Integrand size = 19, antiderivative size = 116 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^2} \, dx=\frac {2 \sqrt {1-a^2 x^2}}{3 c^2 x (1-a x)^2}+\frac {14 a \sqrt {1-a^2 x^2}}{3 c^2 (1-a x)}-\frac {5 \sqrt {1-a^2 x^2}}{3 c^2 x (1-a x)}-\frac {3 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{c^2} \] Output:
2/3*(-a^2*x^2+1)^(1/2)/c^2/x/(-a*x+1)^2+14/3*a*(-a^2*x^2+1)^(1/2)/c^2/(-a* x+1)-5/3*(-a^2*x^2+1)^(1/2)/c^2/x/(-a*x+1)-3*a*arctanh((-a^2*x^2+1)^(1/2)) /c^2
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^2} \, dx=\frac {3-16 a x-5 a^2 x^2+14 a^3 x^3-9 a x (-1+a x) \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{3 c^2 x (-1+a x) \sqrt {1-a^2 x^2}} \] Input:
Integrate[E^ArcTanh[a*x]/(x^2*(c - a*c*x)^2),x]
Output:
(3 - 16*a*x - 5*a^2*x^2 + 14*a^3*x^3 - 9*a*x*(-1 + a*x)*Sqrt[1 - a^2*x^2]* ArcTanh[Sqrt[1 - a^2*x^2]])/(3*c^2*x*(-1 + a*x)*Sqrt[1 - a^2*x^2])
Time = 0.45 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {6678, 27, 570, 532, 25, 2336, 27, 534, 243, 73, 221}
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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^2} \, dx\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle c \int \frac {\sqrt {1-a^2 x^2}}{c^3 x^2 (1-a x)^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {1-a^2 x^2}}{x^2 (1-a x)^3}dx}{c^2}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle \frac {\int \frac {(a x+1)^3}{x^2 \left (1-a^2 x^2\right )^{5/2}}dx}{c^2}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {\frac {4 a (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int -\frac {8 a^2 x^2+9 a x+3}{x^2 \left (1-a^2 x^2\right )^{3/2}}dx}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {8 a^2 x^2+9 a x+3}{x^2 \left (1-a^2 x^2\right )^{3/2}}dx+\frac {4 a (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {a (11 a x+9)}{\sqrt {1-a^2 x^2}}-\int -\frac {3 (3 a x+1)}{x^2 \sqrt {1-a^2 x^2}}dx\right )+\frac {4 a (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \int \frac {3 a x+1}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {a (11 a x+9)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (3 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{x}\right )+\frac {a (11 a x+9)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {3}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2}}{x}\right )+\frac {a (11 a x+9)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (-\frac {3 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2}}{x}\right )+\frac {a (11 a x+9)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (-3 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{x}\right )+\frac {a (11 a x+9)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
Input:
Int[E^ArcTanh[a*x]/(x^2*(c - a*c*x)^2),x]
Output:
((4*a*(1 + a*x))/(3*(1 - a^2*x^2)^(3/2)) + ((a*(9 + 11*a*x))/Sqrt[1 - a^2* x^2] + 3*(-(Sqrt[1 - a^2*x^2]/x) - 3*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/3)/c^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*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)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.02
| method | result | size |
| default | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{x}-3 a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {11 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{c^{2}}\) | \(118\) |
| risch | \(\frac {a^{2} x^{2}-1}{x \sqrt {-a^{2} x^{2}+1}\, c^{2}}-\frac {a \left (3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {2 \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{a}+\frac {3 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}\right )}{c^{2}}\) | \(178\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c)^2,x,method=_RETURNVERBOSE)
Output:
1/c^2*(-(-a^2*x^2+1)^(1/2)/x-3*a*arctanh(1/(-a^2*x^2+1)^(1/2))+2/3/a/(x-1/ a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-11/3/(x-1/a)*(-(x-1/a)^2*a^2-2*a*( x-1/a))^(1/2))
Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.02 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^2} \, dx=\frac {13 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 13 \, a x + 9 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (14 \, a^{2} x^{2} - 19 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c)^2,x, algorithm="fricas ")
Output:
1/3*(13*a^3*x^3 - 26*a^2*x^2 + 13*a*x + 9*(a^3*x^3 - 2*a^2*x^2 + a*x)*log( (sqrt(-a^2*x^2 + 1) - 1)/x) - (14*a^2*x^2 - 19*a*x + 3)*sqrt(-a^2*x^2 + 1) )/(a^2*c^2*x^3 - 2*a*c^2*x^2 + c^2*x)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^2} \, dx=\frac {\int \frac {a x}{a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a x^{3} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a x^{3} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2/(-a*c*x+c)**2,x)
