Integrand size = 25, antiderivative size = 31 \[ \int \frac {e^{6 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{19}} \, dx=-\frac {1-6 a x}{210 a^3 c^{19} (1-a x)^{21} (1+a x)^{15}} \] Output:
-1/210*(-6*a*x+1)/a^3/c^19/(-a*x+1)^21/(a*x+1)^15
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {e^{6 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{19}} \, dx=\frac {1-6 a x}{210 a^3 c^{19} (-1+a x)^{21} (1+a x)^{15}} \] Input:
Integrate[(E^(6*ArcTanh[a*x])*x^2)/(c - a^2*c*x^2)^19,x]
Output:
(1 - 6*a*x)/(210*a^3*c^19*(-1 + a*x)^21*(1 + a*x)^15)
Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6700, 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.
| \(\displaystyle \int \frac {x^2 e^{6 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{19}} \, dx\) |
| \(\Big \downarrow \) 6700 |
| \(\displaystyle \frac {\int \frac {x^2}{(1-a x)^{22} (a x+1)^{16}}dx}{c^{19}}\) |
| \(\Big \downarrow \) 91 |
| \(\displaystyle -\frac {1-6 a x}{210 a^3 c^{19} (1-a x)^{21} (a x+1)^{15}}\) |
Input:
Int[(E^(6*ArcTanh[a*x])*x^2)/(c - a^2*c*x^2)^19,x]
Output:
-1/210*(1 - 6*a*x)/(a^3*c^19*(1 - a*x)^21*(1 + a*x)^15)
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]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Time = 0.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06
| method | result | size |
| gosper | \(-\frac {6 a x -1}{210 c^{19} \left (a x -1\right )^{6} \left (a^{2} x^{2}-1\right )^{15} a^{3}}\) | \(33\) |
| risch | \(\frac {-\frac {x}{35 a^{2}}+\frac {1}{210 a^{3}}}{c^{19} \left (a x -1\right )^{6} \left (a^{2} x^{2}-1\right )^{15}}\) | \(35\) |
| parallelrisch | \(\frac {a^{33} x^{36}-6 a^{32} x^{35}+70 x^{33} a^{30}-105 x^{32} a^{29}-336 x^{31} a^{28}+896 a^{27} x^{30}+720 x^{29} a^{26}-3900 x^{28} a^{25}+280 x^{27} a^{24}+10752 x^{26} a^{23}-6552 x^{25} a^{22}-20020 x^{24} a^{21}+21840 x^{23} a^{20}+24960 x^{22} a^{19}-43472 x^{21} a^{18}-18018 x^{20} a^{17}+60060 a^{16} x^{19}-60060 x^{17} a^{14}+18018 x^{16} a^{13}+43472 x^{15} a^{12}-24960 x^{14} a^{11}-21840 x^{13} a^{10}+20020 x^{12} a^{9}+6552 x^{11} a^{8}-10752 x^{10} a^{7}-280 a^{6} x^{9}+3900 a^{5} x^{8}-720 a^{4} x^{7}-896 a^{3} x^{6}+336 a^{2} x^{5}+105 a \,x^{4}-70 x^{3}}{210 c^{19} \left (a x -1\right )^{6} \left (a^{2} x^{2}-1\right )^{15}}\) | \(275\) |
