3.1368 \(\int \frac {e^{6 \tanh ^{-1}(a x)} x^2}{(c-a^2 c x^2)^{19}} \, dx\)

Optimal. Leaf size=31 \[ -\frac {1-6 a x}{210 a^3 c^{19} (1-a x)^{21} (a x+1)^{15}} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 81} \[ -\frac {1-6 a x}{210 a^3 c^{19} (1-a x)^{21} (a x+1)^{15}} \]

Antiderivative was successfully verified.

[In]

Int[(E^(6*ArcTanh[a*x])*x^2)/(c - a^2*c*x^2)^19,x]

[Out]

-(1 - 6*a*x)/(210*a^3*c^19*(1 - a*x)^21*(1 + 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 6150

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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])

Rubi steps

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

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Mathematica [A]  time = 1.27, size = 30, normalized size = 0.97 \[ \frac {1-6 a x}{210 a^3 c^{19} (a x-1)^{21} (a x+1)^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(6*ArcTanh[a*x])*x^2)/(c - a^2*c*x^2)^19,x]

[Out]

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

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fricas [B]  time = 1.48, size = 379, normalized size = 12.23 \[ -\frac {6 \, a x - 1}{210 \, {\left (a^{39} c^{19} x^{36} - 6 \, a^{38} c^{19} x^{35} + 70 \, a^{36} c^{19} x^{33} - 105 \, a^{35} c^{19} x^{32} - 336 \, a^{34} c^{19} x^{31} + 896 \, a^{33} c^{19} x^{30} + 720 \, a^{32} c^{19} x^{29} - 3900 \, a^{31} c^{19} x^{28} + 280 \, a^{30} c^{19} x^{27} + 10752 \, a^{29} c^{19} x^{26} - 6552 \, a^{28} c^{19} x^{25} - 20020 \, a^{27} c^{19} x^{24} + 21840 \, a^{26} c^{19} x^{23} + 24960 \, a^{25} c^{19} x^{22} - 43472 \, a^{24} c^{19} x^{21} - 18018 \, a^{23} c^{19} x^{20} + 60060 \, a^{22} c^{19} x^{19} - 60060 \, a^{20} c^{19} x^{17} + 18018 \, a^{19} c^{19} x^{16} + 43472 \, a^{18} c^{19} x^{15} - 24960 \, a^{17} c^{19} x^{14} - 21840 \, a^{16} c^{19} x^{13} + 20020 \, a^{15} c^{19} x^{12} + 6552 \, a^{14} c^{19} x^{11} - 10752 \, a^{13} c^{19} x^{10} - 280 \, a^{12} c^{19} x^{9} + 3900 \, a^{11} c^{19} x^{8} - 720 \, a^{10} c^{19} x^{7} - 896 \, a^{9} c^{19} x^{6} + 336 \, a^{8} c^{19} x^{5} + 105 \, a^{7} c^{19} x^{4} - 70 \, a^{6} c^{19} x^{3} + 6 \, a^{4} c^{19} x - a^{3} c^{19}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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
^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 - 1075
2*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^1
9*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)

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giac [B]  time = 0.25, size = 299, normalized size = 9.65 \[ -\frac {358229025 \, a^{14} x^{14} + 5340869100 \, a^{13} x^{13} + 37114698075 \, a^{12} x^{12} + 159416118225 \, a^{11} x^{11} + 473088806190 \, a^{10} x^{10} + 1026819468675 \, a^{9} x^{9} + 1682288472150 \, a^{8} x^{8} + 2115551402250 \, a^{7} x^{7} + 2054435046125 \, a^{6} x^{6} + 1535397250002 \, a^{5} x^{5} + 870854759775 \, a^{4} x^{4} + 364307533205 \, a^{3} x^{3} + 106553746740 \, a^{2} x^{2} + 19571887695 \, a x + 1710785408}{901943132160 \, {\left (a x + 1\right )}^{15} a^{3} c^{19}} + \frac {358229025 \, a^{20} x^{20} - 7555375800 \, a^{19} x^{19} + 75901131600 \, a^{18} x^{18} - 483051354975 \, a^{17} x^{17} + 2184946607340 \, a^{16} x^{16} - 7469205450840 \, a^{15} x^{15} + 20031221295000 \, a^{14} x^{14} - 43177004037300 \, a^{13} x^{13} + 76013078916950 \, a^{12} x^{12} - 110448380006328 \, a^{11} x^{11} + 133277726128008 \, a^{10} x^{10} - 133908931763530 \, a^{9} x^{9} + 111933156213900 \, a^{8} x^{8} - 77492989590120 \, a^{7} x^{7} + 44041557267624 \, a^{6} x^{6} - 20244576347604 \, a^{5} x^{5} + 7349182966545 \, a^{4} x^{4} - 2026362494800 \, a^{3} x^{3} + 396520754280 \, a^{2} x^{2} - 48177926223 \, a x + 2584181888}{901943132160 \, {\left (a x - 1\right )}^{21} a^{3} c^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^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*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 + 205443504612
5*a^6*x^6 + 1535397250002*a^5*x^5 + 870854759775*a^4*x^4 + 364307533205*a^3*x^3 + 106553746740*a^2*x^2 + 19571
887695*a*x + 1710785408)/((a*x + 1)^15*a^3*c^19) + 1/901943132160*(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 + 2003122
1295000*a^14*x^14 - 43177004037300*a^13*x^13 + 76013078916950*a^12*x^12 - 110448380006328*a^11*x^11 + 13327772
6128008*a^10*x^10 - 133908931763530*a^9*x^9 + 111933156213900*a^8*x^8 - 77492989590120*a^7*x^7 + 4404155726762
4*a^6*x^6 - 20244576347604*a^5*x^5 + 7349182966545*a^4*x^4 - 2026362494800*a^3*x^3 + 396520754280*a^2*x^2 - 48
177926223*a*x + 2584181888)/((a*x - 1)^21*a^3*c^19)