Output:
(Integral(a*x/(a**2*x**4*sqrt(-a**2*x**2 + 1) - 2*a*x**3*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**2*x**4*sqrt(-a**2*x* *2 + 1) - 2*a*x**3*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x))/ c**2
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^2} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )}^{2} x^{2}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c)^2,x, algorithm="maxima ")
Output:
integrate((a*x + 1)/(sqrt(-a^2*x^2 + 1)*(a*c*x - c)^2*x^2), x)
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.65 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^2} \, dx=\frac {{\left (9 \, a^{2} \log \left (2\right ) - 18 \, a^{2} \log \left (i + 1\right ) + 28 i \, a^{2}\right )} \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right ) + \frac {18 \, a^{2} \log \left (\sqrt {-\frac {2 \, c}{a c x - c} - 1} + 1\right )}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )} - \frac {18 \, a^{2} \log \left ({\left | \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 1 \right |}\right )}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )} - \frac {6 \, a^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1}}{{\left (\frac {c}{a c x - c} + 1\right )} \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )} - \frac {2 \, {\left (a^{2} {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a c x - c}\right )^{2} \mathrm {sgn}\left (a\right )^{2} \mathrm {sgn}\left (c\right )^{2} + 12 \, a^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} \mathrm {sgn}\left (\frac {1}{a c x - c}\right )^{2} \mathrm {sgn}\left (a\right )^{2} \mathrm {sgn}\left (c\right )^{2}\right )}}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right )^{3} \mathrm {sgn}\left (a\right )^{3} \mathrm {sgn}\left (c\right )^{3}}}{6 \, c^{2} {\left | a \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c)^2,x, algorithm="giac")
Output:
1/6*((9*a^2*log(2) - 18*a^2*log(I + 1) + 28*I*a^2)*sgn(1/(a*c*x - c))*sgn( a)*sgn(c) + 18*a^2*log(sqrt(-2*c/(a*c*x - c) - 1) + 1)/(sgn(1/(a*c*x - c)) *sgn(a)*sgn(c)) - 18*a^2*log(abs(sqrt(-2*c/(a*c*x - c) - 1) - 1))/(sgn(1/( a*c*x - c))*sgn(a)*sgn(c)) - 6*a^2*sqrt(-2*c/(a*c*x - c) - 1)/((c/(a*c*x - c) + 1)*sgn(1/(a*c*x - c))*sgn(a)*sgn(c)) - 2*(a^2*(-2*c/(a*c*x - c) - 1) ^(3/2)*sgn(1/(a*c*x - c))^2*sgn(a)^2*sgn(c)^2 + 12*a^2*sqrt(-2*c/(a*c*x - c) - 1)*sgn(1/(a*c*x - c))^2*sgn(a)^2*sgn(c)^2)/(sgn(1/(a*c*x - c))^3*sgn( a)^3*sgn(c)^3))/(c^2*abs(a))
Time = 13.99 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^2} \, dx=\frac {2\,a^3\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}-\frac {\sqrt {1-a^2\,x^2}}{c^2\,x}+\frac {11\,a^2\,\sqrt {1-a^2\,x^2}}{3\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {a\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{c^2} \] Input:
int((a*x + 1)/(x^2*(1 - a^2*x^2)^(1/2)*(c - a*c*x)^2),x)
Output:
(2*a^3*(1 - a^2*x^2)^(1/2))/(3*(a^2*c^2 - 2*a^3*c^2*x + a^4*c^2*x^2)) - (1 - a^2*x^2)^(1/2)/(c^2*x) + (a*atan((1 - a^2*x^2)^(1/2)*1i)*3i)/c^2 + (11* a^2*(1 - a^2*x^2)^(1/2))/(3*(-a^2)^(1/2)*(c^2*x*(-a^2)^(1/2) - (c^2*(-a^2) ^(1/2))/a))
Time = 0.15 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.43 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^2} \, dx=\frac {a \left (18 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{4}-54 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}+54 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}-18 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+3 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{5}-27 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{4}+48 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}-43 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+3\right )}{6 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right ) c^{2} \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}-3 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}+3 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c)^2,x)
Output:
(a*(18*log(tan(asin(a*x)/2))*tan(asin(a*x)/2)**4 - 54*log(tan(asin(a*x)/2) )*tan(asin(a*x)/2)**3 + 54*log(tan(asin(a*x)/2))*tan(asin(a*x)/2)**2 - 18* log(tan(asin(a*x)/2))*tan(asin(a*x)/2) + 3*tan(asin(a*x)/2)**5 - 27*tan(as in(a*x)/2)**4 + 48*tan(asin(a*x)/2)**2 - 43*tan(asin(a*x)/2) + 3))/(6*tan( asin(a*x)/2)*c**2*(tan(asin(a*x)/2)**3 - 3*tan(asin(a*x)/2)**2 + 3*tan(asi n(a*x)/2) - 1))