| orering | \(\frac {x^{3} \left (a^{33} x^{33}-6 a^{32} x^{32}+70 a^{30} x^{30}-105 a^{29} x^{29}-336 a^{28} x^{28}+896 a^{27} x^{27}+720 a^{26} x^{26}-3900 a^{25} x^{25}+280 a^{24} x^{24}+10752 a^{23} x^{23}-6552 a^{22} x^{22}-20020 a^{21} x^{21}+21840 a^{20} x^{20}+24960 a^{19} x^{19}-43472 a^{18} x^{18}-18018 a^{17} x^{17}+60060 a^{16} x^{16}-60060 a^{14} x^{14}+18018 a^{13} x^{13}+43472 a^{12} x^{12}-24960 a^{11} x^{11}-21840 a^{10} x^{10}+20020 a^{9} x^{9}+6552 a^{8} x^{8}-10752 a^{7} x^{7}-280 x^{6} a^{6}+3900 a^{5} x^{5}-720 a^{4} x^{4}-896 a^{3} x^{3}+336 a^{2} x^{2}+105 a x -70\right ) \left (a x -1\right ) \left (a x +1\right )^{7}}{210 \left (-a^{2} x^{2}+1\right )^{3} \left (-a^{2} c \,x^{2}+c \right )^{19}}\) | \(288\) |
| default | \(\frac {-\frac {3411705}{8589934592 a^{3} \left (a x +1\right )}+\frac {334305}{939524096 a^{3} \left (a x -1\right )^{7}}-\frac {111435}{268435456 a^{3} \left (a x -1\right )^{6}}+\frac {1938969}{4294967296 a^{3} \left (a x -1\right )^{5}}-\frac {3991995}{8589934592 a^{3} \left (a x -1\right )^{4}}+\frac {1964315}{4294967296 a^{3} \left (a x -1\right )^{3}}-\frac {930465}{2147483648 a^{3} \left (a x -1\right )^{2}}-\frac {1}{65536 a^{3} \left (a x -1\right )^{19}}-\frac {312455}{2147483648 a^{3} \left (a x +1\right )^{6}}-\frac {858429}{4294967296 a^{3} \left (a x +1\right )^{5}}-\frac {2211105}{8589934592 a^{3} \left (a x +1\right )^{4}}-\frac {1344005}{4294967296 a^{3} \left (a x +1\right )^{3}}-\frac {1550775}{4294967296 a^{3} \left (a x +1\right )^{2}}-\frac {13}{16777216 a^{3} \left (a x +1\right )^{13}}-\frac {1}{62914560 a^{3} \left (a x +1\right )^{15}}+\frac {3411705}{8589934592 \left (a x -1\right ) a^{3}}-\frac {1}{1376256 a^{3} \left (a x -1\right )^{21}}+\frac {3}{655360 a^{3} \left (a x -1\right )^{20}}+\frac {7}{196608 a^{3} \left (a x -1\right )^{18}}-\frac {17}{262144 a^{3} \left (a x -1\right )^{17}}-\frac {9}{58720256 a^{3} \left (a x +1\right )^{14}}-\frac {275}{100663296 a^{3} \left (a x +1\right )^{12}}-\frac {253}{33554432 a^{3} \left (a x +1\right )^{11}}-\frac {5819}{335544320 a^{3} \left (a x +1\right )^{10}}-\frac {13915}{402653184 a^{3} \left (a x +1\right )^{9}}-\frac {16445}{268435456 a^{3} \left (a x +1\right )^{8}}-\frac {740025}{7516192768 a^{3} \left (a x +1\right )^{7}}+\frac {51}{524288 a^{3} \left (a x -1\right )^{16}}-\frac {323}{2621440 a^{3} \left (a x -1\right )^{15}}+\frac {969}{7340032 a^{3} \left (a x -1\right )^{14}}-\frac {969}{8388608 a^{3} \left (a x -1\right )^{13}}+\frac {3553}{50331648 a^{3} \left (a x -1\right )^{12}}-\frac {7429}{83886080 a^{3} \left (a x -1\right )^{10}}+\frac {37145}{201326592 a^{3} \left (a x -1\right )^{9}}-\frac {37145}{134217728 a^{3} \left (a x -1\right )^{8}}}{c^{19}}\) | \(426\) |
