∫ e6iArcTan(ax)x2 ________________________

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

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Rubi [A]
time = 0.05, 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 = {5190, 82} \begin {gather*} -\frac {6 a x+i}{210 a^3 c^{19} (1-i a x)^{21} (1+i a x)^{15}} \end {gather*}

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)

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

\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 = 0.78, size = 36, normalized size = 0.95 \begin {gather*} \frac {i+6 a x}{210 a^3 c^{19} (-i+a x)^{15} (i+a x)^{21}} \end {gather*}

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method	result	size

default	\(\frac {\frac {x}{35 a^{2}}+\frac {i}{210 a^{3}}}{c^{19} \left (a x +i\right )^{21} \left (a x -i\right )^{15}}\)	\(34\)
risch	\(\frac {\frac {x}{35 a^{2}}+\frac {i}{210 a^{3}}}{c^{19} \left (a x +i\right )^{21} \left (a x -i\right )^{15}}\)	\(34\)
gosper	\(\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (6 a x +i\right ) \left (i a x +1\right )^{6}}{210 a^{3} \left (a^{2} x^{2}+1\right )^{22} c^{19}}\)	\(49\)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (30) = 60\).
time = 0.62, size = 292, normalized size = 7.68 \begin {gather*} \frac {6 \, a^{7} x^{7} - 35 i \, a^{6} x^{6} - 84 \, a^{5} x^{5} + 105 i \, a^{4} x^{4} + 70 \, a^{3} x^{3} - 21 i \, a^{2} x^{2} - i}{210 \, {\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 )}} \end {gather*}

1/210*(6*a^7*x^7 - 35*I*a^6*x^6 - 84*a^5*x^5 + 105*I*a^4*x^4 + 70*a^3*x^3 - 21*I*a^2*x^2 - 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^1

9*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^

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (30) = 60\).
time = 9.67, size = 379, normalized size = 9.97 \begin {gather*} \frac {6 \, a x + i}{210 \, {\left (a^{39} c^{19} x^{36} + 6 i \, a^{38} c^{19} x^{35} + 70 i \, a^{36} c^{19} x^{33} - 105 \, a^{35} c^{19} x^{32} + 336 i \, a^{34} c^{19} x^{31} - 896 \, a^{33} c^{19} x^{30} + 720 i \, a^{32} c^{19} x^{29} - 3900 \, a^{31} c^{19} x^{28} - 280 i \, a^{30} c^{19} x^{27} - 10752 \, a^{29} c^{19} x^{26} - 6552 i \, a^{28} c^{19} x^{25} - 20020 \, a^{27} c^{19} x^{24} - 21840 i \, a^{26} c^{19} x^{23} - 24960 \, a^{25} c^{19} x^{22} - 43472 i \, a^{24} c^{19} x^{21} - 18018 \, a^{23} c^{19} x^{20} - 60060 i \, a^{22} c^{19} x^{19} - 60060 i \, a^{20} c^{19} x^{17} + 18018 \, a^{19} c^{19} x^{16} - 43472 i \, a^{18} c^{19} x^{15} + 24960 \, a^{17} c^{19} x^{14} - 21840 i \, a^{16} c^{19} x^{13} + 20020 \, a^{15} c^{19} x^{12} - 6552 i \, a^{14} c^{19} x^{11} + 10752 \, a^{13} c^{19} x^{10} - 280 i \, a^{12} c^{19} x^{9} + 3900 \, a^{11} c^{19} x^{8} + 720 i \, a^{10} c^{19} x^{7} + 896 \, a^{9} c^{19} x^{6} + 336 i \, a^{8} c^{19} x^{5} + 105 \, a^{7} c^{19} x^{4} + 70 i \, a^{6} c^{19} x^{3} + 6 i \, a^{4} c^{19} x - a^{3} c^{19}\right )}} \end {gather*}

1/210*(6*a*x + I)/(a^39*c^19*x^36 + 6*I*a^38*c^19*x^35 + 70*I*a^36*c^19*x^33 - 105*a^35*c^19*x^32 + 336*I*a^34

*c^19*x^31 - 896*a^33*c^19*x^30 + 720*I*a^32*c^19*x^29 - 3900*a^31*c^19*x^28 - 280*I*a^30*c^19*x^27 - 10752*a^

29*c^19*x^26 - 6552*I*a^28*c^19*x^25 - 20020*a^27*c^19*x^24 - 21840*I*a^26*c^19*x^23 - 24960*a^25*c^19*x^22 -

