3.375 \(\int \frac {e^{6 i \tan ^{-1}(a x)} x^2}{(c+a^2 c x^2)^{19}} \, dx\)

Optimal. Leaf size=38 \[ -\frac {6 a x+i}{210 a^3 c^{19} (1-i a x)^{21} (1+i a x)^{15}} \]

[Out]

1/210*(-I-6*a*x)/a^3/c^19/(1-I*a*x)^21/(1+I*a*x)^15

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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} \[ -\frac {6 a x+i}{210 a^3 c^{19} (1-i a x)^{21} (1+i a x)^{15}} \]

Antiderivative was successfully verified.

[In]

Int[(E^((6*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^19,x]

[Out]

-(I + 6*a*x)/(210*a^3*c^19*(1 - I*a*x)^21*(1 + I*a*x)^15)

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^{6 i \tan ^{-1}(a x)} x^2}{\left (c+a^2 c x^2\right )^{19}} \, dx &=\frac {\int \frac {x^2}{(1-i a x)^{22} (1+i a x)^{16}} \, dx}{c^{19}}\\ &=-\frac {i+6 a x}{210 a^3 c^{19} (1-i a x)^{21} (1+i a x)^{15}}\\ \end {align*}

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Mathematica [A]  time = 1.23, size = 36, normalized size = 0.95 \[ \frac {6 a x+i}{210 a^3 c^{19} (a x-i)^{15} (a x+i)^{21}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^((6*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^19,x]

[Out]

(I + 6*a*x)/(210*a^3*c^19*(-I + a*x)^15*(I + a*x)^21)

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fricas [B]  time = 1.34, size = 379, normalized size = 9.97 \[ \frac {6 \, a x + i}{210 \, a^{39} c^{19} x^{36} + 1260 i \, a^{38} c^{19} x^{35} + 14700 i \, a^{36} c^{19} x^{33} - 22050 \, a^{35} c^{19} x^{32} + 70560 i \, a^{34} c^{19} x^{31} - 188160 \, a^{33} c^{19} x^{30} + 151200 i \, a^{32} c^{19} x^{29} - 819000 \, a^{31} c^{19} x^{28} - 58800 i \, a^{30} c^{19} x^{27} - 2257920 \, a^{29} c^{19} x^{26} - 1375920 i \, a^{28} c^{19} x^{25} - 4204200 \, a^{27} c^{19} x^{24} - 4586400 i \, a^{26} c^{19} x^{23} - 5241600 \, a^{25} c^{19} x^{22} - 9129120 i \, a^{24} c^{19} x^{21} - 3783780 \, a^{23} c^{19} x^{20} - 12612600 i \, a^{22} c^{19} x^{19} - 12612600 i \, a^{20} c^{19} x^{17} + 3783780 \, a^{19} c^{19} x^{16} - 9129120 i \, a^{18} c^{19} x^{15} + 5241600 \, a^{17} c^{19} x^{14} - 4586400 i \, a^{16} c^{19} x^{13} + 4204200 \, a^{15} c^{19} x^{12} - 1375920 i \, a^{14} c^{19} x^{11} + 2257920 \, a^{13} c^{19} x^{10} - 58800 i \, a^{12} c^{19} x^{9} + 819000 \, a^{11} c^{19} x^{8} + 151200 i \, a^{10} c^{19} x^{7} + 188160 \, a^{9} c^{19} x^{6} + 70560 i \, a^{8} c^{19} x^{5} + 22050 \, a^{7} c^{19} x^{4} + 14700 i \, a^{6} c^{19} x^{3} + 1260 i \, a^{4} c^{19} x - 210 \, a^{3} c^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^6/(a^2*x^2+1)^3*x^2/(a^2*c*x^2+c)^19,x, algorithm="fricas")

[Out]