Input:
int((a*x+1)^6/(-a^2*x^2+1)^3*x^2/(-a^2*c*x^2+c)^19,x,method=_RETURNVERBOSE )
Output:
-1/210*(6*a*x-1)/c^19/(a*x-1)^6/(a^2*x^2-1)^15/a^3
Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 379, normalized size of antiderivative = 12.23 \[ \int \frac {e^{6 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{19}} \, dx=-\frac {6 \, a x - 1}{210 \, {\left (a^{39} c^{19} x^{36} - 6 \, a^{38} c^{19} x^{35} + 70 \, a^{36} c^{19} x^{33} - 105 \, a^{35} c^{19} x^{32} - 336 \, a^{34} c^{19} x^{31} + 896 \, a^{33} c^{19} x^{30} + 720 \, a^{32} c^{19} x^{29} - 3900 \, a^{31} c^{19} x^{28} + 280 \, a^{30} c^{19} x^{27} + 10752 \, a^{29} c^{19} x^{26} - 6552 \, a^{28} c^{19} x^{25} - 20020 \, a^{27} c^{19} x^{24} + 21840 \, a^{26} c^{19} x^{23} + 24960 \, a^{25} c^{19} x^{22} - 43472 \, a^{24} c^{19} x^{21} - 18018 \, a^{23} c^{19} x^{20} + 60060 \, a^{22} c^{19} x^{19} - 60060 \, a^{20} c^{19} x^{17} + 18018 \, a^{19} c^{19} x^{16} + 43472 \, a^{18} c^{19} x^{15} - 24960 \, a^{17} c^{19} x^{14} - 21840 \, a^{16} c^{19} x^{13} + 20020 \, a^{15} c^{19} x^{12} + 6552 \, a^{14} c^{19} x^{11} - 10752 \, a^{13} c^{19} x^{10} - 280 \, a^{12} c^{19} x^{9} + 3900 \, a^{11} c^{19} x^{8} - 720 \, a^{10} c^{19} x^{7} - 896 \, a^{9} c^{19} x^{6} + 336 \, a^{8} c^{19} x^{5} + 105 \, a^{7} c^{19} x^{4} - 70 \, a^{6} c^{19} x^{3} + 6 \, a^{4} c^{19} x - a^{3} c^{19}\right )}} \] Input:
integrate((a*x+1)^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 - 1)/(a^39*c^19*x^36 - 6*a^38*c^19*x^35 + 70*a^36*c^19*x^33 - 105*a^35*c^19*x^32 - 336*a^34*c^19*x^31 + 896*a^33*c^19*x^30 + 720*a^32* c^19*x^29 - 3900*a^31*c^19*x^28 + 280*a^30*c^19*x^27 + 10752*a^29*c^19*x^2 6 - 6552*a^28*c^19*x^25 - 20020*a^27*c^19*x^24 + 21840*a^26*c^19*x^23 + 24 960*a^25*c^19*x^22 - 43472*a^24*c^19*x^21 - 18018*a^23*c^19*x^20 + 60060*a ^22*c^19*x^19 - 60060*a^20*c^19*x^17 + 18018*a^19*c^19*x^16 + 43472*a^18*c ^19*x^15 - 24960*a^17*c^19*x^14 - 21840*a^16*c^19*x^13 + 20020*a^15*c^19*x ^12 + 6552*a^14*c^19*x^11 - 10752*a^13*c^19*x^10 - 280*a^12*c^19*x^9 + 390 0*a^11*c^19*x^8 - 720*a^10*c^19*x^7 - 896*a^9*c^19*x^6 + 336*a^8*c^19*x^5 + 105*a^7*c^19*x^4 - 70*a^6*c^19*x^3 + 6*a^4*c^19*x - a^3*c^19)
Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (27) = 54\).