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maple [B]  time = 0.06, size = 426, normalized size = 13.74 \[ \frac {-\frac {3991995}{8589934592 a^{3} \left (a x -1\right )^{4}}+\frac {1964315}{4294967296 a^{3} \left (a x -1\right )^{3}}-\frac {930465}{2147483648 a^{3} \left (a x -1\right )^{2}}-\frac {1}{65536 a^{3} \left (a x -1\right )^{19}}-\frac {858429}{4294967296 a^{3} \left (a x +1\right )^{5}}-\frac {2211105}{8589934592 a^{3} \left (a x +1\right )^{4}}-\frac {1344005}{4294967296 a^{3} \left (a x +1\right )^{3}}-\frac {1550775}{4294967296 a^{3} \left (a x +1\right )^{2}}-\frac {13}{16777216 a^{3} \left (a x +1\right )^{13}}-\frac {1}{62914560 a^{3} \left (a x +1\right )^{15}}-\frac {9}{58720256 a^{3} \left (a x +1\right )^{14}}-\frac {275}{100663296 a^{3} \left (a x +1\right )^{12}}-\frac {253}{33554432 a^{3} \left (a x +1\right )^{11}}-\frac {5819}{335544320 a^{3} \left (a x +1\right )^{10}}-\frac {13915}{402653184 a^{3} \left (a x +1\right )^{9}}-\frac {16445}{268435456 a^{3} \left (a x +1\right )^{8}}-\frac {740025}{7516192768 a^{3} \left (a x +1\right )^{7}}-\frac {312455}{2147483648 a^{3} \left (a x +1\right )^{6}}-\frac {1}{1376256 a^{3} \left (a x -1\right )^{21}}+\frac {3}{655360 a^{3} \left (a x -1\right )^{20}}+\frac {7}{196608 a^{3} \left (a x -1\right )^{18}}-\frac {17}{262144 a^{3} \left (a x -1\right )^{17}}+\frac {51}{524288 a^{3} \left (a x -1\right )^{16}}-\frac {323}{2621440 a^{3} \left (a x -1\right )^{15}}+\frac {969}{7340032 a^{3} \left (a x -1\right )^{14}}-\frac {969}{8388608 a^{3} \left (a x -1\right )^{13}}+\frac {3553}{50331648 a^{3} \left (a x -1\right )^{12}}-\frac {7429}{83886080 a^{3} \left (a x -1\right )^{10}}+\frac {37145}{201326592 a^{3} \left (a x -1\right )^{9}}-\frac {37145}{134217728 a^{3} \left (a x -1\right )^{8}}+\frac {334305}{939524096 a^{3} \left (a x -1\right )^{7}}-\frac {111435}{268435456 a^{3} \left (a x -1\right )^{6}}+\frac {1938969}{4294967296 a^{3} \left (a x -1\right )^{5}}+\frac {3411705}{8589934592 a^{3} \left (a x -1\right )}-\frac {3411705}{8589934592 a^{3} \left (a x +1\right )}}{c^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^19*(-3991995/8589934592/a^3/(a*x-1)^4+1964315/4294967296/a^3/(a*x-1)^3-930465/2147483648/a^3/(a*x-1)^2-1/6
5536/a^3/(a*x-1)^19-858429/4294967296/a^3/(a*x+1)^5-2211105/8589934592/a^3/(a*x+1)^4-1344005/4294967296/a^3/(a
*x+1)^3-1550775/4294967296/a^3/(a*x+1)^2-13/16777216/a^3/(a*x+1)^13-1/62914560/a^3/(a*x+1)^15-9/58720256/a^3/(
a*x+1)^14-275/100663296/a^3/(a*x+1)^12-253/33554432/a^3/(a*x+1)^11-5819/335544320/a^3/(a*x+1)^10-13915/4026531
84/a^3/(a*x+1)^9-16445/268435456/a^3/(a*x+1)^8-740025/7516192768/a^3/(a*x+1)^7-312455/2147483648/a^3/(a*x+1)^6
-1/1376256/a^3/(a*x-1)^21+3/655360/a^3/(a*x-1)^20+7/196608/a^3/(a*x-1)^18-17/262144/a^3/(a*x-1)^17+51/524288/a
^3/(a*x-1)^16-323/2621440/a^3/(a*x-1)^15+969/7340032/a^3/(a*x-1)^14-969/8388608/a^3/(a*x-1)^13+3553/50331648/a
^3/(a*x-1)^12-7429/83886080/a^3/(a*x-1)^10+37145/201326592/a^3/(a*x-1)^9-37145/134217728/a^3/(a*x-1)^8+334305/
939524096/a^3/(a*x-1)^7-111435/268435456/a^3/(a*x-1)^6+1938969/4294967296/a^3/(a*x-1)^5+3411705/8589934592/a^3
/(a*x-1)-3411705/8589934592/a^3/(a*x+1))

