\(\int \frac {e^{6 i \arctan (a x)} x^2}{(c+a^2 c x^2)^{19}} \, dx\) [389]

Optimal result

Integrand size = 26, antiderivative size = 38 \[ \int \frac {e^{6 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{19}} \, dx=-\frac {i+6 a x}{210 a^3 c^{19} (1-i a x)^{21} (1+i a x)^{15}} \] Output:

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

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {e^{6 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{19}} \, dx=\frac {i+6 a x}{210 a^3 c^{19} (-i+a x)^{15} (i+a x)^{21}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (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 = {5605, 91}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2 e^{6 i \arctan (a x)}}{\left (a^2 c x^2+c\right )^{19}} \, dx\)

\(\Big \downarrow \) 5605

\(\displaystyle \frac {\int \frac {x^2}{(1-i a x)^{22} (i a x+1)^{16}}dx}{c^{19}}\)

\(\Big \downarrow \) 91

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> 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 5605

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[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] && (Integer 
Q[p] || GtQ[c, 0])
 

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89

method	result	size
default	\(\frac {\frac {i}{210 a^{3}}+\frac {x}{35 a^{2}}}{c^{19} \left (a x +i\right )^{21} \left (a x -i\right )^{15}}\)	\(34\)
risch	\(\frac {\frac {i}{210 a^{3}}+\frac {x}{35 a^{2}}}{c^{19} \left (a x +i\right )^{21} \left (a x -i\right )^{15}}\)	\(34\)
gosper	\(\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (6 a x +i\right ) \left (i a x +1\right )^{6}}{210 a^{3} \left (a^{2} x^{2}+1\right )^{22} c^{19}}\)	\(49\)
parallelrisch	\(\frac {i x^{42} a^{39}+21 i x^{40} a^{37}+210 i x^{38} a^{35}+1330 i x^{36} a^{33}+5985 i x^{34} a^{31}+20349 i x^{32} a^{29}+54264 i x^{30} a^{27}+116280 i x^{28} a^{25}+203490 i x^{26} a^{23}+293930 i x^{24} a^{21}+352716 i x^{22} a^{19}+352716 i x^{20} a^{17}+293930 i x^{18} a^{15}+203490 i x^{16} a^{13}+116280 i x^{14} a^{11}+54264 i x^{12} a^{9}+20349 i x^{10} a^{7}+5985 i x^{8} a^{5}+6 a^{4} x^{7}+1295 i x^{6} a^{3}-84 a^{2} x^{5}+315 i x^{4} a +70 x^{3}}{210 c^{19} \left (a^{2} x^{2}+1\right )^{21}}\)	\(217\)
orering	\(-\frac {i x^{3} \left (a^{33} x^{33}+6 i a^{32} x^{32}+70 i a^{30} x^{30}-105 a^{29} x^{29}+336 i a^{28} x^{28}-896 a^{27} x^{27}+720 i a^{26} x^{26}-3900 a^{25} x^{25}-280 i a^{24} x^{24}-10752 a^{23} x^{23}-6552 i a^{22} x^{22}-20020 a^{21} x^{21}-21840 i a^{20} x^{20}-24960 a^{19} x^{19}-43472 i a^{18} x^{18}-18018 a^{17} x^{17}-60060 i a^{16} x^{16}-60060 i a^{14} x^{14}+18018 a^{13} x^{13}-43472 i a^{12} x^{12}+24960 a^{11} x^{11}-21840 i a^{10} x^{10}+20020 a^{9} x^{9}-6552 i a^{8} x^{8}+10752 a^{7} x^{7}-280 i a^{6} x^{6}+3900 a^{5} x^{5}+720 i x^{4} a^{4}+896 a^{3} x^{3}+336 i x^{2} a^{2}+105 a x +70 i\right ) \left (i a x +1\right )^{6}}{210 \left (a^{2} x^{2}+1\right )^{2} \left (a^{2} c \,x^{2}+c \right )^{19}}\)	\(301\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

