\(\int \frac {e^{-2 i \arctan (a x)} x^2}{(c+a^2 c x^2)^3} \, dx\) [378]

Optimal result

Integrand size = 26, antiderivative size = 38 \[ \int \frac {e^{-2 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {i-2 a x}{6 a^3 c^3 (1-i a x) (1+i a x)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5190, 82} \[ \int \frac {e^{-2 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-2 a x+i}{6 a^3 c^3 (1-i a x) (1+i a x)^3} \]

[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 82

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 5190

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-2 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-i+2 a x}{6 a^3 c^3 (-i+a x)^3 (i+a x)} \]

[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))

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89

method	result	size
risch	\(\frac {\frac {x}{3 a^{2}}-\frac {i}{6 a^{3}}}{c^{3} \left (a x -i\right )^{3} \left (a x +i\right )}\)	\(34\)
parallelrisch	\(-\frac {i x^{4} a +2 x^{3}}{6 c^{3} \left (-a x +i\right )^{2} \left (a^{2} x^{2}+1\right )}\)	\(39\)
norman	\(\frac {\frac {x^{3}}{3 c}-\frac {i a \,x^{4}}{2 c}-\frac {i a^{3} x^{6}}{6 c}}{\left (a^{2} x^{2}+1\right )^{3} c^{2}}\)	\(47\)
gosper	\(-\frac {\left (-2 a x +i\right ) \left (a x +i\right ) \left (-a x +i\right )}{6 \left (a^{2} x^{2}+1\right )^{2} c^{3} \left (i a x +1\right )^{2} a^{3}}\)	\(49\)
default	\(\frac {-\frac {i}{8 a^{3} \left (-a x +i\right )^{2}}-\frac {1}{12 a^{3} \left (-a x +i\right )^{3}}-\frac {1}{16 a^{3} \left (-a x +i\right )}-\frac {1}{16 a^{3} \left (a x +i\right )}}{c^{3}}\)	\(62\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int \frac {e^{-2 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {2 \, a x - i}{6 \, {\left (a^{7} c^{3} x^{4} - 2 i \, a^{6} c^{3} x^{3} - 2 i \, a^{4} c^{3} x - a^{3} c^{3}\right )}} \]

[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]

1/6*(2*a*x - I)/(a^7*c^3*x^4 - 2*I*a^6*c^3*x^3 - 2*I*a^4*c^3*x - a^3*c^3)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.39 \[ \int \frac {e^{-2 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^3} \, dx=- \frac {- 2 a x + i}{6 a^{7} c^{3} x^{4} - 12 i a^{6} c^{3} x^{3} - 12 i a^{4} c^{3} x - 6 a^{3} c^{3}} \]

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {e^{-2 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]

[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: expt: undefined: 0 to a negative exponent.

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (30) = 60\).

Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.11 \[ \int \frac {e^{-2 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {1}{32 \, a^{3} c^{3} {\left (\frac {2 i}{i \, a x + 1} - i\right )}} - \frac {-\frac {3 i \, a^{3} c^{6}}{i \, a x + 1} - \frac {6 i \, a^{3} c^{6}}{{\left (i \, a x + 1\right )}^{2}} + \frac {4 i \, a^{3} c^{6}}{{\left (i \, a x + 1\right )}^{3}}}{48 \, a^{6} c^{9}} \]

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

Mupad [B] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.71 \[ \int \frac {e^{-2 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {2\,a^3\,x^3+a^2\,x^2\,3{}\mathrm {i}+1{}\mathrm {i}}{6\,a^9\,c^3\,x^6+18\,a^7\,c^3\,x^4+18\,a^5\,c^3\,x^2+6\,a^3\,c^3} \]

[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)