Integrand size = 20, antiderivative size = 77 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=-\frac {8 c (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {1-a^2 x^2}}{a}-\frac {4 c \arcsin (a x)}{a}+\frac {c \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{a} \] Output:
-8*c*(-a*x+1)/a/(-a^2*x^2+1)^(1/2)-c*(-a^2*x^2+1)^(1/2)/a-4*c*arcsin(a*x)/ a+c*arctanh((-a^2*x^2+1)^(1/2))/a
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.17 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.03 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c \left (10 \left (4 (-1+a x) \sqrt {1+a x}+2 \sqrt {1-a x} (1+a x) \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )+\sqrt {1+a x} \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )+\sqrt {2} (-1+a x)^3 (1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},\frac {1}{2} (1-a x)\right )\right )}{10 a \sqrt {1-a x} (1+a x)} \] Input:
Integrate[(c - c/(a*x))/E^(3*ArcTanh[a*x]),x]
Output:
(c*(10*(4*(-1 + a*x)*Sqrt[1 + a*x] + 2*Sqrt[1 - a*x]*(1 + a*x)*ArcSin[Sqrt [1 - a*x]/Sqrt[2]] + Sqrt[1 + a*x]*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2* x^2]]) + Sqrt[2]*(-1 + a*x)^3*(1 + a*x)*Hypergeometric2F1[3/2, 5/2, 7/2, ( 1 - a*x)/2]))/(10*a*Sqrt[1 - a*x]*(1 + a*x))
Time = 0.47 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.86, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {6681, 6678, 528, 2340, 25, 27, 538, 223, 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.
| \(\displaystyle \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx\) |
| \(\Big \downarrow \) 6681 |
| \(\displaystyle -\frac {c \int \frac {e^{-3 \text {arctanh}(a x)} (1-a x)}{x}dx}{a}\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle -\frac {c \int \frac {(1-a x)^4}{x \left (1-a^2 x^2\right )^{3/2}}dx}{a}\) |
| \(\Big \downarrow \) 528 |
| \(\displaystyle -\frac {c \left (\int \frac {-a^2 x^2+4 a x+1}{x \sqrt {1-a^2 x^2}}dx+\frac {8 (1-a x)}{\sqrt {1-a^2 x^2}}\right )}{a}\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {c \left (-\frac {\int -\frac {a^2 (4 a x+1)}{x \sqrt {1-a^2 x^2}}dx}{a^2}+\frac {8 (1-a x)}{\sqrt {1-a^2 x^2}}+\sqrt {1-a^2 x^2}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {c \left (\frac {\int \frac {a^2 (4 a x+1)}{x \sqrt {1-a^2 x^2}}dx}{a^2}+\frac {8 (1-a x)}{\sqrt {1-a^2 x^2}}+\sqrt {1-a^2 x^2}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c \left (\int \frac {4 a x+1}{x \sqrt {1-a^2 x^2}}dx+\frac {8 (1-a x)}{\sqrt {1-a^2 x^2}}+\sqrt {1-a^2 x^2}\right )}{a}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -\frac {c \left (4 a \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\int \frac {1}{x \sqrt {1-a^2 x^2}}dx+\frac {8 (1-a x)}{\sqrt {1-a^2 x^2}}+\sqrt {1-a^2 x^2}\right )}{a}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {c \left (\int \frac {1}{x \sqrt {1-a^2 x^2}}dx+\frac {8 (1-a x)}{\sqrt {1-a^2 x^2}}+\sqrt {1-a^2 x^2}+4 \arcsin (a x)\right )}{a}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {c \left (\frac {1}{2} \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2+\frac {8 (1-a x)}{\sqrt {1-a^2 x^2}}+\sqrt {1-a^2 x^2}+4 \arcsin (a x)\right )}{a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c \left (-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}+\frac {8 (1-a x)}{\sqrt {1-a^2 x^2}}+\sqrt {1-a^2 x^2}+4 \arcsin (a x)\right )}{a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+\frac {8 (1-a x)}{\sqrt {1-a^2 x^2}}+\sqrt {1-a^2 x^2}+4 \arcsin (a x)\right )}{a}\) |
Input:
Int[(c - c/(a*x))/E^(3*ArcTanh[a*x]),x]
Output:
-((c*((8*(1 - a*x))/Sqrt[1 - a^2*x^2] + Sqrt[1 - a^2*x^2] + 4*ArcSin[a*x] - ArcTanh[Sqrt[1 - a^2*x^2]]))/a)
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[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[(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)^(3/2), x_Sy mbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b *x^2])), x] + Simp[c^2/a Int[(x^m/Sqrt[a + b*x^2])*ExpandToSum[((c + d*x) ^(n - 1) - (2^(n - 1)*c^(m + n - 1))/(d^m*x^m))/(c - d*x), x], x], x] /; Fr eeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 )*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m + q + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) *Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ [Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
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]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol ] :> Simp[d^p Int[u*(1 + c*(x/d))^p*(E^(n*ArcTanh[a*x])/x^p), x], x] /; F reeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(497\) vs. \(2(71)=142\).
