Integrand size = 12, antiderivative size = 91 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {9}{2} a^2 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a-\frac {1}{x}\right )^3}-\frac {3 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (a-\frac {1}{x}\right )}+\frac {9}{2} a^2 \csc ^{-1}(a x) \]
-a^5*(1-1/a^2/x^2)^(5/2)/(a-1/x)^3-3/2*a^3*(1-1/a^2/x^2)^(3/2)/(a-1/x)+9/2 *a^2*arccsc(a*x)-9/2*a^2*(1-1/a^2/x^2)^(1/2)
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{2} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (1+5 a x-14 a^2 x^2\right )}{x (-1+a x)}+9 a \arcsin \left (\frac {1}{a x}\right )\right ) \]
(a*((Sqrt[1 - 1/(a^2*x^2)]*(1 + 5*a*x - 14*a^2*x^2))/(x*(-1 + a*x)) + 9*a* ArcSin[1/(a*x)]))/2
Time = 0.72 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.09, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {6719, 2164, 25, 27, 2027, 2164, 25, 27, 563, 25, 2346, 27, 455, 223}
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.
| \(\displaystyle \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx\) |
| \(\Big \downarrow \) 6719 |
| \(\displaystyle -\int \frac {\left (1+\frac {1}{a x}\right )^2}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right ) x}d\frac {1}{x}\) |
| \(\Big \downarrow \) 2164 |
| \(\displaystyle \frac {\int -\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {a}{x}+\frac {1}{x^2}\right )}{\left (a-\frac {1}{x}\right )^2}d\frac {1}{x}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {a}{x}+\frac {1}{x^2}\right )}{\left (a-\frac {1}{x}\right )^2}d\frac {1}{x}}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -a \int \frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {a}{x}+\frac {1}{x^2}\right )}{\left (a-\frac {1}{x}\right )^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle -a \int \frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right )}{\left (a-\frac {1}{x}\right )^2 x}d\frac {1}{x}\) |
| \(\Big \downarrow \) 2164 |
| \(\displaystyle a^2 \int -\frac {a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^3 x}d\frac {1}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a^2 \int \frac {a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^3 x}d\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -a^3 \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^3 x}d\frac {1}{x}\) |
| \(\Big \downarrow \) 563 |
| \(\displaystyle -a^3 \left (\frac {\int -\frac {4 a^2+\frac {3 a}{x}+\frac {1}{x^2}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a^4}+\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a^3 \left (\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {\int \frac {4 a^2+\frac {3 a}{x}+\frac {1}{x^2}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a^4}\right )\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle -a^3 \left (\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {-\frac {1}{2} a^2 \int -\frac {3 \left (3 a+\frac {2}{x}\right )}{a \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}}{a^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -a^3 \left (\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {\frac {3}{2} a \int \frac {3 a+\frac {2}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}}{a^4}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -a^3 \left (\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {\frac {3}{2} a \left (3 a \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-2 a^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}}{a^4}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -a^3 \left (\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {\frac {3}{2} a \left (3 a^2 \arcsin \left (\frac {1}{a x}\right )-2 a^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}}{a^4}\right )\) |
-(a^3*((4*Sqrt[1 - 1/(a^2*x^2)])/(a - x^(-1)) - (-1/2*(a^2*Sqrt[1 - 1/(a^2 *x^2)])/x + (3*a*(-2*a^2*Sqrt[1 - 1/(a^2*x^2)] + 3*a^2*ArcSin[1/(a*x)]))/2 )/a^4))
3.1.23.3.1 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_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(-(-c)^(m - n - 2))*d^(2*n - m + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)* b^(n + 2)*(c + d*x))), x] - Simp[d^(2*n - m + 2)/b^(n + 1) Int[(1/Sqrt[a + b*x^2])*ExpandToSum[(2^(-n - 1)*(-c)^(m - n - 1) - d^m*x^m*(-c + d*x)^(-n - 1))/(c + d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2 , 0] && IGtQ[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*e Int[(d + e*x)^(m - 1)*PolynomialQuotient[Pq, a*e + b*d*x, x]* (a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + b*d*x, x], 0 ]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x , 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2] && IntegerQ[m]
Time = 0.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.37
| method | result | size |
| risch | \(-\frac {\left (a x -1\right ) \left (6 a x +1\right )}{2 x^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {9 a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}-\frac {4 a \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{x -\frac {1}{a}}\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(125\) |
| default | \(\frac {-6 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{5} x^{5}+6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+21 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+9 a^{4} x^{4} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}-6 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-11 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-24 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-18 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-12 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-4 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+12 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+12 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+4 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+9 a^{2} x^{2} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-6 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}}{2 \sqrt {a^{2}}\, x^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(641\) |
-1/2*(a*x-1)*(6*a*x+1)/x^2/((a*x-1)/(a*x+1))^(1/2)+(9/2*a^2*arctan(1/(a^2* x^2-1)^(1/2))-4*a/(x-1/a)*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2))/(a*x+1)/((a*x -1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)
Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {18 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (14 \, a^{3} x^{3} + 9 \, a^{2} x^{2} - 6 \, a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a x^{3} - x^{2}\right )}} \]
-1/2*(18*(a^3*x^3 - a^2*x^2)*arctan(sqrt((a*x - 1)/(a*x + 1))) + (14*a^3*x ^3 + 9*a^2*x^2 - 6*a*x - 1)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^3 - x^2)
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=\int \frac {1}{x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.21 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=-{\left (9 \, a \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {\frac {15 \, {\left (a x - 1\right )} a}{a x + 1} + \frac {9 \, {\left (a x - 1\right )}^{2} a}{{\left (a x + 1\right )}^{2}} + 4 \, a}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + \sqrt {\frac {a x - 1}{a x + 1}}}\right )} a \]
-(9*a*arctan(sqrt((a*x - 1)/(a*x + 1))) + (15*(a*x - 1)*a/(a*x + 1) + 9*(a *x - 1)^2*a/(a*x + 1)^2 + 4*a)/(((a*x - 1)/(a*x + 1))^(5/2) + 2*((a*x - 1) /(a*x + 1))^(3/2) + sqrt((a*x - 1)/(a*x + 1))))*a
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=\int { \frac {1}{x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.91 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{2\,x^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}-\frac {7\,a^2}{\sqrt {\frac {a\,x-1}{a\,x+1}}}-9\,a^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+\frac {5\,a}{2\,x\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]