Integrand size = 22, antiderivative size = 171 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=-\frac {4 \left (a+\frac {1}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {21 a+\frac {31}{x}}{35 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {3 \left (35 a+\frac {47}{x}\right )}{35 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {35 a+\frac {53}{x}}{35 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3} \] Output:
1/7*(-4*a-4/x)/a^2/c^3/(1-1/a^2/x^2)^(7/2)-1/35*(21*a+31/x)/a^2/c^3/(1-1/a ^2/x^2)^(5/2)-3/35*(35*a+47/x)/a^2/c^3/(1-1/a^2/x^2)^(1/2)-1/35*(35*a+53/x )/a^2/c^3/(1-1/a^2/x^2)^(3/2)+(1-1/a^2/x^2)^(1/2)*x/c^3+3*arctanh((1-1/a^2 /x^2)^(1/2))/a/c^3
Time = 0.94 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.59 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (176-423 a x+125 a^2 x^2+368 a^3 x^3-286 a^4 x^4+35 a^5 x^5\right )}{35 (-1+a x)^4 (1+a x)}+3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^3} \] Input:
Integrate[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^3,x]
Output:
((a*Sqrt[1 - 1/(a^2*x^2)]*x*(176 - 423*a*x + 125*a^2*x^2 + 368*a^3*x^3 - 2 86*a^4*x^4 + 35*a^5*x^5))/(35*(-1 + a*x)^4*(1 + a*x)) + 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^3)
Time = 0.74 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.43, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.864, Rules used = {6748, 114, 25, 27, 169, 25, 27, 169, 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.
| \(\displaystyle \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -\frac {\int \frac {x^2}{\left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{c^3}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {-\int -\frac {\left (3 a+\frac {5}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\left (3 a+\frac {5}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\left (3 a+\frac {5}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}-\frac {1}{7} a \int -\frac {\left (21 a+\frac {32}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} a \int \frac {\left (21 a+\frac {32}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \int \frac {\left (21 a+\frac {32}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {53 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}-\frac {1}{5} a \int -\frac {3 \left (35 a+\frac {53}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {3}{5} \int \frac {\left (35 a+\frac {53}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {53 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {88 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}-\frac {1}{3} a \int -\frac {\left (105 a+\frac {176}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}\right )+\frac {53 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} a \int \frac {\left (105 a+\frac {176}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {88 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {53 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \int \frac {\left (105 a+\frac {176}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {88 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {53 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {281 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-a \int -\frac {\left (105 a+\frac {281}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}\right )+\frac {88 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {53 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (a \int \frac {\left (105 a+\frac {281}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {281 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}\right )+\frac {88 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {53 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\int \frac {\left (105 a+\frac {281}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {281 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}\right )+\frac {88 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {53 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (a \int \frac {105 x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {176 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {281 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}\right )+\frac {88 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {53 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (105 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {176 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {281 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}\right )+\frac {88 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {53 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (-105 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )-\frac {176 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {281 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}\right )+\frac {88 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {53 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (-105 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {176 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {281 a}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}\right )+\frac {88 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {53 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}}{c^3}\) |
Input:
Int[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^3,x]
Output:
-((-(x/((1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)])) + ((8*a)/(7*(1 - 1/(a*x))^ (7/2)*Sqrt[1 + 1/(a*x)]) + ((53*a)/(5*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x) ]) + (3*((88*a)/(3*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]) + ((281*a)/(Sqrt [1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (176*a*Sqrt[1 - 1/(a*x)])/Sqrt[1 + 1/(a *x)] - 105*a*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/3))/5)/7)/a^2)/ c^3)
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]
Time = 0.17 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.75
| method | result | size |
| risch | \(\frac {a x -1}{a \,c^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{6} \sqrt {a^{2}}}-\frac {2931 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{560 a^{8} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{14 a^{11} \left (x -\frac {1}{a}\right )^{4}}-\frac {71 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{140 a^{10} \left (x -\frac {1}{a}\right )^{3}}-\frac {477 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{280 a^{9} \left (x -\frac {1}{a}\right )^{2}}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{16 a^{8} \left (x +\frac {1}{a}\right )}\right ) a^{6} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(299\) |
