Integrand size = 22, antiderivative size = 149 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {9 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{5 c^2 \left (a-\frac {1}{x}\right )^3}-\frac {43 a \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^2 \left (a-\frac {1}{x}\right )^2}-\frac {118 \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^2 \left (a-\frac {1}{x}\right )}+\frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}} x}{c^2 \left (a-\frac {1}{x}\right )^3}+\frac {5 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2} \] Output:
-9/5*a^2*(1-1/a^2/x^2)^(1/2)/c^2/(a-1/x)^3-43/15*a*(1-1/a^2/x^2)^(1/2)/c^2 /(a-1/x)^2-118/15*(1-1/a^2/x^2)^(1/2)/c^2/(a-1/x)+a^3*(1-1/a^2/x^2)^(1/2)* x/c^2/(a-1/x)^3+5*arctanh((1-1/a^2/x^2)^(1/2))/a/c^2
Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.70 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {-118+161 a x+91 a^2 x^2-173 a^3 x^3+15 a^4 x^4+75 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{15 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2} \] Input:
Integrate[E^(3*ArcCoth[a*x])/(c - c/(a*x))^2,x]
Output:
(-118 + 161*a*x + 91*a^2*x^2 - 173*a^3*x^3 + 15*a^4*x^4 + 75*a*Sqrt[1 - 1/ (a^2*x^2)]*x*(-1 + a*x)^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(15*a^2*c^2*Sqrt [1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2)
Time = 1.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.96, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {6731, 27, 570, 532, 25, 2336, 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.
| \(\displaystyle \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -c^3 \int \frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{c^5 \left (a-\frac {1}{x}\right )^5}d\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^5 \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{\left (a-\frac {1}{x}\right )^5}d\frac {1}{x}}{c^2}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle -\frac {\int \frac {\left (a+\frac {1}{x}\right )^5 x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}}d\frac {1}{x}}{a^5 c^2}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle -\frac {\frac {16 a^3 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} \int -\frac {\left (5 a^5+\frac {25 a^4}{x}+\frac {39 a^3}{x^2}-\frac {5 a^2}{x^3}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}}{a^5 c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{5} \int \frac {\left (5 a^5+\frac {25 a^4}{x}+\frac {39 a^3}{x^2}-\frac {5 a^2}{x^3}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^5 c^2}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {4 a^3 \left (5 a+\frac {11}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {1}{3} \int -\frac {\left (15 a^5+\frac {75 a^4}{x}+\frac {88 a^3}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^5 c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (15 a^5+\frac {75 a^4}{x}+\frac {88 a^3}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {4 a^3 \left (5 a+\frac {11}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^5 c^2}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (75 a+\frac {103}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}-\int -\frac {15 a^4 \left (a+\frac {5}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )+\frac {4 a^3 \left (5 a+\frac {11}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^5 c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a^4 \int \frac {\left (a+\frac {5}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {a^3 \left (75 a+\frac {103}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a^3 \left (5 a+\frac {11}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^5 c^2}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a^4 \left (5 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^3 \left (75 a+\frac {103}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a^3 \left (5 a+\frac {11}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^5 c^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a^4 \left (\frac {5}{2} \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^3 \left (75 a+\frac {103}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a^3 \left (5 a+\frac {11}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^5 c^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a^4 \left (-5 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^3 \left (75 a+\frac {103}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a^3 \left (5 a+\frac {11}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^5 c^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {16 a^3 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {1}{5} \left (\frac {4 a^3 \left (5 a+\frac {11}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {1}{3} \left (15 a^4 \left (-5 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^3 \left (75 a+\frac {103}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )\right )}{a^5 c^2}\) |
Input:
Int[E^(3*ArcCoth[a*x])/(c - c/(a*x))^2,x]
Output:
-(((16*a^3*(a + x^(-1)))/(5*(1 - 1/(a^2*x^2))^(5/2)) + ((4*a^3*(5*a + 11/x ))/(3*(1 - 1/(a^2*x^2))^(3/2)) + ((a^3*(75*a + 103/x))/Sqrt[1 - 1/(a^2*x^2 )] + 15*a^4*(-(a*Sqrt[1 - 1/(a^2*x^2)]*x) - 5*ArcTanh[Sqrt[1 - 1/(a^2*x^2) ]]))/3)/5)/(a^5*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^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S imp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Time = 0.13 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.51
| method | result | size |