Time = 1.78 (sec) , antiderivative size = 405, normalized size of antiderivative = 13.06 \[ \int \frac {e^{6 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{19}} \, dx=\frac {- 6 a x + 1}{210 a^{39} c^{19} x^{36} - 1260 a^{38} c^{19} x^{35} + 14700 a^{36} c^{19} x^{33} - 22050 a^{35} c^{19} x^{32} - 70560 a^{34} c^{19} x^{31} + 188160 a^{33} c^{19} x^{30} + 151200 a^{32} c^{19} x^{29} - 819000 a^{31} c^{19} x^{28} + 58800 a^{30} c^{19} x^{27} + 2257920 a^{29} c^{19} x^{26} - 1375920 a^{28} c^{19} x^{25} - 4204200 a^{27} c^{19} x^{24} + 4586400 a^{26} c^{19} x^{23} + 5241600 a^{25} c^{19} x^{22} - 9129120 a^{24} c^{19} x^{21} - 3783780 a^{23} c^{19} x^{20} + 12612600 a^{22} c^{19} x^{19} - 12612600 a^{20} c^{19} x^{17} + 3783780 a^{19} c^{19} x^{16} + 9129120 a^{18} c^{19} x^{15} - 5241600 a^{17} c^{19} x^{14} - 4586400 a^{16} c^{19} x^{13} + 4204200 a^{15} c^{19} x^{12} + 1375920 a^{14} c^{19} x^{11} - 2257920 a^{13} c^{19} x^{10} - 58800 a^{12} c^{19} x^{9} + 819000 a^{11} c^{19} x^{8} - 151200 a^{10} c^{19} x^{7} - 188160 a^{9} c^{19} x^{6} + 70560 a^{8} c^{19} x^{5} + 22050 a^{7} c^{19} x^{4} - 14700 a^{6} c^{19} x^{3} + 1260 a^{4} c^{19} x - 210 a^{3} c^{19}} \] Input:
integrate((a*x+1)**6/(-a**2*x**2+1)**3*x**2/(-a**2*c*x**2+c)**19,x)
Output:
(-6*a*x + 1)/(210*a**39*c**19*x**36 - 1260*a**38*c**19*x**35 + 14700*a**36 *c**19*x**33 - 22050*a**35*c**19*x**32 - 70560*a**34*c**19*x**31 + 188160* a**33*c**19*x**30 + 151200*a**32*c**19*x**29 - 819000*a**31*c**19*x**28 + 58800*a**30*c**19*x**27 + 2257920*a**29*c**19*x**26 - 1375920*a**28*c**19* x**25 - 4204200*a**27*c**19*x**24 + 4586400*a**26*c**19*x**23 + 5241600*a* *25*c**19*x**22 - 9129120*a**24*c**19*x**21 - 3783780*a**23*c**19*x**20 + 12612600*a**22*c**19*x**19 - 12612600*a**20*c**19*x**17 + 3783780*a**19*c* *19*x**16 + 9129120*a**18*c**19*x**15 - 5241600*a**17*c**19*x**14 - 458640 0*a**16*c**19*x**13 + 4204200*a**15*c**19*x**12 + 1375920*a**14*c**19*x**1 1 - 2257920*a**13*c**19*x**10 - 58800*a**12*c**19*x**9 + 819000*a**11*c**1 9*x**8 - 151200*a**10*c**19*x**7 - 188160*a**9*c**19*x**6 + 70560*a**8*c** 19*x**5 + 22050*a**7*c**19*x**4 - 14700*a**6*c**19*x**3 + 1260*a**4*c**19* x - 210*a**3*c**19)
Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (28) = 56\).
Time = 0.20 (sec) , antiderivative size = 379, normalized size of antiderivative = 12.23 \[ \int \frac {e^{6 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{19}} \, dx=-\frac {6 \, a x - 1}{210 \, {\left (a^{39} c^{19} x^{36} - 6 \, a^{38} c^{19} x^{35} + 70 \, a^{36} c^{19} x^{33} - 105 \, a^{35} c^{19} x^{32} - 336 \, a^{34} c^{19} x^{31} + 896 \, a^{33} c^{19} x^{30} + 720 \, a^{32} c^{19} x^{29} - 3900 \, a^{31} c^{19} x^{28} + 280 \, a^{30} c^{19} x^{27} + 10752 \, a^{29} c^{19} x^{26} - 6552 \, a^{28} c^{19} x^{25} - 20020 \, a^{27} c^{19} x^{24} + 