Integrand size = 23, antiderivative size = 123 \[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {1+a x}{5 a^7 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {15+11 a x}{15 a^7 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {45+23 a x}{15 a^7 c^3 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a^7 c^3}-\frac {\arcsin (a x)}{a^7 c^3} \] Output:
1/5*(a*x+1)/a^7/c^3/(-a^2*x^2+1)^(5/2)-1/15*(11*a*x+15)/a^7/c^3/(-a^2*x^2+ 1)^(3/2)+1/15*(23*a*x+45)/a^7/c^3/(-a^2*x^2+1)^(1/2)+(-a^2*x^2+1)^(1/2)/a^ 7/c^3-arcsin(a*x)/a^7/c^3
Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {48-33 a x-87 a^2 x^2+52 a^3 x^3+38 a^4 x^4-15 a^5 x^5-15 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2} \arcsin (a x)}{15 a^7 c^3 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^ArcTanh[a*x]*x^6)/(c - a^2*c*x^2)^3,x]
Output:
(48 - 33*a*x - 87*a^2*x^2 + 52*a^3*x^3 + 38*a^4*x^4 - 15*a^5*x^5 - 15*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(15*a^7*c^3*(-1 + a*x)^2 *(1 + a*x)*Sqrt[1 - a^2*x^2])
Time = 0.80 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6698, 529, 2345, 2345, 27, 455, 223}
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.
\(\displaystyle \int \frac {x^6 e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \frac {\int \frac {x^6 (a x+1)}{\left (1-a^2 x^2\right )^{7/2}}dx}{c^3}\) |
\(\Big \downarrow \) 529 |
\(\displaystyle \frac {\frac {a x+1}{5 a^7 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{5} \int \frac {\frac {5 x^5}{a}+\frac {5 x^4}{a^2}+\frac {5 x^3}{a^3}+\frac {5 x^2}{a^4}+\frac {5 x}{a^5}+\frac {1}{a^6}}{\left (1-a^2 x^2\right )^{5/2}}dx}{c^3}\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {\frac {15 x^3}{a^3}+\frac {15 x^2}{a^4}+\frac {30 x}{a^5}+\frac {8}{a^6}}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {11 a x+15}{3 a^7 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^7 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {23 a x+45}{a^7 \sqrt {1-a^2 x^2}}-\int \frac {15 (a x+1)}{a^6 \sqrt {1-a^2 x^2}}dx\right )-\frac {11 a x+15}{3 a^7 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^7 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {23 a x+45}{a^7 \sqrt {1-a^2 x^2}}-\frac {15 \int \frac {a x+1}{\sqrt {1-a^2 x^2}}dx}{a^6}\right )-\frac {11 a x+15}{3 a^7 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^7 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {23 a x+45}{a^7 \sqrt {1-a^2 x^2}}-\frac {15 \left (\int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^6}\right )-\frac {11 a x+15}{3 a^7 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^7 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\frac {a x+1}{5 a^7 \left (1-a^2 x^2\right )^{5/2}}+\frac {1}{5} \left (\frac {1}{3} \left (\frac {23 a x+45}{a^7 \sqrt {1-a^2 x^2}}-\frac {15 \left (\frac {\arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^6}\right )-\frac {11 a x+15}{3 a^7 \left (1-a^2 x^2\right )^{3/2}}\right )}{c^3}\) |
Input:
Int[(E^ArcTanh[a*x]*x^6)/(c - a^2*c*x^2)^3,x]
Output:
((1 + a*x)/(5*a^7*(1 - a^2*x^2)^(5/2)) + (-1/3*(15 + 11*a*x)/(a^7*(1 - a^2 *x^2)^(3/2)) + ((45 + 23*a*x)/(a^7*Sqrt[1 - a^2*x^2]) - (15*(-(Sqrt[1 - a^ 2*x^2]/a) + ArcSin[a*x]/a))/a^6)/3)/5)/c^3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ (2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b* x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* c^2 + a*d^2, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[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] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[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] || GtQ[c, 0]) && IGtQ[(n + 1)/2, 0] && !IntegerQ[p - n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(258\) vs. \(2(109)=218\).
