Integrand size = 20, antiderivative size = 104 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {a+\frac {1}{x}}{3 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {3 a+\frac {5}{x}}{3 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2} \] Output:
-1/3*(a+1/x)/a^2/c^2/(1-1/a^2/x^2)^(3/2)-1/3*(3*a+5/x)/a^2/c^2/(1-1/a^2/x^ 2)^(1/2)+(1-1/a^2/x^2)^(1/2)*x/c^2+arctanh((1-1/a^2/x^2)^(1/2))/a/c^2
Time = 0.46 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (8-5 a x-7 a^2 x^2+3 a^3 x^3\right )}{3 (-1+a x)^2 (1+a x)}+\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^2} \] Input:
Integrate[E^ArcCoth[a*x]/(c - c/(a^2*x^2))^2,x]
Output:
((a*Sqrt[1 - 1/(a^2*x^2)]*x*(8 - 5*a*x - 7*a^2*x^2 + 3*a^3*x^3))/(3*(-1 + a*x)^2*(1 + a*x)) + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^2)
Time = 0.63 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.63, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6748, 114, 25, 27, 169, 25, 27, 169, 25, 27, 169, 27, 103, 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.
| \(\displaystyle \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -\frac {\int \frac {x^2}{\left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{c^2}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {-\int -\frac {\left (a+\frac {3}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\left (a+\frac {3}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\left (a+\frac {3}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{c^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}-\frac {1}{3} a \int -\frac {\left (3 a+\frac {8}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} a \int \frac {\left (3 a+\frac {8}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \int \frac {\left (3 a+\frac {8}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{c^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {11 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-a \int -\frac {\left (3 a+\frac {11}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}\right )+\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (a \int \frac {\left (3 a+\frac {11}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {11 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}\right )+\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\int \frac {\left (3 a+\frac {11}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {11 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}\right )+\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{c^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (a \int \frac {3 x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {8 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {11 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}\right )+\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (3 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {8 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {11 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}\right )+\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{c^2}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (-3 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )-\frac {8 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {11 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}\right )+\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{c^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (-3 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {8 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {11 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}\right )+\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}}{c^2}\) |
Input:
Int[E^ArcCoth[a*x]/(c - c/(a^2*x^2))^2,x]
Output:
-((-(x/((1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)])) + ((4*a)/(3*(1 - 1/(a*x))^ (3/2)*Sqrt[1 + 1/(a*x)]) + ((11*a)/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (8*a*Sqrt[1 - 1/(a*x)])/Sqrt[1 + 1/(a*x)] - 3*a*ArcTanh[Sqrt[1 - 1/(a*x)] *Sqrt[1 + 1/(a*x)]])/3)/a^2)/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[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
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[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Simp[-c^p Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/a)^(p + n/2)/x^2), x], x , 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[ n/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegersQ[2*p, p + n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(217\) vs. \(2(92)=184\).
Time = 0.16 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.10
| method | result | size |
| risch | \(\frac {a x -1}{a \,c^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}-\frac {19 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{12 a^{6} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{6 a^{7} \left (x -\frac {1}{a}\right )^{2}}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{4 a^{6} \left (x +\frac {1}{a}\right )}\right ) a^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(218\) |
| default | \(-\frac {-45 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-24 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+21 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{3} x^{3}+45 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}+24 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+11 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+90 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+48 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-5 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -90 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-48 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-19 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-45 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x -24 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +45 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+24 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{24 a \left (a x +1\right )^{2} \sqrt {a^{2}}\, \left (a x -1\right )^{2} c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}\) | \(530\) |
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^2,x,method=_RETURNVERBOSE)
Output:
1/a*(a*x-1)/c^2/((a*x-1)/(a*x+1))^(1/2)+(1/a^4*ln(a^2*x/(a^2)^(1/2)+(a^2*x ^2-1)^(1/2))/(a^2)^(1/2)-19/12/a^6/(x-1/a)*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/ 2)-1/6/a^7/(x-1/a)^2*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)+1/4/a^6/(x+1/a)*(a^ 2*(x+1/a)^2-2*a*(x+1/a))^(1/2))*a^4/c^2/((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*( a*x+1))^(1/2)/(a*x+1)
Time = 0.09 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 5 \, a x + 8\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^2,x, algorithm="fricas")
Output:
1/3*(3*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 3*(a^2*x ^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (3*a^3*x^3 - 7*a^2*x^ 2 - 5*a*x + 8)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c ^2)
\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {a^{4} \int \frac {x^{4}}{a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 2 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{2}} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)/(c-c/a**2/x**2)**2,x)
Output:
a**4*Integral(x**4/(a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - 2*a**2*x **2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) + sqrt(a*x/(a*x + 1) - 1/(a*x + 1))) , x)/c**2
Time = 0.03 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.54 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {1}{12} \, a {\left (\frac {\frac {17 \, {\left (a x - 1\right )}}{a x + 1} - \frac {42 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}} + \frac {3 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^2,x, algorithm="maxima")
Output:
1/12*a*((17*(a*x - 1)/(a*x + 1) - 42*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c^2 *((a*x - 1)/(a*x + 1))^(5/2) - a^2*c^2*((a*x - 1)/(a*x + 1))^(3/2)) + 12*l og(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^2) - 12*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^2) + 3*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^2))
\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^2,x, algorithm="giac")
Output:
integrate(1/((c - c/(a^2*x^2))^2*sqrt((a*x - 1)/(a*x + 1))), x)
Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.23 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{4\,a\,c^2}-\frac {\frac {17\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {14\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}+\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^2} \] Input:
int(1/((c - c/(a^2*x^2))^2*((a*x - 1)/(a*x + 1))^(1/2)),x)
Output:
((a*x - 1)/(a*x + 1))^(1/2)/(4*a*c^2) - ((17*(a*x - 1))/(3*(a*x + 1)) - (1 4*(a*x - 1)^2)/(a*x + 1)^2 + 1/3)/(4*a*c^2*((a*x - 1)/(a*x + 1))^(3/2) - 4 *a*c^2*((a*x - 1)/(a*x + 1))^(5/2)) + (2*atanh(((a*x - 1)/(a*x + 1))^(1/2) ))/(a*c^2)
Time = 0.15 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {24 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{2} x^{2}-24 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )-11 \sqrt {a x -1}\, a^{2} x^{2}+11 \sqrt {a x -1}+12 \sqrt {a x +1}\, a^{3} x^{3}-28 \sqrt {a x +1}\, a^{2} x^{2}-20 \sqrt {a x +1}\, a x +32 \sqrt {a x +1}}{12 \sqrt {a x -1}\, a \,c^{2} \left (a^{2} x^{2}-1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^2,x)
Output:
(24*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**2*x**2 - 24*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) - 11*sqrt(a *x - 1)*a**2*x**2 + 11*sqrt(a*x - 1) + 12*sqrt(a*x + 1)*a**3*x**3 - 28*sqr t(a*x + 1)*a**2*x**2 - 20*sqrt(a*x + 1)*a*x + 32*sqrt(a*x + 1))/(12*sqrt(a *x - 1)*a*c**2*(a**2*x**2 - 1))