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

________________________________________________________________________________________

Rubi [A]  time = 0.0914841, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, 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 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])

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]

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*}

Mathematica [A]  time = 0.974625, 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)

________________________________________________________________________________________

Maple [B]  time = 0.059, size = 426, normalized size = 13.7 \begin{align*} \text{result too large to display} \end{align*}

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*(-13/16777216/a^3/(a*x+1)^13+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+334305/93952
4096/a^3/(a*x-1)^7-111435/268435456/a^3/(a*x-1)^6+1938969/4294967296/a^3/(a*x-1)^5-3991995/8589934592/a^3/(a*x
-1)^4+1964315/4294967296/a^3/(a*x-1)^3-930465/2147483648/a^3/(a*x-1)^2+3411705/8589934592/a^3/(a*x-1)-1/65536/
a^3/(a*x-1)^19-3411705/8589934592/a^3/(a*x+1)-1/62914560/a^3/(a*x+1)^15-9/58720256/a^3/(a*x+1)^14-275/10066329
6/a^3/(a*x+1)^12-253/33554432/a^3/(a*x+1)^11-5819/335544320/a^3/(a*x+1)^10-13915/402653184/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)^16-323/2621
440/a^3/(a*x-1)^15-858429/4294967296/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)

________________________________________________________________________________________

Maxima [B]  time = 2.00376, size = 512, normalized size = 16.52 \begin{align*} -\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 )}} \end{align*}

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)

________________________________________________________________________________________

Fricas [B]  time = 5.52788, size = 987, normalized size = 31.84 \begin{align*} -\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 )}} \end{align*}

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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.22541, size = 404, normalized size = 13.03 \begin{align*} -\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}} \end{align*}

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)