Time = 0.22 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.11
method | result | size |
risch | \(-\frac {a^{2} x^{2}-1}{a^{7} \sqrt {-a^{2} x^{2}+1}\, c^{3}}-\frac {\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{6} \sqrt {a^{2}}}+\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{20 a^{10} \left (x -\frac {1}{a}\right )^{3}}+\frac {23 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{60 a^{9} \left (x -\frac {1}{a}\right )^{2}}+\frac {493 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{240 a^{8} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{24 a^{9} \left (x +\frac {1}{a}\right )^{2}}-\frac {25 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{48 a^{8} \left (x +\frac {1}{a}\right )}}{c^{3}}\) | \(259\) |
default | \(-\frac {\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{6} \sqrt {a^{2}}}-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{7}}-\frac {9 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 a^{8} \left (x +\frac {1}{a}\right )}+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{4 a^{9}}+\frac {\frac {5 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{12 a \left (x -\frac {1}{a}\right )^{2}}-\frac {5 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{12 \left (x -\frac {1}{a}\right )}}{a^{8}}+\frac {39 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{16 a^{8} \left (x -\frac {1}{a}\right )}-\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}}{8 a^{8}}}{c^{3}}\) | \(417\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^6/(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOS E)
Output:
-1/a^7*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)/c^3-(1/a^6/(a^2)^(1/2)*arctan((a^2)^ (1/2)*x/(-a^2*x^2+1)^(1/2))+1/20/a^10/(x-1/a)^3*(-(x-1/a)^2*a^2-2*a*(x-1/a ))^(1/2)+23/60/a^9/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+493/240/a^ 8/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+1/24/a^9/(x+1/a)^2*(-a^2*(x+1 /a)^2+2*a*(x+1/a))^(1/2)-25/48/a^8/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1 /2))/c^3
Time = 0.08 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.73 \[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {48 \, a^{5} x^{5} - 48 \, a^{4} x^{4} - 96 \, a^{3} x^{3} + 96 \, a^{2} x^{2} + 48 \, a x + 30 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{5} x^{5} - 38 \, a^{4} x^{4} - 52 \, a^{3} x^{3} + 87 \, a^{2} x^{2} + 33 \, a x - 48\right )} \sqrt {-a^{2} x^{2} + 1} - 48}{15 \, {\left (a^{12} c^{3} x^{5} - a^{11} c^{3} x^{4} - 2 \, a^{10} c^{3} x^{3} + 2 \, a^{9} c^{3} x^{2} + a^{8} c^{3} x - a^{7} c^{3}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^6/(-a^2*c*x^2+c)^3,x, algorithm="fr icas")
Output:
1/15*(48*a^5*x^5 - 48*a^4*x^4 - 96*a^3*x^3 + 96*a^2*x^2 + 48*a*x + 30*(a^5 *x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^5*x^5 - 38*a^4*x^4 - 52*a^3*x^3 + 87*a^2*x^2 + 33*a *x - 48)*sqrt(-a^2*x^2 + 1) - 48)/(a^12*c^3*x^5 - a^11*c^3*x^4 - 2*a^10*c^ 3*x^3 + 2*a^9*c^3*x^2 + a^8*c^3*x - a^7*c^3)
\[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{6}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{7}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**6/(-a**2*c*x**2+c)**3,x)
Output:
(Integral(x**6/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x **2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**7/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2* x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/ c**3