(6*a*x + I)/(210*a^39*c^19*x^36 + 1260*I*a^38*c^19*x^35 + 14700*I*a^36*c^19*x^33 - 22050*a^35*c^19*x^32 + 7056
0*I*a^34*c^19*x^31 - 188160*a^33*c^19*x^30 + 151200*I*a^32*c^19*x^29 - 819000*a^31*c^19*x^28 - 58800*I*a^30*c^
19*x^27 - 2257920*a^29*c^19*x^26 - 1375920*I*a^28*c^19*x^25 - 4204200*a^27*c^19*x^24 - 4586400*I*a^26*c^19*x^2
3 - 5241600*a^25*c^19*x^22 - 9129120*I*a^24*c^19*x^21 - 3783780*a^23*c^19*x^20 - 12612600*I*a^22*c^19*x^19 - 1
2612600*I*a^20*c^19*x^17 + 3783780*a^19*c^19*x^16 - 9129120*I*a^18*c^19*x^15 + 5241600*a^17*c^19*x^14 - 458640
0*I*a^16*c^19*x^13 + 4204200*a^15*c^19*x^12 - 1375920*I*a^14*c^19*x^11 + 2257920*a^13*c^19*x^10 - 58800*I*a^12
*c^19*x^9 + 819000*a^11*c^19*x^8 + 151200*I*a^10*c^19*x^7 + 188160*a^9*c^19*x^6 + 70560*I*a^8*c^19*x^5 + 22050
*a^7*c^19*x^4 + 14700*I*a^6*c^19*x^3 + 1260*I*a^4*c^19*x - 210*a^3*c^19)

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giac [B]  time = 0.14, size = 318, normalized size = 8.37 \[ -\frac {358229025 \, a^{14} x^{14} - 5340869100 \, a^{13} i x^{13} - 37114698075 \, a^{12} x^{12} + 159416118225 \, a^{11} i x^{11} + 473088806190 \, a^{10} x^{10} - 1026819468675 \, a^{9} i x^{9} - 1682288472150 \, a^{8} x^{8} + 2115551402250 \, a^{7} i x^{7} + 2054435046125 \, a^{6} x^{6} - 1535397250002 \, a^{5} i x^{5} - 870854759775 \, a^{4} x^{4} + 364307533205 \, a^{3} i x^{3} + 106553746740 \, a^{2} x^{2} - 19571887695 \, a i x - 1710785408}{901943132160 \, {\left (a x - i\right )}^{15} a^{3} c^{19}} + \frac {358229025 \, a^{20} x^{20} + 7555375800 \, a^{19} i x^{19} - 75901131600 \, a^{18} x^{18} - 483051354975 \, a^{17} i x^{17} + 2184946607340 \, a^{16} x^{16} + 7469205450840 \, a^{15} i x^{15} - 20031221295000 \, a^{14} x^{14} - 43177004037300 \, a^{13} i x^{13} + 76013078916950 \, a^{12} x^{12} + 110448380006328 \, a^{11} i x^{11} - 133277726128008 \, a^{10} x^{10} - 133908931763530 \, a^{9} i x^{9} + 111933156213900 \, a^{8} x^{8} + 77492989590120 \, a^{7} i x^{7} - 44041557267624 \, a^{6} x^{6} - 20244576347604 \, a^{5} i x^{5} + 7349182966545 \, a^{4} x^{4} + 2026362494800 \, a^{3} i x^{3} - 396520754280 \, a^{2} x^{2} - 48177926223 \, a i x + 2584181888}{901943132160 \, {\left (a x + i\right )}^{21} a^{3} c^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^6/(a^2*x^2+1)^3*x^2/(a^2*c*x^2+c)^19,x, algorithm="giac")

[Out]

