \frac {1}{32 \, a^{3} c^{3} {\left (i - \frac {2 \, i}{a i x + 1}\right )}} + \frac {\frac {3 \, a^{3} c^{6} i^{2}}{a i x + 1} + \frac {4 \, a^{3} c^{6} i^{4}}{{\left (a i x + 1\right )}^{3}} + \frac {6 \, a^{3} c^{6} i^{2}}{{\left (a i x + 1\right )}^{2}}}{48 \, a^{6} c^{9} i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32/(a^3*c^3*(i - 2*i/(a*i*x + 1))) + 1/48*(3*a^3*c^6*i^2/(a*i*x + 1) + 4*a^3*c^6*i^4/(a*i*x + 1)^3 + 6*a^3*c
^6*i^2/(a*i*x + 1)^2)/(a^6*c^9*i)

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maple [A]  time = 0.10, size = 62, normalized size = 1.63 \[ \frac {-\frac {1}{16 a^{3} \left (a x +i\right )}-\frac {i}{8 a^{3} \left (-a x +i\right )^{2}}-\frac {1}{12 a^{3} \left (-a x +i\right )^{3}}-\frac {1}{16 a^{3} \left (-a x +i\right )}}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(-1/16/a^3/(I+a*x)-1/8*I/a^3/(-a*x+I)^2-1/12/a^3/(-a*x+I)^3-1/16/a^3/(-a*x+I))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 0.59, size = 65, normalized size = 1.71 \[ \frac {2\,a^3\,x^3+a^2\,x^2\,3{}\mathrm {i}+1{}\mathrm {i}}{6\,a^9\,c^3\,x^6+18\,a^7\,c^3\,x^4+18\,a^5\,c^3\,x^2+6\,a^3\,c^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*x^2*3i + 2*a^3*x^3 + 1i)/(6*a^3*c^3 + 18*a^5*c^3*x^2 + 18*a^7*c^3*x^4 + 6*a^9*c^3*x^6)

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sympy [A]  time = 0.40, size = 53, normalized size = 1.39 \[ - \frac {2 a x - i}{- 6 a^{7} c^{3} x^{4} + 12 i a^{6} c^{3} x^{3} + 12 i a^{4} c^{3} x + 6 a^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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