| default | \(-\frac {-3675 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{7} x^{7}-3360 \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^{8} x^{7}+2555 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}+11025 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{6} x^{6}+10080 \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^{7} x^{6}-1873 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-3675 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-3360 \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}-4426 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{3} x^{3}-18375 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}-16800 \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}+3350 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+18375 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+16800 \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}+2511 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +3675 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+3360 \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}-1957 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-11025 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x -10080 \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 +3675 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+3360 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 )}{1120 a \sqrt {a^{2}}\, \left (a x -1\right )^{3} c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right )^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(714\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^3,x,method=_RETURNVERBOSE)
Output:
1/a*(a*x-1)/c^3/((a*x-1)/(a*x+1))^(1/2)+(3/a^6*ln(a^2*x/(a^2)^(1/2)+(a^2*x ^2-1)^(1/2))/(a^2)^(1/2)-2931/560/a^8/(x-1/a)*((x-1/a)^2*a^2+2*a*(x-1/a))^ (1/2)-1/14/a^11/(x-1/a)^4*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-71/140/a^10/(x -1/a)^3*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-477/280/a^9/(x-1/a)^2*((x-1/a)^2 *a^2+2*a*(x-1/a))^(1/2)+1/16/a^8/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2) )*a^6/c^3/((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x+1)
Time = 0.11 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.19 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {105 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (35 \, a^{5} x^{5} - 286 \, a^{4} x^{4} + 368 \, a^{3} x^{3} + 125 \, a^{2} x^{2} - 423 \, a x + 176\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^3,x, algorithm="fricas")
Output:
1/35*(105*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(sqrt((a*x - 1) /(a*x + 1)) + 1) - 105*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(s qrt((a*x - 1)/(a*x + 1)) - 1) + (35*a^5*x^5 - 286*a^4*x^4 + 368*a^3*x^3 + 125*a^2*x^2 - 423*a*x + 176)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*c^3*x^4 - 4*a ^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {a^{6} \int \frac {x^{6}}{\frac {a^{7} x^{7} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {3 a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {3 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {3 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{3}} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)/(c-c/a**2/x**2)**3,x)
Output:
a**6*Integral(x**6/(a**7*x**7*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - a**6*x**6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - 3*a**5*x**5*sqrt (a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) + 3*a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) + 3*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) - a*x*sq rt(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**3
Time = 0.03 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.12 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {1}{560} \, a {\left (\frac {\frac {51 \, {\left (a x - 1\right )}}{a x + 1} + \frac {294 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2170 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {3640 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 5}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}} + \frac {1680 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {1680 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}} + \frac {35 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^3,x, algorithm="maxima")
Output:
1/560*a*((51*(a*x - 1)/(a*x + 1) + 294*(a*x - 1)^2/(a*x + 1)^2 + 2170*(a*x - 1)^3/(a*x + 1)^3 - 3640*(a*x - 1)^4/(a*x + 1)^4 + 5)/(a^2*c^3*((a*x - 1 )/(a*x + 1))^(9/2) - a^2*c^3*((a*x - 1)/(a*x + 1))^(7/2)) + 1680*log(sqrt( (a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 1680*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^3) + 35*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^3))
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^3,x, algorithm="giac")
Output:
integrate(1/((c - c/(a^2*x^2))^3*((a*x - 1)/(a*x + 1))^(3/2)), x)
Time = 0.05 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.94 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{16\,a\,c^3}-\frac {\frac {42\,{\left (a\,x-1\right )}^2}{5\,{\left (a\,x+1\right )}^2}+\frac {62\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {104\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}+\frac {51\,\left (a\,x-1\right )}{35\,\left (a\,x+1\right )}+\frac {1}{7}}{16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}-16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}+\frac {6\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^3} \] Input:
int(1/((c - c/(a^2*x^2))^3*((a*x - 1)/(a*x + 1))^(3/2)),x)
Output:
((a*x - 1)/(a*x + 1))^(1/2)/(16*a*c^3) - ((42*(a*x - 1)^2)/(5*(a*x + 1)^2) + (62*(a*x - 1)^3)/(a*x + 1)^3 - (104*(a*x - 1)^4)/(a*x + 1)^4 + (51*(a*x - 1))/(35*(a*x + 1)) + 1/7)/(16*a*c^3*((a*x - 1)/(a*x + 1))^(7/2) - 16*a* c^3*((a*x - 1)/(a*x + 1))^(9/2)) + (6*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/ (a*c^3)
Time = 0.17 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.65 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {840 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{4} x^{4}-1680 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{3} x^{3}+1680 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x -840 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+549 \sqrt {a x -1}\, a^{4} x^{4}-1098 \sqrt {a x -1}\, a^{3} x^{3}+1098 \sqrt {a x -1}\, a x -549 \sqrt {a x -1}+140 \sqrt {a x +1}\, a^{5} x^{5}-1144 \sqrt {a x +1}\, a^{4} x^{4}+1472 \sqrt {a x +1}\, a^{3} x^{3}+500 \sqrt {a x +1}\, a^{2} x^{2}-1692 \sqrt {a x +1}\, a x +704 \sqrt {a x +1}}{140 \sqrt {a x -1}\, a \,c^{3} \left (a^{4} x^{4}-2 a^{3} x^{3}+2 a x -1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^3,x)
Output:
(840*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**4*x**4 - 1680*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**3*x** 3 + 1680*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x - 840*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 549*sqrt( a*x - 1)*a**4*x**4 - 1098*sqrt(a*x - 1)*a**3*x**3 + 1098*sqrt(a*x - 1)*a*x - 549*sqrt(a*x - 1) + 140*sqrt(a*x + 1)*a**5*x**5 - 1144*sqrt(a*x + 1)*a* *4*x**4 + 1472*sqrt(a*x + 1)*a**3*x**3 + 500*sqrt(a*x + 1)*a**2*x**2 - 169 2*sqrt(a*x + 1)*a*x + 704*sqrt(a*x + 1))/(140*sqrt(a*x - 1)*a*c**3*(a**4*x **4 - 2*a**3*x**3 + 2*a*x - 1))