21840 \, a^{26} c^{19} x^{23} + 24960 \, a^{25} c^{19} x^{22} - 43472 \, a^{24} c^{19} x^{21} - 18018 \, a^{23} c^{19} x^{20} + 60060 \, a^{22} c^{19} x^{19} - 60060 \, a^{20} c^{19} x^{17} + 18018 \, a^{19} c^{19} x^{16} + 43472 \, a^{18} c^{19} x^{15} - 24960 \, a^{17} c^{19} x^{14} - 21840 \, a^{16} c^{19} x^{13} + 20020 \, a^{15} c^{19} x^{12} + 6552 \, a^{14} c^{19} x^{11} - 10752 \, a^{13} c^{19} x^{10} - 280 \, a^{12} c^{19} x^{9} + 3900 \, a^{11} c^{19} x^{8} - 720 \, a^{10} c^{19} x^{7} - 896 \, a^{9} c^{19} x^{6} + 336 \, a^{8} c^{19} x^{5} + 105 \, a^{7} c^{19} x^{4} - 70 \, a^{6} c^{19} x^{3} + 6 \, a^{4} c^{19} x - a^{3} c^{19}\right )}} \] Input:
integrate((a*x+1)^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*x - 1)/(a^39*c^19*x^36 - 6*a^38*c^19*x^35 + 70*a^36*c^19*x^33 - 105*a^35*c^19*x^32 - 336*a^34*c^19*x^31 + 896*a^33*c^19*x^30 + 720*a^32* c^19*x^29 - 3900*a^31*c^19*x^28 + 280*a^30*c^19*x^27 + 10752*a^29*c^19*x^2 6 - 6552*a^28*c^19*x^25 - 20020*a^27*c^19*x^24 + 21840*a^26*c^19*x^23 + 24 960*a^25*c^19*x^22 - 43472*a^24*c^19*x^21 - 18018*a^23*c^19*x^20 + 60060*a ^22*c^19*x^19 - 60060*a^20*c^19*x^17 + 18018*a^19*c^19*x^16 + 43472*a^18*c ^19*x^15 - 24960*a^17*c^19*x^14 - 21840*a^16*c^19*x^13 + 20020*a^15*c^19*x ^12 + 6552*a^14*c^19*x^11 - 10752*a^13*c^19*x^10 - 280*a^12*c^19*x^9 + 390 0*a^11*c^19*x^8 - 720*a^10*c^19*x^7 - 896*a^9*c^19*x^6 + 336*a^8*c^19*x^5 + 105*a^7*c^19*x^4 - 70*a^6*c^19*x^3 + 6*a^4*c^19*x - a^3*c^19)
Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (28) = 56\).
Time = 0.13 (sec) , antiderivative size = 299, normalized size of antiderivative = 9.65 \[ \int \frac {e^{6 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{19}} \, dx=-\frac {358229025 \, a^{14} x^{14} + 5340869100 \, a^{13} x^{13} + 37114698075 \, a^{12} x^{12} + 159416118225 \, a^{11} x^{11} + 473088806190 \, a^{10} x^{10} + 1026819468675 \, a^{9} x^{9} + 1682288472150 \, a^{8} x^{8} + 2115551402250 \, a^{7} x^{7} + 2054435046125 \, a^{6} x^{6} + 1535397250002 \, a^{5} x^{5} + 870854759775 \, a^{4} x^{4} + 364307533205 \, a^{3} x^{3} + 106553746740 \, a^{2} x^{2} + 19571887695 \, a x + 1710785408}{901943132160 \, {\left (a x + 1\right )}^{15} a^{3} c^{19}} + \frac {358229025 \, a^{20} x^{20} - 7555375800 \, a^{19} x^{19} + 75901131600 \, a^{18} x^{18} - 483051354975 \, a^{17} x^{17} + 2184946607340 \, a^{16} x^{16} - 7469205450840 \, a^{15} x^{15} + 20031221295000 \, a^{14} x^{14} - 43177004037300 \, a^{13} x^{13} + 76013078916950 \, a^{12} x^{12} - 110448380006328 \, a^{11} x^{11} + 133277726128008 \, a^{10} x^{10} - 133908931763530 \, a^{9} x^{9} + 111933156213900 \, a^{8} x^{8} - 77492989590120 \, a^{7} x^{7} + 44041557267624 \, a^{6} x^{6} - 20244576347604 \, a^{5} x^{5} + 7349182966545 \, a^{4} x^{4} - 2026362494800 \, a^{3} x^{3} + 396520754280 \, a^{2} x^{2} - 48177926223 \, a x + 2584181888}{901943132160 \, {\left (a x - 1\right )}^{21} a^{3} c^{19}} \] Input:
integrate((a*x+1)^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*a^13*x^13 + 37114698075* a^12*x^12 + 159416118225*a^11*x^11 + 473088806190*a^10*x^10 + 102681946867 5*a^9*x^9 + 1682288472150*a^8*x^8 + 2115551402250*a^7*x^7 + 2054435046125* a^6*x^6 + 1535397250002*a^5*x^5 + 870854759775*a^4*x^4 + 364307533205*a^3* x^3 + 106553746740*a^2*x^2 + 19571887695*a*x + 1710785408)/((a*x + 1)^15*a ^3*c^19) + 1/901943132160*(358229025*a^20*x^20 - 7555375800*a^19*x^19 + 75 901131600*a^18*x^18 - 483051354975*a^17*x^17 + 2184946607340*a^16*x^16 - 7 469205450840*a^15*x^15 + 20031221295000*a^14*x^14 - 43177004037300*a^13*x^ 13 + 76013078916950*a^12*x^12 - 110448380006328*a^11*x^11 + 13327772612800 8*a^10*x^10 - 133908931763530*a^9*x^9 + 111933156213900*a^8*x^8 - 77492989 590120*a^7*x^7 + 44041557267624*a^6*x^6 - 20244576347604*a^5*x^5 + 7349182 966545*a^4*x^4 - 2026362494800*a^3*x^3 + 396520754280*a^2*x^2 - 4817792622 3*a*x + 2584181888)/((a*x - 1)^21*a^3*c^19)
Timed out. \[ \int \frac {e^{6 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{19}} \, dx=\text {Hanged} \] Input:
int(-(x^2*(a*x + 1)^6)/((c - a^2*c*x^2)^19*(a^2*x^2 - 1)^3),x)
Output:
\text{Hanged}
Time = 0.15 (sec) , antiderivative size = 277, normalized size of antiderivative = 8.94 \[ \int \frac {e^{6 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{19}} \, dx=\frac {-6 a x +1}{210 a^{3} c^{19} \left (a^{36} x^{36}-6 a^{35} x^{35}+70 a^{33} x^{33}-105 a^{32} x^{32}-336 a^{31} x^{31}+896 a^{30} x^{30}+720 a^{29} x^{29}-3900 a^{28} x^{28}+280 a^{27} x^{27}+10752 a^{26} x^{26}-6552 a^{25} x^{25}-20020 a^{24} x^{24}+21840 a^{23} x^{23}+24960 a^{22} x^{22}-43472 a^{21} x^{21}-18018 a^{20} x^{20}+60060 a^{19} x^{19}-60060 a^{17} x^{17}+18018 a^{16} x^{16}+43472 a^{15} x^{15}-24960 a^{14} x^{14}-21840 a^{13} x^{13}+20020 a^{12} x^{12}+6552 a^{11} x^{11}-10752 a^{10} x^{10}-280 a^{9} x^{9}+3900 a^{8} x^{8}-720 a^{7} x^{7}-896 a^{6} x^{6}+336 a^{5} x^{5}+105 a^{4} x^{4}-70 a^{3} x^{3}+6 a x -1\right )} \] Input:
int((a*x+1)^6/(-a^2*x^2+1)^3*x^2/(-a^2*c*x^2+c)^19,x)
Output:
( - 6*a*x + 1)/(210*a**3*c**19*(a**36*x**36 - 6*a**35*x**35 + 70*a**33*x** 33 - 105*a**32*x**32 - 336*a**31*x**31 + 896*a**30*x**30 + 720*a**29*x**29 - 3900*a**28*x**28 + 280*a**27*x**27 + 10752*a**26*x**26 - 6552*a**25*x** 25 - 20020*a**24*x**24 + 21840*a**23*x**23 + 24960*a**22*x**22 - 43472*a** 21*x**21 - 18018*a**20*x**20 + 60060*a**19*x**19 - 60060*a**17*x**17 + 180 18*a**16*x**16 + 43472*a**15*x**15 - 24960*a**14*x**14 - 21840*a**13*x**13 + 20020*a**12*x**12 + 6552*a**11*x**11 - 10752*a**10*x**10 - 280*a**9*x** 9 + 3900*a**8*x**8 - 720*a**7*x**7 - 896*a**6*x**6 + 336*a**5*x**5 + 105*a **4*x**4 - 70*a**3*x**3 + 6*a*x - 1))