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 = 0.54 (sec) , antiderivative size = 379, normalized size of antiderivative = 9.97 \[ \int \frac {e^{6 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{19}} \, dx=\frac {6 \, a x + i}{210 \, {\left (a^{39} c^{19} x^{36} + 6 i \, a^{38} c^{19} x^{35} + 70 i \, a^{36} c^{19} x^{33} - 105 \, a^{35} c^{19} x^{32} + 336 i \, a^{34} c^{19} x^{31} - 896 \, a^{33} c^{19} x^{30} + 720 i \, a^{32} c^{19} x^{29} - 3900 \, a^{31} c^{19} x^{28} - 280 i \, a^{30} c^{19} x^{27} - 10752 \, a^{29} c^{19} x^{26} - 6552 i \, a^{28} c^{19} x^{25} - 20020 \, a^{27} c^{19} x^{24} - 21840 i \, a^{26} c^{19} x^{23} - 24960 \, a^{25} c^{19} x^{22} - 43472 i \, a^{24} c^{19} x^{21} - 18018 \, a^{23} c^{19} x^{20} - 60060 i \, a^{22} c^{19} x^{19} - 60060 i \, a^{20} c^{19} x^{17} + 18018 \, a^{19} c^{19} x^{16} - 43472 i \, a^{18} c^{19} x^{15} + 24960 \, a^{17} c^{19} x^{14} - 21840 i \, a^{16} c^{19} x^{13} + 20020 \, a^{15} c^{19} x^{12} - 6552 i \, a^{14} c^{19} x^{11} + 10752 \, a^{13} c^{19} x^{10} - 280 i \, a^{12} c^{19} x^{9} + 3900 \, a^{11} c^{19} x^{8} + 720 i \, a^{10} c^{19} x^{7} + 896 \, a^{9} c^{19} x^{6} + 336 i \, a^{8} c^{19} x^{5} + 105 \, a^{7} c^{19} x^{4} + 70 i \, a^{6} c^{19} x^{3} + 6 i \, a^{4} c^{19} x - a^{3} c^{19}\right )}} \] Input:

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

Output:

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 - 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)
 

Sympy [B] (verification not implemented)

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 = 2.76 (sec) , antiderivative size = 439, normalized size of antiderivative = 11.55 \[ \int \frac {e^{6 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{19}} \, dx=- \frac {- 6 a x - i}{210 a^{39} c^{19} x^{36} + 1260 i a^{38} c^{19} x^{35} + 14700 i a^{36} c^{19} x^{33} - 22050 a^{35} c^{19} x^{32} + 70560 i a^{34} c^{19} x^{31} - 188160 a^{33} c^{19} x^{30} + 151200 i a^{32} c^{19} x^{29} - 819000 a^{31} c^{19} x^{28} - 58800 i a^{30} c^{19} x^{27} - 2257920 a^{29} c^{19} x^{26} - 1375920 i a^{28} c^{19} x^{25} - 4204200 a^{27} c^{19} x^{24} - 4586400 i a^{26} c^{19} x^{23} - 5241600 a^{25} c^{19} x^{22} - 9129120 i a^{24} c^{19} x^{21} - 3783780 a^{23} c^{19} x^{20} - 12612600 i a^{22} c^{19} x^{19} - 12612600 i a^{20} c^{19} x^{17} + 3783780 a^{19} c^{19} x^{16} - 9129120 i a^{18} c^{19} x^{15} + 5241600 a^{17} c^{19} x^{14} - 4586400 i a^{16} c^{19} x^{13} + 4204200 a^{15} c^{19} x^{12} - 1375920 i a^{14} c^{19} x^{11} + 2257920 a^{13} c^{19} x^{10} - 58800 i a^{12} c^{19} x^{9} + 819000 a^{11} c^{19} x^{8} + 151200 i a^{10} c^{19} x^{7} + 188160 a^{9} c^{19} x^{6} + 70560 i a^{8} c^{19} x^{5} + 22050 a^{7} c^{19} x^{4} + 14700 i a^{6} c^{19} x^{3} + 1260 i a^{4} c^{19} x - 210 a^{3} c^{19}} \] Input:

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

Output:

-(-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)
 

Maxima [B] (verification not implemented)

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 = 0.19 (sec) , antiderivative size = 292, normalized size of antiderivative = 7.68 \[ \int \frac {e^{6 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{19}} \, dx=\frac {6 \, a^{7} x^{7} - 35 i \, a^{6} x^{6} - 84 \, a^{5} x^{5} + 105 i \, a^{4} x^{4} + 70 \, a^{3} x^{3} - 21 i \, a^{2} x^{2} - i}{210 \, {\left (a^{45} c^{19} x^{42} + 21 \, a^{43} c^{19} x^{40} + 210 \, a^{41} c^{19} x^{38} + 1330 \, a^{39} c^{19} x^{36} + 5985 \, a^{37} c^{19} x^{34} + 20349 \, a^{35} c^{19} x^{32} + 54264 \, a^{33} c^{19} x^{30} + 116280 \, a^{31} c^{19} x^{28} + 203490 \, a^{29} c^{19} x^{26} + 293930 \, a^{27} c^{19} x^{24} + 352716 \, a^{25} c^{19} x^{22} + 352716 \, a^{23} c^{19} x^{20} + 293930 \, a^{21} c^{19} x^{18} + 203490 \, a^{19} c^{19} x^{16} + 116280 \, a^{17} c^{19} x^{14} + 54264 \, a^{15} c^{19} x^{12} + 20349 \, a^{13} c^{19} x^{10} + 5985 \, a^{11} c^{19} x^{8} + 1330 \, a^{9} c^{19} x^{6} + 210 \, a^{7} c^{19} x^{4} + 21 \, a^{5} c^{19} x^{2} + a^{3} c^{19}\right )}} \] Input:

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

Output:

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 + 54 
264*a^33*c^19*x^30 + 116280*a^31*c^19*x^28 + 203490*a^29*c^19*x^26 + 29393 
0*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^1 
5*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)
 

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. 299 vs. \(2 (30) = 60\).

Time = 0.13 (sec) , antiderivative size = 299, normalized size of antiderivative = 7.87 \[ \int \frac {e^{6 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{19}} \, dx=-\frac {358229025 \, a^{14} x^{14} - 5340869100 i \, a^{13} x^{13} - 37114698075 \, a^{12} x^{12} + 159416118225 i \, a^{11} x^{11} + 473088806190 \, a^{10} x^{10} - 1026819468675 i \, a^{9} x^{9} - 1682288472150 \, a^{8} x^{8} + 2115551402250 i \, a^{7} x^{7} + 2054435046125 \, a^{6} x^{6} - 1535397250002 i \, a^{5} x^{5} - 870854759775 \, a^{4} x^{4} + 364307533205 i \, a^{3} x^{3} + 106553746740 \, a^{2} x^{2} - 19571887695 i \, a x - 1710785408}{901943132160 \, {\left (a x - i\right )}^{15} a^{3} c^{19}} + \frac {358229025 \, a^{20} x^{20} + 7555375800 i \, a^{19} x^{19} - 75901131600 \, a^{18} x^{18} - 483051354975 i \, a^{17} x^{17} + 2184946607340 \, a^{16} x^{16} + 7469205450840 i \, a^{15} x^{15} - 20031221295000 \, a^{14} x^{14} - 43177004037300 i \, a^{13} x^{13} + 76013078916950 \, a^{12} x^{12} + 110448380006328 i \, a^{11} x^{11} - 133277726128008 \, a^{10} x^{10} - 133908931763530 i \, a^{9} x^{9} + 111933156213900 \, a^{8} x^{8} + 77492989590120 i \, a^{7} x^{7} - 44041557267624 \, a^{6} x^{6} - 20244576347604 i \, a^{5} x^{5} + 7349182966545 \, a^{4} x^{4} + 2026362494800 i \, a^{3} x^{3} - 396520754280 \, a^{2} x^{2} - 48177926223 i \, a x + 2584181888}{901943132160 \, {\left (a x + i\right )}^{21} a^{3} c^{19}} \] Input:

