Integrand size = 12, antiderivative size = 47 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=a \sqrt {1-\frac {1}{a^2 x^2}}+\frac {4 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}+3 a \csc ^{-1}(a x) \] Output:
a*(1-1/a^2/x^2)^(1/2)+4*(a-1/x)/(1-1/a^2/x^2)^(1/2)+3*a*arccsc(a*x)
Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=\frac {a \sqrt {1-\frac {1}{a^2 x^2}} (1+5 a x)}{1+a x}+3 a \arcsin \left (\frac {1}{a x}\right ) \] Input:
Integrate[1/(E^(3*ArcCoth[a*x])*x^2),x]
Output:
(a*Sqrt[1 - 1/(a^2*x^2)]*(1 + 5*a*x))/(1 + a*x) + 3*a*ArcSin[1/(a*x)]
Time = 0.45 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6719, 711, 25, 27, 671, 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.
\(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx\) |
\(\Big \downarrow \) 6719 |
\(\displaystyle -\int \frac {\left (1-\frac {1}{a x}\right )^2}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1+\frac {1}{a x}\right )}d\frac {1}{x}\) |
\(\Big \downarrow \) 711 |
\(\displaystyle a^4 \int -\frac {a-\frac {3}{x}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right )}d\frac {1}{x}+a \sqrt {1-\frac {1}{a^2 x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle a \sqrt {1-\frac {1}{a^2 x^2}}-a^4 \int \frac {a-\frac {3}{x}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right )}d\frac {1}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a \sqrt {1-\frac {1}{a^2 x^2}}-\int \frac {a-\frac {3}{x}}{\sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right )}d\frac {1}{x}\) |
\(\Big \downarrow \) 671 |
\(\displaystyle 3 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}+a \sqrt {1-\frac {1}{a^2 x^2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}+a \sqrt {1-\frac {1}{a^2 x^2}}+3 a \arcsin \left (\frac {1}{a x}\right )\) |
Input:
Int[1/(E^(3*ArcCoth[a*x])*x^2),x]
Output:
a*Sqrt[1 - 1/(a^2*x^2)] + (4*a^2*Sqrt[1 - 1/(a^2*x^2)])/(a + x^(-1)) + 3*a *ArcSin[1/(a*x)]
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[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m + p + 1))), x] + Simp[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m + p + 1)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((a_.) + (c_.)*(x_ )^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + c*x^2)^(p + 1) /(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m + n + 2*p + 1)) Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^n*(m + n + 2*p + 1)*(f + g*x) ^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - 2*e*g^n*(m + p + n)*(d + e*x)^(n - 2)*(a*e - c*d*x), x], x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && Eq Q[c*d^2 + a*e^2, 0] && IGtQ[n, 0] && NeQ[m + n + 2*p + 1, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(43)=86\).
Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.32
method | result | size |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{x}+\frac {\left (3 a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\frac {4 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a x -1}\) | \(109\) |
default | \(-\frac {\left (-\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}+\sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{2} x^{2}-5 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-3 a^{3} \sqrt {a^{2}}\, x^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-\ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+2 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -7 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-6 a^{2} \sqrt {a^{2}}\, x^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+2 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-2 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -3 a \sqrt {a^{2}}\, x \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x -\ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x \right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{\sqrt {a^{2}}\, x \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}\) | \(592\) |
Input:
int(((a*x-1)/(a*x+1))^(3/2)/x^2,x,method=_RETURNVERBOSE)
Output:
(a*x+1)/x*((a*x-1)/(a*x+1))^(1/2)+(3*a*arctan(1/(a^2*x^2-1)^(1/2))+4/(x+1/ a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2))*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a* x+1))^(1/2)/(a*x-1)
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {6 \, a x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (5 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{x} \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/x^2,x, algorithm="fricas")
Output:
-(6*a*x*arctan(sqrt((a*x - 1)/(a*x + 1))) - (5*a*x + 1)*sqrt((a*x - 1)/(a* x + 1)))/x
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=\int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{2}}\, dx \] Input:
integrate(((a*x-1)/(a*x+1))**(3/2)/x**2,x)
Output:
Integral(((a*x - 1)/(a*x + 1))**(3/2)/x**2, x)
Time = 0.10 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.53 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=2 \, a {\left (2 \, \sqrt {\frac {a x - 1}{a x + 1}} + \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\frac {a x - 1}{a x + 1} + 1} - 3 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/x^2,x, algorithm="maxima")
Output:
2*a*(2*sqrt((a*x - 1)/(a*x + 1)) + sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)/(a *x + 1) + 1) - 3*arctan(sqrt((a*x - 1)/(a*x + 1))))
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{2}} \,d x } \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/x^2,x, algorithm="giac")
Output:
undef
Time = 23.82 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.26 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}+5\,a\,x\,\sqrt {\frac {a\,x-1}{a\,x+1}}-6\,a\,x\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{x} \] Input:
int(((a*x - 1)/(a*x + 1))^(3/2)/x^2,x)
Output:
(((a*x - 1)/(a*x + 1))^(1/2) + 5*a*x*((a*x - 1)/(a*x + 1))^(1/2) - 6*a*x*a tan(((a*x - 1)/(a*x + 1))^(1/2)))/x
Time = 0.16 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.91 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=\frac {-6 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{2} x^{2}-6 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a x +6 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{2} x^{2}+6 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a x +5 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +\sqrt {a x +1}\, \sqrt {a x -1}+3 a^{2} x^{2}+3 a x}{x \left (a x +1\right )} \] Input:
int(((a*x-1)/(a*x+1))^(3/2)/x^2,x)
Output:
( - 6*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**2*x**2 - 6*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a*x + 6*atan(sqrt(a*x - 1) + sqrt(a*x + 1) + 1)* a**2*x**2 + 6*atan(sqrt(a*x - 1) + sqrt(a*x + 1) + 1)*a*x + 5*sqrt(a*x + 1 )*sqrt(a*x - 1)*a*x + sqrt(a*x + 1)*sqrt(a*x - 1) + 3*a**2*x**2 + 3*a*x)/( x*(a*x + 1))