-1/901943132160*(358229025*a^14*x^14 - 5340869100*a^13*i*x^13 - 37114698075*a^12*x^12 + 159416118225*a^11*i*x^
11 + 473088806190*a^10*x^10 - 1026819468675*a^9*i*x^9 - 1682288472150*a^8*x^8 + 2115551402250*a^7*i*x^7 + 2054
435046125*a^6*x^6 - 1535397250002*a^5*i*x^5 - 870854759775*a^4*x^4 + 364307533205*a^3*i*x^3 + 106553746740*a^2
*x^2 - 19571887695*a*i*x - 1710785408)/((a*x - i)^15*a^3*c^19) + 1/901943132160*(358229025*a^20*x^20 + 7555375
800*a^19*i*x^19 - 75901131600*a^18*x^18 - 483051354975*a^17*i*x^17 + 2184946607340*a^16*x^16 + 7469205450840*a
^15*i*x^15 - 20031221295000*a^14*x^14 - 43177004037300*a^13*i*x^13 + 76013078916950*a^12*x^12 + 11044838000632
8*a^11*i*x^11 - 133277726128008*a^10*x^10 - 133908931763530*a^9*i*x^9 + 111933156213900*a^8*x^8 + 774929895901
20*a^7*i*x^7 - 44041557267624*a^6*x^6 - 20244576347604*a^5*i*x^5 + 7349182966545*a^4*x^4 + 2026362494800*a^3*i
*x^3 - 396520754280*a^2*x^2 - 48177926223*a*i*x + 2584181888)/((a*x + i)^21*a^3*c^19)

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maple [A]  time = 0.37, size = 34, normalized size = 0.89 \[ \frac {\frac {x}{35 a^{2}}+\frac {i}{210 a^{3}}}{c^{19} \left (a x +i\right )^{21} \left (a x -i\right )^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+I*a*x)^6/(a^2*x^2+1)^3*x^2/(a^2*c*x^2+c)^19,x)

[Out]

1/c^19*(1/35*x/a^2+1/210*I/a^3)/(I+a*x)^21/(a*x-I)^15

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maxima [B]  time = 0.54, size = 292, normalized size = 7.68 \[ \frac {20775821962641408 \, a^{7} x^{7} - 121192294782074880 i \, a^{6} x^{6} - 290861507476979712 \, a^{5} x^{5} + 363576884346224640 i \, a^{4} x^{4} + 242384589564149760 \, a^{3} x^{3} - 72715376869244928 i \, a^{2} x^{2} - 3462636993773568 i}{727153768692449280 \, {\left (a^{45} c^{19} x^{42} + 21 \, a^{43} c^{19} x^{40} + 210 \, a^{41} c^{19} x^{38} + 1330 \, a^{39} c^{19} x^{36} + 5985 \, a^{37} c^{19} x^{34} + 20349 \, a^{35} c^{19} x^{32} + 54264 \, a^{33} c^{19} x^{30} + 116280 \, a^{31} c^{19} x^{28} + 203490 \, a^{29} c^{19} x^{26} + 293930 \, a^{27} c^{19} x^{24} + 352716 \, a^{25} c^{19} x^{22} + 352716 \, a^{23} c^{19} x^{20} + 293930 \, a^{21} c^{19} x^{18} + 203490 \, a^{19} c^{19} x^{16} + 116280 \, a^{17} c^{19} x^{14} + 54264 \, a^{15} c^{19} x^{12} + 20349 \, a^{13} c^{19} x^{10} + 5985 \, a^{11} c^{19} x^{8} + 1330 \, a^{9} c^{19} x^{6} + 210 \, a^{7} c^{19} x^{4} + 21 \, a^{5} c^{19} x^{2} + a^{3} c^{19}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^6/(a^2*x^2+1)^3*x^2/(a^2*c*x^2+c)^19,x, algorithm="maxima")

[Out]

1/727153768692449280*(20775821962641408*a^7*x^7 - 121192294782074880*I*a^6*x^6 - 290861507476979712*a^5*x^5 +
363576884346224640*I*a^4*x^4 + 242384589564149760*a^3*x^3 - 72715376869244928*I*a^2*x^2 - 3462636993773568*I)/
(a^45*c^19*x^42 + 21*a^43*c^19*x^40 + 210*a^41*c^19*x^38 + 1330*a^39*c^19*x^36 + 5985*a^37*c^19*x^34 + 20349*a
^35*c^19*x^32 + 54264*a^33*c^19*x^30 + 116280*a^31*c^19*x^28 + 203490*a^29*c^19*x^26 + 293930*a^27*c^19*x^24 +
352716*a^25*c^19*x^22 + 352716*a^23*c^19*x^20 + 293930*a^21*c^19*x^18 + 203490*a^19*c^19*x^16 + 116280*a^17*c
^19*x^14 + 54264*a^15*c^19*x^12 + 20349*a^13*c^19*x^10 + 5985*a^11*c^19*x^8 + 1330*a^9*c^19*x^6 + 210*a^7*c^19
*x^4 + 21*a^5*c^19*x^2 + a^3*c^19)

