3.376 \(\int \frac{e^{4 i \tan ^{-1}(a x)} x^2}{(c+a^2 c x^2)^9} \, dx\)

Optimal. Leaf size=38 \[ -\frac{4 a x+i}{60 a^3 c^9 (1-i a x)^{10} (1+i a x)^6} \]

[Out]

-(I + 4*a*x)/(60*a^3*c^9*(1 - I*a*x)^10*(1 + I*a*x)^6)

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Rubi [A]  time = 0.0775599, 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} \[ -\frac{4 a x+i}{60 a^3 c^9 (1-i a x)^{10} (1+i a x)^6} \]

Antiderivative was successfully verified.

[In]

Int[(E^((4*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^9,x]

[Out]

-(I + 4*a*x)/(60*a^3*c^9*(1 - I*a*x)^10*(1 + I*a*x)^6)

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^{4 i \tan ^{-1}(a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx &=\frac{\int \frac{x^2}{(1-i a x)^{11} (1+i a x)^7} \, dx}{c^9}\\ &=-\frac{i+4 a x}{60 a^3 c^9 (1-i a x)^{10} (1+i a x)^6}\\ \end{align*}

Mathematica [A]  time = 0.19339, size = 36, normalized size = 0.95 \[ -\frac{4 a x+i}{60 a^3 c^9 (a x-i)^6 (a x+i)^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^((4*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^9,x]

[Out]

-(I + 4*a*x)/(60*a^3*c^9*(-I + a*x)^6*(I + a*x)^10)

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Maple [A]  time = 0.13, size = 35, normalized size = 0.9 \begin{align*} -{\frac{1}{{c}^{9} \left ( ax+i \right ) ^{10} \left ( ax-i \right ) ^{6}} \left ({\frac{{\frac{i}{60}}}{{a}^{3}}}+{\frac{x}{15\,{a}^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+I*a*x)^4/(a^2*x^2+1)^2*x^2/(a^2*c*x^2+c)^9,x)

[Out]

-1/c^9*(1/60*I/a^3+1/15*x/a^2)/(a*x+I)^10/(a*x-I)^6

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Maxima [B]  time = 1.58723, size = 209, normalized size = 5.5 \begin{align*} -\frac{5505024 \, a^{5} x^{5} - 20643840 i \, a^{4} x^{4} - 27525120 \, a^{3} x^{3} + 13762560 i \, a^{2} x^{2} + 1376256 i}{82575360 \,{\left (a^{23} c^{9} x^{20} + 10 \, a^{21} c^{9} x^{18} + 45 \, a^{19} c^{9} x^{16} + 120 \, a^{17} c^{9} x^{14} + 210 \, a^{15} c^{9} x^{12} + 252 \, a^{13} c^{9} x^{10} + 210 \, a^{11} c^{9} x^{8} + 120 \, a^{9} c^{9} x^{6} + 45 \, a^{7} c^{9} x^{4} + 10 \, a^{5} c^{9} x^{2} + a^{3} c^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^4/(a^2*x^2+1)^2*x^2/(a^2*c*x^2+c)^9,x, algorithm="maxima")

[Out]

-1/82575360*(5505024*a^5*x^5 - 20643840*I*a^4*x^4 - 27525120*a^3*x^3 + 13762560*I*a^2*x^2 + 1376256*I)/(a^23*c
^9*x^20 + 10*a^21*c^9*x^18 + 45*a^19*c^9*x^16 + 120*a^17*c^9*x^14 + 210*a^15*c^9*x^12 + 252*a^13*c^9*x^10 + 21
0*a^11*c^9*x^8 + 120*a^9*c^9*x^6 + 45*a^7*c^9*x^4 + 10*a^5*c^9*x^2 + a^3*c^9)

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Fricas [B]  time = 2.49146, size = 429, normalized size = 11.29 \begin{align*} -\frac{4 \, a x + i}{60 \, a^{19} c^{9} x^{16} + 240 i \, a^{18} c^{9} x^{15} + 1200 i \, a^{16} c^{9} x^{13} - 1200 \, a^{15} c^{9} x^{12} + 2160 i \, a^{14} c^{9} x^{11} - 3840 \, a^{13} c^{9} x^{10} + 1200 i \, a^{12} c^{9} x^{9} - 5400 \, a^{11} c^{9} x^{8} - 1200 i \, a^{10} c^{9} x^{7} - 3840 \, a^{9} c^{9} x^{6} - 2160 i \, a^{8} c^{9} x^{5} - 1200 \, a^{7} c^{9} x^{4} - 1200 i \, a^{6} c^{9} x^{3} - 240 i \, a^{4} c^{9} x + 60 \, a^{3} c^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^4/(a^2*x^2+1)^2*x^2/(a^2*c*x^2+c)^9,x, algorithm="fricas")

