Integrand size = 12, antiderivative size = 53 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=-a \sqrt {1-\frac {1}{a^2 x^2}}-\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+3 a \csc ^{-1}(a x) \] Output:
-a*(1-1/a^2/x^2)^(1/2)-4*a^2*(1-1/a^2/x^2)^(1/2)/(a-1/x)+3*a*arccsc(a*x)
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=\frac {a \sqrt {1-\frac {1}{a^2 x^2}} (1-5 a x)}{-1+a x}+3 a \arcsin \left (\frac {1}{a x}\right ) \] Input:
Integrate[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 = 0.46 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.08, 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 |
\(\displaystyle a^4 \left (-\int \frac {a+\frac {3}{x}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}d\frac {1}{x}\right )-a \sqrt {1-\frac {1}{a^2 x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )-\int \frac {a+\frac {3}{x}}{\sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}d\frac {1}{x}\) |
\(\Big \downarrow \) 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[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. \(115\) vs. \(2(49)=98\).
Time = 0.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.19
method | result | size |
risch | \(-\frac {a x -1}{x \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (3 a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\frac {4 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(116\) |
default | \(\frac {-\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}+\sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{2} x^{2}+5 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+3 a^{3} \sqrt {a^{2}}\, x^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-\ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-2 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -7 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}-2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-6 a^{2} \sqrt {a^{2}}\, x^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-2 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+2 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+3 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +3 a \sqrt {a^{2}}\, x \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x -\ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x}{\sqrt {a^{2}}\, x \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(593\) |
Input:
int(1/((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)*((x-1/a)^2*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 = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.40 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {6 \, {\left (a^{2} x^{2} - a x\right )} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x^{2} - x} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/x^2,x, algorithm="fricas")
Output:
-(6*(a^2*x^2 - a*x)*arctan(sqrt((a*x - 1)/(a*x + 1))) + (5*a^2*x^2 + 4*a*x - 1)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^2 - x)
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=\int \frac {1}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)/x**2,x)
Output:
Integral(1/(x**2*((a*x - 1)/(a*x + 1))**(3/2)), x)
Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.36 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=-2 \, a {\left (\frac {\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + 2}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + \sqrt {\frac {a x - 1}{a x + 1}}} + 3 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/x^2,x, algorithm="maxima")
Output:
-2*a*((3*(a*x - 1)/(a*x + 1) + 2)/(((a*x - 1)/(a*x + 1))^(3/2) + sqrt((a*x - 1)/(a*x + 1))) + 3*arctan(sqrt((a*x - 1)/(a*x + 1))))
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=\int { \frac {1}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/x^2,x, algorithm="giac")
Output:
undef
Time = 23.44 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.08 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=\frac {1}{x\,\sqrt {\frac {a\,x-1}{a\,x+1}}}-6\,a\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-\frac {5\,a}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \] Input:
int(1/(x^2*((a*x - 1)/(a*x + 1))^(3/2)),x)
Output:
1/(x*((a*x - 1)/(a*x + 1))^(1/2)) - 6*a*atan(((a*x - 1)/(a*x + 1))^(1/2)) - (5*a)/((a*x - 1)/(a*x + 1))^(1/2)
Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.68 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=\frac {-6 \sqrt {a x -1}\, \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a x +6 \sqrt {a x -1}\, \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a x -3 \sqrt {a x -1}\, a x -5 \sqrt {a x +1}\, a x +\sqrt {a x +1}}{\sqrt {a x -1}\, x} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/x^2,x)
Output:
( - 6*sqrt(a*x - 1)*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a*x + 6*sqrt(a *x - 1)*atan(sqrt(a*x - 1) + sqrt(a*x + 1) + 1)*a*x - 3*sqrt(a*x - 1)*a*x - 5*sqrt(a*x + 1)*a*x + sqrt(a*x + 1))/(sqrt(a*x - 1)*x)