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maxima [B]  time = 0.57, size = 379, normalized size = 12.23 \[ -\frac {6 \, a x - 1}{210 \, {\left (a^{39} c^{19} x^{36} - 6 \, a^{38} c^{19} x^{35} + 70 \, a^{36} c^{19} x^{33} - 105 \, a^{35} c^{19} x^{32} - 336 \, a^{34} c^{19} x^{31} + 896 \, a^{33} c^{19} x^{30} + 720 \, a^{32} c^{19} x^{29} - 3900 \, a^{31} c^{19} x^{28} + 280 \, a^{30} c^{19} x^{27} + 10752 \, a^{29} c^{19} x^{26} - 6552 \, a^{28} c^{19} x^{25} - 20020 \, a^{27} c^{19} x^{24} + 21840 \, a^{26} c^{19} x^{23} + 24960 \, a^{25} c^{19} x^{22} - 43472 \, a^{24} c^{19} x^{21} - 18018 \, a^{23} c^{19} x^{20} + 60060 \, a^{22} c^{19} x^{19} - 60060 \, a^{20} c^{19} x^{17} + 18018 \, a^{19} c^{19} x^{16} + 43472 \, a^{18} c^{19} x^{15} - 24960 \, a^{17} c^{19} x^{14} - 21840 \, a^{16} c^{19} x^{13} + 20020 \, a^{15} c^{19} x^{12} + 6552 \, a^{14} c^{19} x^{11} - 10752 \, a^{13} c^{19} x^{10} - 280 \, a^{12} c^{19} x^{9} + 3900 \, a^{11} c^{19} x^{8} - 720 \, a^{10} c^{19} x^{7} - 896 \, a^{9} c^{19} x^{6} + 336 \, a^{8} c^{19} x^{5} + 105 \, a^{7} c^{19} x^{4} - 70 \, a^{6} c^{19} x^{3} + 6 \, a^{4} c^{19} x - a^{3} c^{19}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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
^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 - 1075
2*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^1
9*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)

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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 + 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 = 3.15, size = 405, normalized size = 13.06 \[ \frac {- 6 a x + 1}{210 a^{39} c^{19} x^{36} - 1260 a^{38} c^{19} x^{35} + 14700 a^{36} c^{19} x^{33} - 22050 a^{35} c^{19} x^{32} - 70560 a^{34} c^{19} x^{31} + 188160 a^{33} c^{19} x^{30} + 151200 a^{32} c^{19} x^{29} - 819000 a^{31} c^{19} x^{28} + 58800 a^{30} c^{19} x^{27} + 2257920 a^{29} c^{19} x^{26} - 1375920 a^{28} c^{19} x^{25} - 4204200 a^{27} c^{19} x^{24} + 4586400 a^{26} c^{19} x^{23} + 5241600 a^{25} c^{19} x^{22} - 9129120 a^{24} c^{19} x^{21} - 3783780 a^{23} c^{19} x^{20} + 12612600 a^{22} c^{19} x^{19} - 12612600 a^{20} c^{19} x^{17} + 3783780 a^{19} c^{19} x^{16} + 9129120 a^{18} c^{19} x^{15} - 5241600 a^{17} c^{19} x^{14} - 4586400 a^{16} c^{19} x^{13} + 4204200 a^{15} c^{19} x^{12} + 1375920 a^{14} c^{19} x^{11} - 2257920 a^{13} c^{19} x^{10} - 58800 a^{12} c^{19} x^{9} + 819000 a^{11} c^{19} x^{8} - 151200 a^{10} c^{19} x^{7} - 188160 a^{9} c^{19} x^{6} + 70560 a^{8} c^{19} x^{5} + 22050 a^{7} c^{19} x^{4} - 14700 a^{6} c^{19} x^{3} + 1260 a^{4} c^{19} x - 210 a^{3} c^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-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**2
0 + 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**1
9*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)

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