3.684 \(\int \frac {e^{-3 \tanh ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^3} \, dx\)

Optimal. Leaf size=159 \[ \frac {(1-a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}-\frac {38 (1-a x)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {137 (1-a x)}{35 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {245-181 a x}{35 a c^3 \sqrt {1-a^2 x^2}}-\frac {3 \sin ^{-1}(a x)}{a c^3} \]

[Out]

1/7*(-a*x+1)^3/a/c^3/(-a^2*x^2+1)^(7/2)-38/35*(-a*x+1)^2/a/c^3/(-a^2*x^2+1)^(5/2)+137/35*(-a*x+1)/a/c^3/(-a^2*
x^2+1)^(3/2)-3*arcsin(a*x)/a/c^3+1/35*(181*a*x-245)/a/c^3/(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2)/a/c^3

________________________________________________________________________________________

Rubi [A]  time = 0.45, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6157, 6149, 1635, 1814, 641, 216} \[ \frac {(1-a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}-\frac {38 (1-a x)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {137 (1-a x)}{35 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {245-181 a x}{35 a c^3 \sqrt {1-a^2 x^2}}-\frac {3 \sin ^{-1}(a x)}{a c^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^3),x]

[Out]

(1 - a*x)^3/(7*a*c^3*(1 - a^2*x^2)^(7/2)) - (38*(1 - a*x)^2)/(35*a*c^3*(1 - a^2*x^2)^(5/2)) + (137*(1 - a*x))/
(35*a*c^3*(1 - a^2*x^2)^(3/2)) - (245 - 181*a*x)/(35*a*c^3*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(a*c^3) - (3
*ArcSin[a*x])/(a*c^3)

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 1635

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, -Simp[(d*f*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*
a*e*(p + 1)), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)*Q
+ f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p +
1/2, 0] && GtQ[m, 0]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 6149

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(x^m*(1 -
a^2*x^2)^(p + n/2))/(1 - a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || G
tQ[c, 0]) && ILtQ[(n - 1)/2, 0] &&  !IntegerQ[p - n/2]

Rule 6157

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[d^p, Int[(u*(1 - a^2*x^
2)^p*E^(n*ArcTanh[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx &=-\frac {a^6 \int \frac {e^{-3 \tanh ^{-1}(a x)} x^6}{\left (1-a^2 x^2\right )^3} \, dx}{c^3}\\ &=-\frac {a^6 \int \frac {x^6 (1-a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^3}\\ &=\frac {(1-a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {a^6 \int \frac {(1-a x)^2 \left (\frac {3}{a^6}-\frac {7 x}{a^5}+\frac {7 x^2}{a^4}-\frac {7 x^3}{a^3}+\frac {7 x^4}{a^2}-\frac {7 x^5}{a}\right )}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^3}\\ &=\frac {(1-a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}-\frac {38 (1-a x)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {a^6 \int \frac {(1-a x) \left (\frac {61}{a^6}-\frac {140 x}{a^5}+\frac {105 x^2}{a^4}-\frac {70 x^3}{a^3}+\frac {35 x^4}{a^2}\right )}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^3}\\ &=\frac {(1-a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}-\frac {38 (1-a x)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {137 (1-a x)}{35 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^6 \int \frac {\frac {228}{a^6}-\frac {630 x}{a^5}+\frac {315 x^2}{a^4}-\frac {105 x^3}{a^3}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{105 c^3}\\ &=\frac {(1-a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}-\frac {38 (1-a x)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {137 (1-a x)}{35 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {245-181 a x}{35 a c^3 \sqrt {1-a^2 x^2}}-\frac {a^6 \int \frac {\frac {315}{a^6}-\frac {105 x}{a^5}}{\sqrt {1-a^2 x^2}} \, dx}{105 c^3}\\ &=\frac {(1-a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}-\frac {38 (1-a x)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {137 (1-a x)}{35 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {245-181 a x}{35 a c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=\frac {(1-a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}-\frac {38 (1-a x)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {137 (1-a x)}{35 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {245-181 a x}{35 a c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {3 \sin ^{-1}(a x)}{a c^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 94, normalized size = 0.59 \[ \frac {35 a^5 x^5+286 a^4 x^4+368 a^3 x^3-125 a^2 x^2-105 (a x+1)^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-423 a x-176}{35 a \sqrt {1-a^2 x^2} (a c x+c)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^3),x]