Time = 0.26 (sec) , antiderivative size = 498, normalized size of antiderivative = 6.47
| method | result | size |
| default | \(\frac {c \left (\frac {\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )}{a}+\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}-\sqrt {-a^{2} x^{2}+1}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {-\frac {2 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-4 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{2}}\right )}{a}\) | \(498\) |
Input:
int((c-c/a/x)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
c/a*(1/a*(1/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)+3*a*(1/3*(-a^2* (x+1/a)^2+2*a*(x+1/a))^(3/2)+a*(-1/4*(-2*a^2*(x+1/a)+2*a)/a^2*(-a^2*(x+1/a )^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^ 2+2*a*(x+1/a))^(1/2)))))+1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+a*(-1/4*(- 2*a^2*(x+1/a)+2*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)* arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)))-1/3*(-a^2*x^2+1) ^(3/2)-(-a^2*x^2+1)^(1/2)+arctanh(1/(-a^2*x^2+1)^(1/2))+2/a^2*(-1/a/(x+1/a )^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-2*a*(1/a/(x+1/a)^2*(-a^2*(x+1/a)^2+ 2*a*(x+1/a))^(5/2)+3*a*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+a*(-1/4*(-2 *a^2*(x+1/a)+2*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)*a rctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)))))))
Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.26 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=-\frac {9 \, a c x - 8 \, {\left (a c x + c\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (a c x + c\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a c x + 9 \, c\right )} + 9 \, c}{a^{2} x + a} \] Input:
integrate((c-c/a/x)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="fricas")
Output:
-(9*a*c*x - 8*(a*c*x + c)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (a*c*x + c)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + sqrt(-a^2*x^2 + 1)*(a*c*x + 9*c) + 9*c)/(a^2*x + a)
\[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c \left (\int \left (- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{4} + 3 a^{2} x^{3} + 3 a x^{2} + x}\right )\, dx + \int \frac {a x \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{4} + 3 a^{2} x^{3} + 3 a x^{2} + x}\, dx + \int \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{4} + 3 a^{2} x^{3} + 3 a x^{2} + x}\, dx + \int \left (- \frac {a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{4} + 3 a^{2} x^{3} + 3 a x^{2} + x}\right )\, dx\right )}{a} \] Input:
integrate((c-c/a/x)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)
Output:
c*(Integral(-sqrt(-a**2*x**2 + 1)/(a**3*x**4 + 3*a**2*x**3 + 3*a*x**2 + x) , x) + Integral(a*x*sqrt(-a**2*x**2 + 1)/(a**3*x**4 + 3*a**2*x**3 + 3*a*x* *2 + x), x) + Integral(a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**3*x**4 + 3*a**2* x**3 + 3*a*x**2 + x), x) + Integral(-a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**3* x**4 + 3*a**2*x**3 + 3*a*x**2 + x), x))/a
\[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}}{{\left (a x + 1\right )}^{3}} \,d x } \] Input:
integrate((c-c/a/x)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxima")
Output:
integrate((-a^2*x^2 + 1)^(3/2)*(c - c/(a*x))/(a*x + 1)^3, x)
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.35 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=-\frac {4 \, c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c}{a} + \frac {16 \, c}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \] Input:
integrate((c-c/a/x)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac")
Output:
-4*c*arcsin(a*x)*sgn(a)/abs(a) + c*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a ) - 2*a)/(a^2*abs(x)))/abs(a) - sqrt(-a^2*x^2 + 1)*c/a + 16*c/(((sqrt(-a^2 *x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a))
Time = 14.83 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.32 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{a}-\frac {c\,\sqrt {1-a^2\,x^2}}{a}-\frac {4\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {8\,c\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \] Input:
int(((c - c/(a*x))*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3,x)
Output:
(c*atanh((1 - a^2*x^2)^(1/2)))/a - (c*(1 - a^2*x^2)^(1/2))/a - (4*c*asinh( x*(-a^2)^(1/2)))/(-a^2)^(1/2) + (8*c*(1 - a^2*x^2)^(1/2))/((x*(-a^2)^(1/2) + (-a^2)^(1/2)/a)*(-a^2)^(1/2))
Time = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.74 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c \left (-4 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+4 \mathit {asin} \left (a x \right ) a x +4 \mathit {asin} \left (a x \right )-\sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right )+\sqrt {-a^{2} x^{2}+1}\, a x +16 \sqrt {-a^{2} x^{2}+1}+\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a x +\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right )+a^{2} x^{2}+a x -16\right )}{a \left (\sqrt {-a^{2} x^{2}+1}-a x -1\right )} \] Input:
int((c-c/a/x)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
Output:
(c*( - 4*sqrt( - a**2*x**2 + 1)*asin(a*x) + 4*asin(a*x)*a*x + 4*asin(a*x) - sqrt( - a**2*x**2 + 1)*log(tan(asin(a*x)/2)) + sqrt( - a**2*x**2 + 1)*a* x + 16*sqrt( - a**2*x**2 + 1) + log(tan(asin(a*x)/2))*a*x + log(tan(asin(a *x)/2)) + a**2*x**2 + a*x - 16))/(a*(sqrt( - a**2*x**2 + 1) - a*x - 1))