Integrand size = 22, antiderivative size = 179 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {2}{a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^2} \]
-arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c^2-2/a/c^2/(1+1/a/x)^(3/2)/(1 -1/a/x)^(1/2)+x/c^2/(1+1/a/x)^(3/2)/(1-1/a/x)^(1/2)+5/3*(1-1/a/x)^(1/2)/a/ c^2/(1+1/a/x)^(3/2)+8/3*(1-1/a/x)^(1/2)/a/c^2/(1+1/a/x)^(1/2)
Time = 0.34 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.47 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-8-5 a x+7 a^2 x^2+3 a^3 x^3\right )}{3 (-1+a x) (1+a x)^2}-\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^2} \]
((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)*(1 + a*x)^2) - Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^2)
Time = 0.35 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.96, 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, 169, 25, 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.
| \(\displaystyle \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -\frac {\int \frac {x^2}{\left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}}{c^2}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {-\int \frac {\left (a-\frac {3}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (a-\frac {3}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {a \left (-\int -\frac {\left (a-\frac {4}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}\right )-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {a \int \frac {\left (a-\frac {4}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (a-\frac {4}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} a \int \frac {\left (3 a-\frac {5}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {5 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \int \frac {\left (3 a-\frac {5}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {5 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (a \int \frac {3 x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {8 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}\right )+\frac {5 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (3 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {8 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}\right )+\frac {5 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (\frac {8 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-3 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )\right )+\frac {5 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (\frac {8 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-3 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )\right )+\frac {5 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^2}\) |
-((-(x/(Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))) - ((-2*a)/(Sqrt[1 - 1/(a*x )]*(1 + 1/(a*x))^(3/2)) + (5*a*Sqrt[1 - 1/(a*x)])/(3*(1 + 1/(a*x))^(3/2)) + ((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)
3.9.11.3.1 Defintions of rubi rules used
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 = 0.23 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.19
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{2}}+\frac {\left (-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}-\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{6 a^{7} \left (x +\frac {1}{a}\right )^{2}}+\frac {19 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{12 a^{6} \left (x +\frac {1}{a}\right )}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{4 a^{6} \left (x -\frac {1}{a}\right )}\right ) a^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{2} \left (a x -1\right )}\) | \(213\) |
| default | \(-\frac {\left (-45 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}+24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+21 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-45 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}+24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-11 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+90 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-48 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-5 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +90 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-48 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+19 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-45 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -45 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+24 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right ) \sqrt {\frac {a x -1}{a x +1}}}{24 a \left (a x -1\right )^{2} \sqrt {a^{2}}\, \left (a x +1\right )^{2} c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}\) | \(530\) |
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)-1/6/a^7/(x+1/a)^2*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1 /2)+19/12/a^6/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)-1/4/a^6/(x-1/a)*(( x-1/a)^2*a^2+2*(x-1/a)*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 = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.66 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (3 \, a^{3} x^{3} + 7 \, a^{2} x^{2} - 5 \, a x - 8\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{2} x^{2} - a c^{2}\right )}} \]
-1/3*(3*(a^2*x^2 - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 3*(a^2*x^2 - 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 - a*c^2)
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {a^{4} \int \frac {x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Time = 0.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {1}{12} \, a {\left (\frac {3 \, {\left (\frac {9 \, {\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{2} c^{2} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 18 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2}} + \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \]
-1/12*a*(3*(9*(a*x - 1)/(a*x + 1) - 1)/(a^2*c^2*((a*x - 1)/(a*x + 1))^(3/2 ) - a^2*c^2*sqrt((a*x - 1)/(a*x + 1))) - (((a*x - 1)/(a*x + 1))^(3/2) + 18 *sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^2) + 12*log(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))
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\int { \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2}} \,d x } \]
Time = 3.87 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {\frac {9\,\left (a\,x-1\right )}{a\,x+1}-1}{4\,a\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}-4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}+\frac {3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,a\,c^2}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{12\,a\,c^2}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,c^2} \]