[Out]

(-176 - 423*a*x - 125*a^2*x^2 + 368*a^3*x^3 + 286*a^4*x^4 + 35*a^5*x^5 - 105*(1 + a*x)^3*Sqrt[1 - a^2*x^2]*Arc
Sin[a*x])/(35*a*(c + a*c*x)^3*Sqrt[1 - a^2*x^2])

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 213, normalized size = 1.34 \[ -\frac {176 \, a^{5} x^{5} + 528 \, a^{4} x^{4} + 352 \, a^{3} x^{3} - 352 \, a^{2} x^{2} - 528 \, a x - 210 \, {\left (a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (35 \, a^{5} x^{5} + 286 \, a^{4} x^{4} + 368 \, a^{3} x^{3} - 125 \, a^{2} x^{2} - 423 \, a x - 176\right )} \sqrt {-a^{2} x^{2} + 1} - 176}{35 \, {\left (a^{6} c^{3} x^{5} + 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} - 3 \, a^{2} c^{3} x - a c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(176*a^5*x^5 + 528*a^4*x^4 + 352*a^3*x^3 - 352*a^2*x^2 - 528*a*x - 210*(a^5*x^5 + 3*a^4*x^4 + 2*a^3*x^3
- 2*a^2*x^2 - 3*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (35*a^5*x^5 + 286*a^4*x^4 + 368*a^3*x^3 - 12
5*a^2*x^2 - 423*a*x - 176)*sqrt(-a^2*x^2 + 1) - 176)/(a^6*c^3*x^5 + 3*a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^3*c^3*
x^2 - 3*a^2*c^3*x - a*c^3)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.07, size = 494, normalized size = 3.11 \[ -\frac {529 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{256 a \,c^{3}}+\frac {5 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{16 a^{5} c^{3} \left (x +\frac {1}{a}\right )^{4}}-\frac {31 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{32 a^{4} c^{3} \left (x +\frac {1}{a}\right )^{3}}-\frac {263 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{128 a^{3} c^{3} \left (x +\frac {1}{a}\right )^{2}}-\frac {1587 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{512 c^{3}}-\frac {1587 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{512 c^{3} \sqrt {a^{2}}}+\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{56 a^{7} c^{3} \left (x +\frac {1}{a}\right )^{6}}-\frac {61 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{560 a^{6} c^{3} \left (x +\frac {1}{a}\right )^{5}}-\frac {5 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{64 a^{3} c^{3} \left (x -\frac {1}{a}\right )^{2}}+\frac {51 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}\, x}{512 c^{3}}+\frac {51 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{512 c^{3} \sqrt {a^{2}}}+\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{64 a^{4} c^{3} \left (x -\frac {1}{a}\right )^{3}}-\frac {17 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{256 a \,c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-529/256/a/c^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+5/16/a^5/c^3/(x+1/a)^4*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-31
/32/a^4/c^3/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-263/128/a^3/c^3/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a)
)^(5/2)-1587/512/c^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)*x-1587/512/c^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*
(x+1/a)^2+2*a*(x+1/a))^(1/2))+1/56/a^7/c^3/(x+1/a)^6*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-61/560/a^6/c^3/(x+1/a)