[Out]

-(4*a*x + I)/(60*a^19*c^9*x^16 + 240*I*a^18*c^9*x^15 + 1200*I*a^16*c^9*x^13 - 1200*a^15*c^9*x^12 + 2160*I*a^14
*c^9*x^11 - 3840*a^13*c^9*x^10 + 1200*I*a^12*c^9*x^9 - 5400*a^11*c^9*x^8 - 1200*I*a^10*c^9*x^7 - 3840*a^9*c^9*
x^6 - 2160*I*a^8*c^9*x^5 - 1200*a^7*c^9*x^4 - 1200*I*a^6*c^9*x^3 - 240*I*a^4*c^9*x + 60*a^3*c^9)

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Sympy [B]  time = 111.484, size = 202, normalized size = 5.32 \begin{align*} - \frac{a^{4} \left (4 a^{121} x + i a^{120}\right )}{60 a^{143} c^{9} x^{16} + 240 i a^{142} c^{9} x^{15} + 1200 i a^{140} c^{9} x^{13} - 1200 a^{139} c^{9} x^{12} + 2160 i a^{138} c^{9} x^{11} - 3840 a^{137} c^{9} x^{10} + 1200 i a^{136} c^{9} x^{9} - 5400 a^{135} c^{9} x^{8} - 1200 i a^{134} c^{9} x^{7} - 3840 a^{133} c^{9} x^{6} - 2160 i a^{132} c^{9} x^{5} - 1200 a^{131} c^{9} x^{4} - 1200 i a^{130} c^{9} x^{3} - 240 i a^{128} c^{9} x + 60 a^{127} c^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)**4/(a**2*x**2+1)**2*x**2/(a**2*c*x**2+c)**9,x)

[Out]

-a**4*(4*a**121*x + I*a**120)/(60*a**143*c**9*x**16 + 240*I*a**142*c**9*x**15 + 1200*I*a**140*c**9*x**13 - 120
0*a**139*c**9*x**12 + 2160*I*a**138*c**9*x**11 - 3840*a**137*c**9*x**10 + 1200*I*a**136*c**9*x**9 - 5400*a**13
5*c**9*x**8 - 1200*I*a**134*c**9*x**7 - 3840*a**133*c**9*x**6 - 2160*I*a**132*c**9*x**5 - 1200*a**131*c**9*x**
4 - 1200*I*a**130*c**9*x**3 - 240*I*a**128*c**9*x + 60*a**127*c**9)

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Giac [B]  time = 1.14383, size = 204, normalized size = 5.37 \begin{align*} -\frac{2145 \, a^{5} x^{5} - 12540 \, a^{4} i x^{4} - 30030 \, a^{3} x^{3} + 37080 \, a^{2} i x^{2} + 23841 \, a x - 6476 \, i}{983040 \,{\left (a x - i\right )}^{6} a^{3} c^{9}} + \frac{2145 \, a^{9} x^{9} + 21780 \, a^{8} i x^{8} - 99660 \, a^{7} x^{7} - 270480 \, a^{6} i x^{6} + 481446 \, a^{5} x^{5} + 584920 \, a^{4} i x^{4} - 486220 \, a^{3} x^{3} - 265680 \, a^{2} i x^{2} + 84065 \, a x + 9908 \, i}{983040 \,{\left (a x + i\right )}^{10} a^{3} c^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^4/(a^2*x^2+1)^2*x^2/(a^2*c*x^2+c)^9,x, algorithm="giac")

[Out]

-1/983040*(2145*a^5*x^5 - 12540*a^4*i*x^4 - 30030*a^3*x^3 + 37080*a^2*i*x^2 + 23841*a*x - 6476*i)/((a*x - i)^6
*a^3*c^9) + 1/983040*(2145*a^9*x^9 + 21780*a^8*i*x^8 - 99660*a^7*x^7 - 270480*a^6*i*x^6 + 481446*a^5*x^5 + 584
920*a^4*i*x^4 - 486220*a^3*x^3 - 265680*a^2*i*x^2 + 84065*a*x + 9908*i)/((a*x + i)^10*a^3*c^9)