Integrand size = 22, antiderivative size = 138 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {4 \left (a+\frac {1}{x}\right )}{5 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {5 a+\frac {7}{x}}{5 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {15 a+\frac {19}{x}}{5 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2} \] Output:
1/5*(-4*a-4/x)/a^2/c^2/(1-1/a^2/x^2)^(5/2)-1/5*(5*a+7/x)/a^2/c^2/(1-1/a^2/ x^2)^(3/2)-1/5*(15*a+19/x)/a^2/c^2/(1-1/a^2/x^2)^(1/2)+(1-1/a^2/x^2)^(1/2) *x/c^2+3*arctanh((1-1/a^2/x^2)^(1/2))/a/c^2
Time = 0.52 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.57 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-24+57 a x-39 a^2 x^2+5 a^3 x^3\right )}{5 (-1+a x)^3}+3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^2} \] Input:
Integrate[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^2,x]
Output:
((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-24 + 57*a*x - 39*a^2*x^2 + 5*a^3*x^3))/(5*(- 1 + a*x)^3) + 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^2)
Time = 0.61 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.25, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {6748, 114, 27, 35, 110, 27, 169, 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.
| \(\displaystyle \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -\frac {\int \frac {x^2}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{c^2}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {-\int -\frac {3 \left (a+\frac {1}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{5/2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3 \int \frac {\left (a+\frac {1}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a^2}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{5/2}}}{c^2}\) |
| \(\Big \downarrow \) 35 |
| \(\displaystyle -\frac {\frac {3 \int \frac {\sqrt {1+\frac {1}{a x}} x}{\left (1-\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}}{a}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{5/2}}}{c^2}\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {2 \sqrt {\frac {1}{a x}+1}}{5 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {2}{5} \int -\frac {\left (5 a+\frac {4}{x}\right ) x}{2 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )}{a}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{5/2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {\left (5 a+\frac {4}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{5 a}+\frac {2 \sqrt {\frac {1}{a x}+1}}{5 \left (1-\frac {1}{a x}\right )^{5/2}}\right )}{a}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{5/2}}}{c^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {\frac {3 a \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{3/2}}-\frac {1}{3} a \int -\frac {3 \left (5 a+\frac {3}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{5 a}+\frac {2 \sqrt {\frac {1}{a x}+1}}{5 \left (1-\frac {1}{a x}\right )^{5/2}}\right )}{a}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{5/2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {\left (5 a+\frac {3}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {3 a \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{3/2}}}{5 a}+\frac {2 \sqrt {\frac {1}{a x}+1}}{5 \left (1-\frac {1}{a x}\right )^{5/2}}\right )}{a}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{5/2}}}{c^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {a \left (-\int -\frac {5 x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )+\frac {8 a \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}+\frac {3 a \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{3/2}}}{5 a}+\frac {2 \sqrt {\frac {1}{a x}+1}}{5 \left (1-\frac {1}{a x}\right )^{5/2}}\right )}{a}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{5/2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {5 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {8 a \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}+\frac {3 a \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{3/2}}}{5 a}+\frac {2 \sqrt {\frac {1}{a x}+1}}{5 \left (1-\frac {1}{a x}\right )^{5/2}}\right )}{a}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{5/2}}}{c^2}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {-5 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )+\frac {8 a \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}+\frac {3 a \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{3/2}}}{5 a}+\frac {2 \sqrt {\frac {1}{a x}+1}}{5 \left (1-\frac {1}{a x}\right )^{5/2}}\right )}{a}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{5/2}}}{c^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {-5 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {8 a \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}+\frac {3 a \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{3/2}}}{5 a}+\frac {2 \sqrt {\frac {1}{a x}+1}}{5 \left (1-\frac {1}{a x}\right )^{5/2}}\right )}{a}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{5/2}}}{c^2}\) |
Input:
Int[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^2,x]
Output:
-((-((Sqrt[1 + 1/(a*x)]*x)/(1 - 1/(a*x))^(5/2)) + (3*((2*Sqrt[1 + 1/(a*x)] )/(5*(1 - 1/(a*x))^(5/2)) + ((3*a*Sqrt[1 + 1/(a*x)])/(1 - 1/(a*x))^(3/2) + (8*a*Sqrt[1 + 1/(a*x)])/Sqrt[1 - 1/(a*x)] - 5*a*ArcTanh[Sqrt[1 - 1/(a*x)] *Sqrt[1 + 1/(a*x)]])/(5*a)))/a)/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[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} , x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && !(IntegerQ[n] && SimplerQ[a + b*x, c + d*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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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]
Time = 0.15 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.63
| method | result | size |