^5*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-5/64/a^3/c^3/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)+51/512/c^3*(-a
^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)*x+51/512/c^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/
2))+1/64/a^4/c^3/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)-17/256/a/c^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(3/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.44, size = 436, normalized size = 2.74 \[ \frac {49\,a\,\sqrt {1-a^2\,x^2}}{24\,\left (a^4\,c^3\,x^2+2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^3\,\sqrt {-a^2}}+\frac {a^3\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^3\,x^2+2\,a^5\,c^3\,x+a^4\,c^3\right )}-\frac {11\,a^6\,\sqrt {1-a^2\,x^2}}{30\,\left (a^9\,c^3\,x^2+2\,a^8\,c^3\,x+a^7\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^3}+\frac {a\,\sqrt {1-a^2\,x^2}}{14\,\left (a^6\,c^3\,x^4+4\,a^5\,c^3\,x^3+6\,a^4\,c^3\,x^2+4\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {2931\,\sqrt {1-a^2\,x^2}}{560\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}+\frac {c^3\,\sqrt {-a^2}}{a}\right )}-\frac {\sqrt {1-a^2\,x^2}}{16\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {71\,\sqrt {1-a^2\,x^2}}{140\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}+\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}+3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(49*a*(1 - a^2*x^2)^(1/2))/(24*(a^2*c^3 + 2*a^3*c^3*x + a^4*c^3*x^2)) - (3*asinh(x*(-a^2)^(1/2)))/(c^3*(-a^2)^
(1/2)) + (a^3*(1 - a^2*x^2)^(1/2))/(35*(a^4*c^3 + 2*a^5*c^3*x + a^6*c^3*x^2)) - (11*a^6*(1 - a^2*x^2)^(1/2))/(
30*(a^7*c^3 + 2*a^8*c^3*x + a^9*c^3*x^2)) - (1 - a^2*x^2)^(1/2)/(a*c^3) + (a*(1 - a^2*x^2)^(1/2))/(14*(a^2*c^3
+ 4*a^3*c^3*x + 6*a^4*c^3*x^2 + 4*a^5*c^3*x^3 + a^6*c^3*x^4)) + (2931*(1 - a^2*x^2)^(1/2))/(560*(-a^2)^(1/2)*
(c^3*x*(-a^2)^(1/2) + (c^3*(-a^2)^(1/2))/a)) - (1 - a^2*x^2)^(1/2)/(16*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) - (c^3
*(-a^2)^(1/2))/a)) + (71*(1 - a^2*x^2)^(1/2))/(140*(-a^2)^(1/2)*(3*c^3*x*(-a^2)^(1/2) + (c^3*(-a^2)^(1/2))/a +
a^2*c^3*x^3*(-a^2)^(1/2) + 3*a*c^3*x^2*(-a^2)^(1/2)))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{6} \left (\int \frac {x^{6} \sqrt {- a^{2} x^{2} + 1}}{a^{9} x^{9} + 3 a^{8} x^{8} - 8 a^{6} x^{6} - 6 a^{5} x^{5} + 6 a^{4} x^{4} + 8 a^{3} x^{3} - 3 a x - 1}\, dx + \int \left (- \frac {a^{2} x^{8} \sqrt {- a^{2} x^{2} + 1}}{a^{9} x^{9} + 3 a^{8} x^{8} - 8 a^{6} x^{6} - 6 a^{5} x^{5} + 6 a^{4} x^{4} + 8 a^{3} x^{3} - 3 a x - 1}\right )\, dx\right )}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**6*(Integral(x**6*sqrt(-a**2*x**2 + 1)/(a**9*x**9 + 3*a**8*x**8 - 8*a**6*x**6 - 6*a**5*x**5 + 6*a**4*x**4 +
8*a**3*x**3 - 3*a*x - 1), x) + Integral(-a**2*x**8*sqrt(-a**2*x**2 + 1)/(a**9*x**9 + 3*a**8*x**8 - 8*a**6*x**6
- 6*a**5*x**5 + 6*a**4*x**4 + 8*a**3*x**3 - 3*a*x - 1), x))/c**3

________________________________________________________________________________________