Integrand size = 10, antiderivative size = 79 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^4} \, dx=\frac {2}{3} a^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a^3 \csc ^{-1}(a x) \] Output:
2/3*a^3*(1-1/a^2/x^2)^(1/2)+1/3*a*(1-1/a^2/x^2)^(1/2)/x^2+1/2*a^2*(1-1/a^2 /x^2)^(1/2)/x-1/2*a^3*arccsc(a*x)
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{6} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (2+3 a x+4 a^2 x^2\right )}{x^2}-3 a^2 \arcsin \left (\frac {1}{a x}\right )\right ) \] Input:
Integrate[E^ArcCoth[a*x]/x^4,x]
Output:
(a*((Sqrt[1 - 1/(a^2*x^2)]*(2 + 3*a*x + 4*a^2*x^2))/x^2 - 3*a^2*ArcSin[1/( a*x)]))/6
Time = 0.46 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6719, 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.
\(\displaystyle \int \frac {e^{\coth ^{-1}(a x)}}{x^4} \, dx\) |
\(\Big \downarrow \) 6719 |
\(\displaystyle -\int \frac {1+\frac {1}{a x}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 533 |
\(\displaystyle \frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a^2 \int \frac {2 a+\frac {3}{x}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}d\frac {1}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} \int \frac {2 a+\frac {3}{x}}{\sqrt {1-\frac {1}{a^2 x^2}} x}d\frac {1}{x}\) |
\(\Big \downarrow \) 533 |
\(\displaystyle \frac {1}{3} \left (\frac {3 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a^2 \int \frac {3 a+\frac {4}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (\frac {3 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a \int \frac {3 a+\frac {4}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {1}{3} \left (\frac {3 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a \left (3 a \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {1}{3} \left (\frac {3 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a \left (3 a^2 \arcsin \left (\frac {1}{a x}\right )-4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}\) |
Input:
Int[E^ArcCoth[a*x]/x^4,x]
Output:
(a*Sqrt[1 - 1/(a^2*x^2)])/(3*x^2) + ((3*a^2*Sqrt[1 - 1/(a^2*x^2)])/(2*x) - (a*(-4*a^2*Sqrt[1 - 1/(a^2*x^2)] + 3*a^2*ArcSin[1/(a*x)]))/2)/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_)^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 = 0.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.18
method | result | size |
risch | \(\frac {\left (a x -1\right ) \left (4 a^{2} x^{2}+3 a x +2\right )}{6 x^{3} \sqrt {\frac {a x -1}{a x +1}}}-\frac {a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(93\) |
default | \(\frac {\left (a x -1\right ) \left (6 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}+6 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-6 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{2} x^{2}-3 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-3 a^{3} \sqrt {a^{2}}\, x^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-3 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{3}}\) | \(284\) |
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)/x^4,x,method=_RETURNVERBOSE)
Output:
1/6*(a*x-1)*(4*a^2*x^2+3*a*x+2)/x^3/((a*x-1)/(a*x+1))^(1/2)-1/2*a^3*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 = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^4} \, dx=\frac {6 \, a^{3} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (4 \, a^{3} x^{3} + 7 \, a^{2} x^{2} + 5 \, a x + 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, x^{3}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/x^4,x, algorithm="fricas")
Output:
1/6*(6*a^3*x^3*arctan(sqrt((a*x - 1)/(a*x + 1))) + (4*a^3*x^3 + 7*a^2*x^2 + 5*a*x + 2)*sqrt((a*x - 1)/(a*x + 1)))/x^3
\[ \int \frac {e^{\coth ^{-1}(a x)}}{x^4} \, dx=\int \frac {1}{x^{4} \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)/x**4,x)
Output:
Integral(1/(x**4*sqrt((a*x - 1)/(a*x + 1))), x)
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (65) = 130\).
Time = 0.12 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.72 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{3} \, {\left (3 \, a^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {3 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 4 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 9 \, a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}\right )} a \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/x^4,x, algorithm="maxima")
Output:
1/3*(3*a^2*arctan(sqrt((a*x - 1)/(a*x + 1))) + (3*a^2*((a*x - 1)/(a*x + 1) )^(5/2) + 4*a^2*((a*x - 1)/(a*x + 1))^(3/2) + 9*a^2*sqrt((a*x - 1)/(a*x + 1)))/(3*(a*x - 1)/(a*x + 1) + 3*(a*x - 1)^2/(a*x + 1)^2 + (a*x - 1)^3/(a*x + 1)^3 + 1))*a
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (65) = 130\).
Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.87 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^4} \, dx=\frac {a^{3} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{\mathrm {sgn}\left (a x + 1\right )} - \frac {3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} a^{3} - 12 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a^{2} {\left | a \right |} - 3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} a^{3} - 4 \, a^{2} {\left | a \right |}}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3} \mathrm {sgn}\left (a x + 1\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/x^4,x, algorithm="giac")
Output:
a^3*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))/sgn(a*x + 1) - 1/3*(3*(x*abs(a) - sqrt(a^2*x^2 - 1))^5*a^3 - 12*(x*abs(a) - sqrt(a^2*x^2 - 1))^2*a^2*abs(a ) - 3*(x*abs(a) - sqrt(a^2*x^2 - 1))*a^3 - 4*a^2*abs(a))/(((x*abs(a) - sqr t(a^2*x^2 - 1))^2 + 1)^3*sgn(a*x + 1))
Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.33 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^4} \, dx=\frac {2\,a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3}+\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,x^3}+a^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+\frac {7\,a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{6\,x}+\frac {5\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{6\,x^2} \] Input:
int(1/(x^4*((a*x - 1)/(a*x + 1))^(1/2)),x)
Output:
(2*a^3*((a*x - 1)/(a*x + 1))^(1/2))/3 + ((a*x - 1)/(a*x + 1))^(1/2)/(3*x^3 ) + a^3*atan(((a*x - 1)/(a*x + 1))^(1/2)) + (7*a^2*((a*x - 1)/(a*x + 1))^( 1/2))/(6*x) + (5*a*((a*x - 1)/(a*x + 1))^(1/2))/(6*x^2)
Time = 0.17 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^4} \, dx=\frac {6 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{3} x^{3}-6 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{3} x^{3}+4 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+3 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +2 \sqrt {a x +1}\, \sqrt {a x -1}-4 a^{3} x^{3}}{6 x^{3}} \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)/x^4,x)
Output:
(6*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**3*x**3 - 6*atan(sqrt(a*x - 1 ) + sqrt(a*x + 1) + 1)*a**3*x**3 + 4*sqrt(a*x + 1)*sqrt(a*x - 1)*a**2*x**2 + 3*sqrt(a*x + 1)*sqrt(a*x - 1)*a*x + 2*sqrt(a*x + 1)*sqrt(a*x - 1) - 4*a **3*x**3)/(6*x**3)