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

Output:

-1/901943132160*(358229025*a^14*x^14 - 5340869100*I*a^13*x^13 - 3711469807 
5*a^12*x^12 + 159416118225*I*a^11*x^11 + 473088806190*a^10*x^10 - 10268194 
68675*I*a^9*x^9 - 1682288472150*a^8*x^8 + 2115551402250*I*a^7*x^7 + 205443 
5046125*a^6*x^6 - 1535397250002*I*a^5*x^5 - 870854759775*a^4*x^4 + 3643075 
33205*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 + 7555375800* 
I*a^19*x^19 - 75901131600*a^18*x^18 - 483051354975*I*a^17*x^17 + 218494660 
7340*a^16*x^16 + 7469205450840*I*a^15*x^15 - 20031221295000*a^14*x^14 - 43 
177004037300*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 + 11193315 
6213900*a^8*x^8 + 77492989590120*I*a^7*x^7 - 44041557267624*a^6*x^6 - 2024 
4576347604*I*a^5*x^5 + 7349182966545*a^4*x^4 + 2026362494800*I*a^3*x^3 - 3 
96520754280*a^2*x^2 - 48177926223*I*a*x + 2584181888)/((a*x + I)^21*a^3*c^ 
19)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{6 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{19}} \, dx=\text {Hanged} \] Input:

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

Output:

\text{Hanged}
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 234, normalized size of antiderivative = 6.16 \[ \int \frac {e^{6 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{19}} \, dx=\frac {6 a^{7} x^{7}-35 a^{6} i \,x^{6}-84 a^{5} x^{5}+105 a^{4} i \,x^{4}+70 a^{3} x^{3}-21 a^{2} i \,x^{2}-i}{210 a^{3} c^{19} \left (a^{42} x^{42}+21 a^{40} x^{40}+210 a^{38} x^{38}+1330 a^{36} x^{36}+5985 a^{34} x^{34}+20349 a^{32} x^{32}+54264 a^{30} x^{30}+116280 a^{28} x^{28}+203490 a^{26} x^{26}+293930 a^{24} x^{24}+352716 a^{22} x^{22}+352716 a^{20} x^{20}+293930 a^{18} x^{18}+203490 a^{16} x^{16}+116280 a^{14} x^{14}+54264 a^{12} x^{12}+20349 a^{10} x^{10}+5985 a^{8} x^{8}+1330 a^{6} x^{6}+210 a^{4} x^{4}+21 a^{2} x^{2}+1\right )} \] Input:

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

Output:

(6*a**7*x**7 - 35*a**6*i*x**6 - 84*a**5*x**5 + 105*a**4*i*x**4 + 70*a**3*x 
**3 - 21*a**2*i*x**2 - i)/(210*a**3*c**19*(a**42*x**42 + 21*a**40*x**40 + 
210*a**38*x**38 + 1330*a**36*x**36 + 5985*a**34*x**34 + 20349*a**32*x**32 
+ 54264*a**30*x**30 + 116280*a**28*x**28 + 203490*a**26*x**26 + 293930*a** 
24*x**24 + 352716*a**22*x**22 + 352716*a**20*x**20 + 293930*a**18*x**18 + 
203490*a**16*x**16 + 116280*a**14*x**14 + 54264*a**12*x**12 + 20349*a**10* 
x**10 + 5985*a**8*x**8 + 1330*a**6*x**6 + 210*a**4*x**4 + 21*a**2*x**2 + 1 
))