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.03 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(a*x*1i + 1)^6)/((c + a^2*c*x^2)^19*(a^2*x^2 + 1)^3),x)

[Out]

\text{Hanged}

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sympy [B]  time = 6.50, size = 437, normalized size = 11.50 \[ - \frac {6 a x + i}{- 210 a^{39} c^{19} x^{36} - 1260 i a^{38} c^{19} x^{35} - 14700 i a^{36} c^{19} x^{33} + 22050 a^{35} c^{19} x^{32} - 70560 i a^{34} c^{19} x^{31} + 188160 a^{33} c^{19} x^{30} - 151200 i a^{32} c^{19} x^{29} + 819000 a^{31} c^{19} x^{28} + 58800 i a^{30} c^{19} x^{27} + 2257920 a^{29} c^{19} x^{26} + 1375920 i a^{28} c^{19} x^{25} + 4204200 a^{27} c^{19} x^{24} + 4586400 i a^{26} c^{19} x^{23} + 5241600 a^{25} c^{19} x^{22} + 9129120 i a^{24} c^{19} x^{21} + 3783780 a^{23} c^{19} x^{20} + 12612600 i a^{22} c^{19} x^{19} + 12612600 i a^{20} c^{19} x^{17} - 3783780 a^{19} c^{19} x^{16} + 9129120 i a^{18} c^{19} x^{15} - 5241600 a^{17} c^{19} x^{14} + 4586400 i a^{16} c^{19} x^{13} - 4204200 a^{15} c^{19} x^{12} + 1375920 i a^{14} c^{19} x^{11} - 2257920 a^{13} c^{19} x^{10} + 58800 i a^{12} c^{19} x^{9} - 819000 a^{11} c^{19} x^{8} - 151200 i a^{10} c^{19} x^{7} - 188160 a^{9} c^{19} x^{6} - 70560 i a^{8} c^{19} x^{5} - 22050 a^{7} c^{19} x^{4} - 14700 i a^{6} c^{19} x^{3} - 1260 i a^{4} c^{19} x + 210 a^{3} c^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)**6/(a**2*x**2+1)**3*x**2/(a**2*c*x**2+c)**19,x)

[Out]

-(6*a*x + I)/(-210*a**39*c**19*x**36 - 1260*I*a**38*c**19*x**35 - 14700*I*a**36*c**19*x**33 + 22050*a**35*c**1
9*x**32 - 70560*I*a**34*c**19*x**31 + 188160*a**33*c**19*x**30 - 151200*I*a**32*c**19*x**29 + 819000*a**31*c**
19*x**28 + 58800*I*a**30*c**19*x**27 + 2257920*a**29*c**19*x**26 + 1375920*I*a**28*c**19*x**25 + 4204200*a**27
*c**19*x**24 + 4586400*I*a**26*c**19*x**23 + 5241600*a**25*c**19*x**22 + 9129120*I*a**24*c**19*x**21 + 3783780
*a**23*c**19*x**20 + 12612600*I*a**22*c**19*x**19 + 12612600*I*a**20*c**19*x**17 - 3783780*a**19*c**19*x**16 +
9129120*I*a**18*c**19*x**15 - 5241600*a**17*c**19*x**14 + 4586400*I*a**16*c**19*x**13 - 4204200*a**15*c**19*x
**12 + 1375920*I*a**14*c**19*x**11 - 2257920*a**13*c**19*x**10 + 58800*I*a**12*c**19*x**9 - 819000*a**11*c**19
*x**8 - 151200*I*a**10*c**19*x**7 - 188160*a**9*c**19*x**6 - 70560*I*a**8*c**19*x**5 - 22050*a**7*c**19*x**4 -
14700*I*a**6*c**19*x**3 - 1260*I*a**4*c**19*x + 210*a**3*c**19)

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