| risch | \(\frac {a x -1}{a \,c^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {5 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{2} \sqrt {a^{2}}}-\frac {143 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{15 a^{4} \left (x -\frac {1}{a}\right )}-\frac {4 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{5 a^{6} \left (x -\frac {1}{a}\right )^{3}}-\frac {52 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{15 a^{5} \left (x -\frac {1}{a}\right )^{2}}\right ) a^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{2} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(225\) |
| default | \(-\frac {-75 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}-75 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+60 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+300 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+300 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-97 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -450 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-450 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+43 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+300 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x +300 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -75 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-75 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{15 a \left (a x -1\right )^{2} \sqrt {a^{2}}\, c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(438\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^2,x,method=_RETURNVERBOSE)
Output:
1/a*(a*x-1)/c^2/((a*x-1)/(a*x+1))^(1/2)+(5/a^2*ln(a^2*x/(a^2)^(1/2)+(a^2*x ^2-1)^(1/2))/(a^2)^(1/2)-143/15/a^4/(x-1/a)*((x-1/a)^2*a^2+2*a*(x-1/a))^(1 /2)-4/5/a^6/(x-1/a)^3*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-52/15/a^5/(x-1/a)^ 2*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2))*a^2/c^2/(a*x+1)/((a*x-1)/(a*x+1))^(1/ 2)*((a*x-1)*(a*x+1))^(1/2)
Time = 0.09 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.14 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {75 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 75 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (15 \, a^{4} x^{4} - 173 \, a^{3} x^{3} + 91 \, a^{2} x^{2} + 161 \, a x - 118\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^2,x, algorithm="fricas")
Output:
1/15*(75*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 75*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (15*a^4*x^4 - 173*a^3*x^3 + 91*a^2*x^2 + 161*a*x - 118)*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)
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {a^{2} \int \frac {x^{2}}{\frac {a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {3 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{2}} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**2,x)
Output:
a**2*Integral(x**2/(a**3*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - 3*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) + 3*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 = 1.03 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {1}{15} \, a {\left (\frac {\frac {17 \, {\left (a x - 1\right )}}{a x + 1} + \frac {100 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {150 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {75 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {75 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^2,x, algorithm="maxima")
Output:
1/15*a*((17*(a*x - 1)/(a*x + 1) + 100*(a*x - 1)^2/(a*x + 1)^2 - 150*(a*x - 1)^3/(a*x + 1)^3 + 3)/(a^2*c^2*((a*x - 1)/(a*x + 1))^(7/2) - a^2*c^2*((a* x - 1)/(a*x + 1))^(5/2)) + 75*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^2) - 75*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^2))
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.42 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {5 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{c^{2} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1}}{a c^{2} \mathrm {sgn}\left (a x + 1\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^2,x, algorithm="giac")
Output:
-5*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 = 13.65 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.81 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {10\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^2}-\frac {\frac {20\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {10\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {17\,\left (a\,x-1\right )}{15\,\left (a\,x+1\right )}+\frac {1}{5}}{a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \] Input:
int(1/((c - c/(a*x))^2*((a*x - 1)/(a*x + 1))^(3/2)),x)
Output:
(10*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^2) - ((20*(a*x - 1)^2)/(3*(a* x + 1)^2) - (10*(a*x - 1)^3)/(a*x + 1)^3 + (17*(a*x - 1))/(15*(a*x + 1)) + 1/5)/(a*c^2*((a*x - 1)/(a*x + 1))^(5/2) - a*c^2*((a*x - 1)/(a*x + 1))^(7/ 2))
Time = 0.19 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.34 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {300 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{2} x^{2}-600 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x +300 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+157 \sqrt {a x -1}\, a^{2} x^{2}-314 \sqrt {a x -1}\, a x +157 \sqrt {a x -1}+30 \sqrt {a x +1}\, a^{3} x^{3}-376 \sqrt {a x +1}\, a^{2} x^{2}+558 \sqrt {a x +1}\, a x -236 \sqrt {a x +1}}{30 \sqrt {a x -1}\, a \,c^{2} \left (a^{2} x^{2}-2 a x +1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^2,x)
Output:
(300*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**2*x**2 - 600*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x + 300 *sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 157*sqrt(a*x - 1)*a**2*x**2 - 314*sqrt(a*x - 1)*a*x + 157*sqrt(a*x - 1) + 30*sqrt(a*x + 1)*a**3*x**3 - 376*sqrt(a*x + 1)*a**2*x**2 + 558*sqrt(a*x + 1)*a*x - 236 *sqrt(a*x + 1))/(30*sqrt(a*x - 1)*a*c**2*(a**2*x**2 - 2*a*x + 1))