\(\int \frac {e^{-\coth ^{-1}(a x)}}{x^3} \, dx\) [39]

Optimal result

Integrand size = 12, antiderivative size = 40 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {1}{x}\right )+\frac {1}{2} a^2 \csc ^{-1}(a x) \]

[Out]

1/2*a^2*arccsc(a*x)+1/2*a*(2*a-1/x)*(1-1/a^2/x^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6304, 794, 222} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {1}{x}\right )+\frac {1}{2} a^2 \csc ^{-1}(a x) \]

[In]

Int[1/(E^ArcCoth[a*x]*x^3),x]

[Out]

(a*Sqrt[1 - 1/(a^2*x^2)]*(2*a - x^(-1)))/2 + (a^2*ArcCsc[a*x])/2

Rule 222

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]

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 6304

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]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x \left (1-\frac {x}{a}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {1}{x}\right )+\frac {1}{2} a \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {1}{x}\right )+\frac {1}{2} a^2 \csc ^{-1}(a x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^3} \, dx=\frac {a \left (\sqrt {1-\frac {1}{a^2 x^2}} (-1+2 a x)+a x \arcsin \left (\frac {1}{a x}\right )\right )}{2 x} \]

[In]

Integrate[1/(E^ArcCoth[a*x]*x^3),x]

[Out]

(a*(Sqrt[1 - 1/(a^2*x^2)]*(-1 + 2*a*x) + a*x*ArcSin[1/(a*x)]))/(2*x)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(84\) vs. \(2(34)=68\).

Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.12

method	result	size
risch	\(\frac {\left (a x +1\right ) \left (2 a x -1\right ) \sqrt {\frac {a x -1}{a x +1}}}{2 x^{2}}+\frac {a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 a x -2}\)	\(85\)
default	\(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-2 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x -\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-a^{2} x^{2} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+2 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{2} \sqrt {a^{2}}}\)	\(260\)

[In]

int(((a*x-1)/(a*x+1))^(1/2)/x^3,x,method=_RETURNVERBOSE)

[Out]

1/2*(a*x+1)*(2*a*x-1)/x^2*((a*x-1)/(a*x+1))^(1/2)+1/2*a^2*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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.50 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^3} \, dx=-\frac {2 \, a^{2} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (2 \, a^{2} x^{2} + a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, x^{2}} \]

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/x^3,x, algorithm="fricas")

[Out]

-1/2*(2*a^2*x^2*arctan(sqrt((a*x - 1)/(a*x + 1))) - (2*a^2*x^2 + a*x - 1)*sqrt((a*x - 1)/(a*x + 1)))/x^2

Sympy [F]

\[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^3} \, dx=\int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{x^{3}}\, dx \]

[In]

integrate(((a*x-1)/(a*x+1))**(1/2)/x**3,x)

[Out]

Integral(sqrt((a*x - 1)/(a*x + 1))/x**3, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (34) = 68\).

Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.32 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^3} \, dx=-{\left (a \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - \frac {3 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + a \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )}}{a x + 1} + \frac {{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}\right )} a \]

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/x^3,x, algorithm="maxima")

[Out]

-(a*arctan(sqrt((a*x - 1)/(a*x + 1))) - (3*a*((a*x - 1)/(a*x + 1))^(3/2) + a*sqrt((a*x - 1)/(a*x + 1)))/(2*(a*
x - 1)/(a*x + 1) + (a*x - 1)^2/(a*x + 1)^2 + 1))*a

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (34) = 68\).

Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 3.92 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^3} \, dx=-a^{2} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right ) + \frac {{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} a^{2} \mathrm {sgn}\left (a x + 1\right ) + 2 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} a^{2} \mathrm {sgn}\left (a x + 1\right ) + 2 \, a {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{2}} \]

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/x^3,x, algorithm="giac")

[Out]

-a^2*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1) + ((x*abs(a) - sqrt(a^2*x^2 - 1))^3*a^2*sgn(a*x + 1) +
2*(x*abs(a) - sqrt(a^2*x^2 - 1))^2*a*abs(a)*sgn(a*x + 1) - (x*abs(a) - sqrt(a^2*x^2 - 1))*a^2*sgn(a*x + 1) +
2*a*abs(a)*sgn(a*x + 1))/((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^2

Mupad [B] (verification not implemented)

Time = 4.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.05 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^3} \, dx=a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,x^2}-a^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+\frac {a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,x} \]

[In]

int(((a*x - 1)/(a*x + 1))^(1/2)/x^3,x)

[Out]

a^2*((a*x - 1)/(a*x + 1))^(1/2) - ((a*x - 1)/(a*x + 1))^(1/2)/(2*x^2) - a^2*atan(((a*x - 1)/(a*x + 1))^(1/2))
+ (a*((a*x - 1)/(a*x + 1))^(1/2))/(2*x)