\[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {{\left (a x + 1\right )} x^{6}}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^6/(-a^2*c*x^2+c)^3,x, algorithm="ma xima")
Output:
-integrate((a*x + 1)*x^6/((a^2*c*x^2 - c)^3*sqrt(-a^2*x^2 + 1)), x)
\[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {{\left (a x + 1\right )} x^{6}}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^6/(-a^2*c*x^2+c)^3,x, algorithm="gi ac")
Output:
integrate(-(a*x + 1)*x^6/((a^2*c*x^2 - c)^3*sqrt(-a^2*x^2 + 1)), x)
Time = 0.09 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.08 \[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {a^6\,\sqrt {1-a^2\,x^2}}{30\,\left (a^{15}\,c^3\,x^2-2\,a^{14}\,c^3\,x+a^{13}\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{24\,\left (a^9\,c^3\,x^2+2\,a^8\,c^3\,x+a^7\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (a^5\,c^3\,\sqrt {-a^2}+3\,a^7\,c^3\,x^2\,\sqrt {-a^2}-a^8\,c^3\,x^3\,\sqrt {-a^2}-3\,a^6\,c^3\,x\,\sqrt {-a^2}\right )}-\frac {5\,\sqrt {1-a^2\,x^2}}{12\,\left (a^9\,c^3\,x^2-2\,a^8\,c^3\,x+a^7\,c^3\right )}-\frac {25\,\sqrt {1-a^2\,x^2}}{48\,\left (a^5\,c^3\,\sqrt {-a^2}+a^6\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {493\,\sqrt {1-a^2\,x^2}}{240\,\left (a^5\,c^3\,\sqrt {-a^2}-a^6\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}}{a^7\,c^3}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^6\,c^3\,\sqrt {-a^2}} \] Input:
int((x^6*(a*x + 1))/((c - a^2*c*x^2)^3*(1 - a^2*x^2)^(1/2)),x)
Output:
(a^6*(1 - a^2*x^2)^(1/2))/(30*(a^13*c^3 - 2*a^14*c^3*x + a^15*c^3*x^2)) - (1 - a^2*x^2)^(1/2)/(24*(a^7*c^3 + 2*a^8*c^3*x + a^9*c^3*x^2)) - (1 - a^2* x^2)^(1/2)/(20*(-a^2)^(1/2)*(a^5*c^3*(-a^2)^(1/2) + 3*a^7*c^3*x^2*(-a^2)^( 1/2) - a^8*c^3*x^3*(-a^2)^(1/2) - 3*a^6*c^3*x*(-a^2)^(1/2))) - (5*(1 - a^2 *x^2)^(1/2))/(12*(a^7*c^3 - 2*a^8*c^3*x + a^9*c^3*x^2)) - (25*(1 - a^2*x^2 )^(1/2))/(48*(a^5*c^3*(-a^2)^(1/2) + a^6*c^3*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (493*(1 - a^2*x^2)^(1/2))/(240*(a^5*c^3*(-a^2)^(1/2) - a^6*c^3*x*(-a^2) ^(1/2))*(-a^2)^(1/2)) + (1 - a^2*x^2)^(1/2)/(a^7*c^3) - asinh(x*(-a^2)^(1/ 2))/(a^6*c^3*(-a^2)^(1/2))
Time = 0.18 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.87 \[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {-15 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{3} x^{3}+15 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{2} x^{2}+15 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x -15 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+33 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-33 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-33 \sqrt {-a^{2} x^{2}+1}\, a x +33 \sqrt {-a^{2} x^{2}+1}-15 a^{5} x^{5}+38 a^{4} x^{4}+52 a^{3} x^{3}-87 a^{2} x^{2}-33 a x +48}{15 \sqrt {-a^{2} x^{2}+1}\, a^{7} c^{3} \left (a^{3} x^{3}-a^{2} x^{2}-a x +1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^6/(-a^2*c*x^2+c)^3,x)
Output:
( - 15*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**3*x**3 + 15*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**2*x**2 + 15*sqrt( - a**2*x**2 + 1)*asin(a*x)*a*x - 15*sqr t( - a**2*x**2 + 1)*asin(a*x) + 33*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 33*s qrt( - a**2*x**2 + 1)*a**2*x**2 - 33*sqrt( - a**2*x**2 + 1)*a*x + 33*sqrt( - a**2*x**2 + 1) - 15*a**5*x**5 + 38*a**4*x**4 + 52*a**3*x**3 - 87*a**2*x **2 - 33*a*x + 48)/(15*sqrt( - a**2*x**2 + 1)*a**7*c**3*(a**3*x**3 - a**2* x**2 - a*x + 1))