43472*I*a^24*c^19*x^21 - 18018*a^23*c^19*x^20 - 60060*I*a^22*c^19*x^19 - 60060*I*a^20*c^19*x^17 + 18018*a^19*c

^19*x^16 - 43472*I*a^18*c^19*x^15 + 24960*a^17*c^19*x^14 - 21840*I*a^16*c^19*x^13 + 20020*a^15*c^19*x^12 - 655

2*I*a^14*c^19*x^11 + 10752*a^13*c^19*x^10 - 280*I*a^12*c^19*x^9 + 3900*a^11*c^19*x^8 + 720*I*a^10*c^19*x^7 + 8

96*a^9*c^19*x^6 + 336*I*a^8*c^19*x^5 + 105*a^7*c^19*x^4 + 70*I*a^6*c^19*x^3 + 6*I*a^4*c^19*x - a^3*c^19)

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (32) = 64\).
time = 3.69, size = 439, normalized size = 11.55 \begin {gather*} - \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}} \end {gather*}

-(-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 +

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (30) = 60\).
time = 0.43, size = 299, normalized size = 7.87 \begin {gather*} -\frac {358229025 \, a^{14} x^{14} - 5340869100 i \, a^{13} x^{13} - 37114698075 \, a^{12} x^{12} + 159416118225 i \, a^{11} x^{11} + 473088806190 \, a^{10} x^{10} - 1026819468675 i \, a^{9} x^{9} - 1682288472150 \, a^{8} x^{8} + 2115551402250 i \, a^{7} x^{7} + 2054435046125 \, a^{6} x^{6} - 1535397250002 i \, a^{5} x^{5} - 870854759775 \, a^{4} x^{4} + 364307533205 i \, a^{3} x^{3} + 106553746740 \, a^{2} x^{2} - 19571887695 i \, a x - 1710785408}{901943132160 \, {\left (a x - i\right )}^{15} a^{3} c^{19}} + \frac {358229025 \, a^{20} x^{20} + 7555375800 i \, a^{19} x^{19} - 75901131600 \, a^{18} x^{18} - 483051354975 i \, a^{17} x^{17} + 2184946607340 \, a^{16} x^{16} + 7469205450840 i \, a^{15} x^{15} - 20031221295000 \, a^{14} x^{14} - 43177004037300 i \, a^{13} x^{13} + 76013078916950 \, a^{12} x^{12} + 110448380006328 i \, a^{11} x^{11} - 133277726128008 \, a^{10} x^{10} - 133908931763530 i \, a^{9} x^{9} + 111933156213900 \, a^{8} x^{8} + 77492989590120 i \, a^{7} x^{7} - 44041557267624 \, a^{6} x^{6} - 20244576347604 i \, a^{5} x^{5} + 7349182966545 \, a^{4} x^{4} + 2026362494800 i \, a^{3} x^{3} - 396520754280 \, a^{2} x^{2} - 48177926223 i \, a x + 2584181888}{901943132160 \, {\left (a x + i\right )}^{21} a^{3} c^{19}} \end {gather*}

-1/901943132160*(358229025*a^14*x^14 - 5340869100*I*a^13*x^13 - 37114698075*a^12*x^12 + 159416118225*I*a^11*x^

11 + 473088806190*a^10*x^10 - 1026819468675*I*a^9*x^9 - 1682288472150*a^8*x^8 + 2115551402250*I*a^7*x^7 + 2054

435046125*a^6*x^6 - 1535397250002*I*a^5*x^5 - 870854759775*a^4*x^4 + 364307533205*I*a^3*x^3 + 106553746740*a^2

*x^2 - 19571887695*I*a*x - 1710785408)/((a*x - I)^15*a^3*c^19) + 1/901943132160*(358229025*a^20*x^20 + 7555375

800*I*a^19*x^19 - 75901131600*a^18*x^18 - 483051354975*I*a^17*x^17 + 2184946607340*a^16*x^16 + 7469205450840*I

*a^15*x^15 - 20031221295000*a^14*x^14 - 43177004037300*I*a^13*x^13 + 76013078916950*a^12*x^12 + 11044838000632

8*I*a^11*x^11 - 133277726128008*a^10*x^10 - 133908931763530*I*a^9*x^9 + 111933156213900*a^8*x^8 + 774929895901

20*I*a^7*x^7 - 44041557267624*a^6*x^6 - 20244576347604*I*a^5*x^5 + 7349182966545*a^4*x^4 + 2026362494800*I*a^3

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \text {Hanged} \end {gather*}

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3.4.75 \(\int \frac {e^{6 i \text {ArcTan}(a x)} x^2}{(c+a^2 c x^2)^{19}} \, dx\) [375]