| risch | \(\frac {a x -1}{a \,c^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{5 a^{8} \left (x -\frac {1}{a}\right )^{3}}-\frac {6 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{5 a^{7} \left (x -\frac {1}{a}\right )^{2}}-\frac {24 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{5 a^{6} \left (x -\frac {1}{a}\right )}+\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}\right ) a^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(225\) |
| default | \(-\frac {-125 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}-120 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+85 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+500 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+480 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-148 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -750 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-720 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+67 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+500 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x +480 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -125 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-120 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{40 a \sqrt {a^{2}}\, \left (a x -1\right )^{2} c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(438\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(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)+(-1/5/a^8/(x-1/a)^3*((x-1/a)^2*a^2 +2*a*(x-1/a))^(1/2)-6/5/a^7/(x-1/a)^2*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-24 /5/a^6/(x-1/a)*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)+3/a^4*ln(a^2*x/(a^2)^(1/2 )+(a^2*x^2-1)^(1/2))/(a^2)^(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 = 0.10 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.23 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {15 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (5 \, a^{4} x^{4} - 34 \, a^{3} x^{3} + 18 \, a^{2} x^{2} + 33 \, a x - 24\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^2,x, algorithm="fricas")
Output:
1/5*(15*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (5*a^4*x^4 - 34*a^3*x^3 + 18*a^2*x^2 + 33*a*x - 24)*sqrt((a*x - 1)/(a *x + 1)))/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {a^{4} \int \frac {x^{4}}{\frac {a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {2 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {2 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{2}} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)/(c-c/a**2/x**2)**2,x)
Output:
a**4*Integral(x**4/(a**5*x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - 2*a**3*x**3*sqrt (a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) + 2*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) + a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x)/c**2
Time = 0.03 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.11 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {1}{20} \, a {\left (\frac {\frac {9 \, {\left (a x - 1\right )}}{a x + 1} + \frac {75 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {125 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}{a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^2,x, algorithm="maxima")
Output:
1/20*a*((9*(a*x - 1)/(a*x + 1) + 75*(a*x - 1)^2/(a*x + 1)^2 - 125*(a*x - 1 )^3/(a*x + 1)^3 + 1)/(a^2*c^2*((a*x - 1)/(a*x + 1))^(7/2) - a^2*c^2*((a*x - 1)/(a*x + 1))^(5/2)) + 60*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^2) - 60*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^2))
Time = 0.17 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.46 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {3 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{c^{2} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1}}{a c^{2} \mathrm {sgn}\left (a x + 1\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^2,x, algorithm="giac")
Output:
-3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(c^2*abs(a)*sgn(a*x + 1)) + sqr t(a^2*x^2 - 1)/(a*c^2*sgn(a*x + 1))
Time = 13.16 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.88 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {6\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^2}-\frac {\frac {15\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {25\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {9\,\left (a\,x-1\right )}{5\,\left (a\,x+1\right )}+\frac {1}{5}}{4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \] Input:
int(1/((c - c/(a^2*x^2))^2*((a*x - 1)/(a*x + 1))^(3/2)),x)
Output:
(6*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^2) - ((15*(a*x - 1)^2)/(a*x + 1)^2 - (25*(a*x - 1)^3)/(a*x + 1)^3 + (9*(a*x - 1))/(5*(a*x + 1)) + 1/5)/( 4*a*c^2*((a*x - 1)/(a*x + 1))^(5/2) - 4*a*c^2*((a*x - 1)/(a*x + 1))^(7/2))
Time = 0.16 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.44 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {60 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{2} x^{2}-120 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x +60 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+33 \sqrt {a x -1}\, a^{2} x^{2}-66 \sqrt {a x -1}\, a x +33 \sqrt {a x -1}+10 \sqrt {a x +1}\, a^{3} x^{3}-78 \sqrt {a x +1}\, a^{2} x^{2}+114 \sqrt {a x +1}\, a x -48 \sqrt {a x +1}}{10 \sqrt {a x -1}\, a \,c^{2} \left (a^{2} x^{2}-2 a x +1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^2,x)
Output:
(60*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**2*x**2 - 120*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x + 60*s qrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 33*sqrt(a*x - 1)*a**2*x**2 - 66*sqrt(a*x - 1)*a*x + 33*sqrt(a*x - 1) + 10*sqrt(a*x + 1)* a**3*x**3 - 78*sqrt(a*x + 1)*a**2*x**2 + 114*sqrt(a*x + 1)*a*x - 48*sqrt(a *x + 1))/(10*sqrt(a*x - 1)*a*c**2*(a**2*x**2 - 2*a*x + 1))