Integrand size = 10, antiderivative size = 88 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^5} \, dx=\frac {1}{24} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a+\frac {9}{x}\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {3}{8} a^4 \csc ^{-1}(a x) \]
-3/8*a^4*arccsc(a*x)+1/24*a^3*(16*a+9/x)*(1-1/a^2/x^2)^(1/2)+1/4*a*(1-1/a^ 2/x^2)^(1/2)/x^3+1/3*a^2*(1-1/a^2/x^2)^(1/2)/x^2
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.67 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^5} \, dx=\frac {1}{24} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (6+8 a x+9 a^2 x^2+16 a^3 x^3\right )}{x^3}-9 a^3 \arcsin \left (\frac {1}{a x}\right )\right ) \]
(a*((Sqrt[1 - 1/(a^2*x^2)]*(6 + 8*a*x + 9*a^2*x^2 + 16*a^3*x^3))/x^3 - 9*a ^3*ArcSin[1/(a*x)]))/24
Time = 0.31 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.36, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6719, 533, 27, 533, 27, 533, 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.
\(\displaystyle \int \frac {e^{\coth ^{-1}(a x)}}{x^5} \, dx\) |
\(\Big \downarrow \) 6719 |
\(\displaystyle -\int \frac {1+\frac {1}{a x}}{\sqrt {1-\frac {1}{a^2 x^2}} x^3}d\frac {1}{x}\) |
\(\Big \downarrow \) 533 |
\(\displaystyle \frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {1}{4} a^2 \int \frac {3 a+\frac {4}{x}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {1}{4} \int \frac {3 a+\frac {4}{x}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 533 |
\(\displaystyle \frac {1}{4} \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a^2 \int \frac {8 a+\frac {9}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}} x}d\frac {1}{x}\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a \int \frac {8 a+\frac {9}{x}}{\sqrt {1-\frac {1}{a^2 x^2}} x}d\frac {1}{x}\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\) |
\(\Big \downarrow \) 533 |
\(\displaystyle \frac {1}{4} \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a \left (\frac {1}{2} a^2 \int \frac {9 a+\frac {16}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {9 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a \left (\frac {1}{2} a \int \frac {9 a+\frac {16}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {9 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {1}{4} \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a \left (\frac {1}{2} a \left (9 a \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-16 a^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {9 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {1}{4} \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a \left (\frac {1}{2} a \left (9 a^2 \arcsin \left (\frac {1}{a x}\right )-16 a^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {9 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\) |
(a*Sqrt[1 - 1/(a^2*x^2)])/(4*x^3) + ((4*a^2*Sqrt[1 - 1/(a^2*x^2)])/(3*x^2) - (a*((-9*a^2*Sqrt[1 - 1/(a^2*x^2)])/(2*x) + (a*(-16*a^2*Sqrt[1 - 1/(a^2* x^2)] + 9*a^2*ArcSin[1/(a*x)]))/2))/3)/4
3.1.9.3.1 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_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
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 = 101, normalized size of antiderivative = 1.15
method | result | size |
risch | \(\frac {\left (a x -1\right ) \left (16 a^{3} x^{3}+9 a^{2} x^{2}+8 a x +6\right )}{24 x^{4} \sqrt {\frac {a x -1}{a x +1}}}-\frac {3 a^{4} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{8 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(101\) |
default | \(-\frac {\left (a x -1\right ) \left (-24 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{5} x^{5}+24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+9 a^{4} x^{4} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+24 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-24 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}-24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+15 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+8 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{24 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} \sqrt {a^{2}}}\) | \(308\) |
1/24*(a*x-1)*(16*a^3*x^3+9*a^2*x^2+8*a*x+6)/x^4/((a*x-1)/(a*x+1))^(1/2)-3/ 8*a^4*arctan(1/(a^2*x^2-1)^(1/2))/((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 = 76, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^5} \, dx=\frac {18 \, a^{4} x^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (16 \, a^{4} x^{4} + 25 \, a^{3} x^{3} + 17 \, a^{2} x^{2} + 14 \, a x + 6\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, x^{4}} \]
1/24*(18*a^4*x^4*arctan(sqrt((a*x - 1)/(a*x + 1))) + (16*a^4*x^4 + 25*a^3* x^3 + 17*a^2*x^2 + 14*a*x + 6)*sqrt((a*x - 1)/(a*x + 1)))/x^4
\[ \int \frac {e^{\coth ^{-1}(a x)}}{x^5} \, dx=\int \frac {1}{x^{5} \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (74) = 148\).
Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.95 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^5} \, dx=\frac {1}{12} \, {\left (9 \, a^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {9 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 49 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 31 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 39 \, a^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )}}{a x + 1} + \frac {6 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 1}\right )} a \]
1/12*(9*a^3*arctan(sqrt((a*x - 1)/(a*x + 1))) + (9*a^3*((a*x - 1)/(a*x + 1 ))^(7/2) + 49*a^3*((a*x - 1)/(a*x + 1))^(5/2) + 31*a^3*((a*x - 1)/(a*x + 1 ))^(3/2) + 39*a^3*sqrt((a*x - 1)/(a*x + 1)))/(4*(a*x - 1)/(a*x + 1) + 6*(a *x - 1)^2/(a*x + 1)^2 + 4*(a*x - 1)^3/(a*x + 1)^3 + (a*x - 1)^4/(a*x + 1)^ 4 + 1))*a
Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (74) = 148\).
Time = 0.27 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.57 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^5} \, dx=\frac {3 \, a^{4} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{4 \, \mathrm {sgn}\left (a x + 1\right )} - \frac {9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{7} a^{4} + 33 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} a^{4} - 48 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a^{3} {\left | a \right |} - 33 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} a^{4} - 64 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a^{3} {\left | a \right |} - 9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} a^{4} - 16 \, a^{3} {\left | a \right |}}{12 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{4} \mathrm {sgn}\left (a x + 1\right )} \]
3/4*a^4*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))/sgn(a*x + 1) - 1/12*(9*(x*ab s(a) - sqrt(a^2*x^2 - 1))^7*a^4 + 33*(x*abs(a) - sqrt(a^2*x^2 - 1))^5*a^4 - 48*(x*abs(a) - sqrt(a^2*x^2 - 1))^4*a^3*abs(a) - 33*(x*abs(a) - sqrt(a^2 *x^2 - 1))^3*a^4 - 64*(x*abs(a) - sqrt(a^2*x^2 - 1))^2*a^3*abs(a) - 9*(x*a bs(a) - sqrt(a^2*x^2 - 1))*a^4 - 16*a^3*abs(a))/(((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^4*sgn(a*x + 1))
Time = 0.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.47 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^5} \, dx=\frac {2\,a^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3}+\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{4\,x^4}+\frac {3\,a^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4}+\frac {17\,a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{24\,x^2}+\frac {25\,a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{24\,x}+\frac {7\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{12\,x^3} \]