\(\int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx\) [40]

Optimal result

Integrand size = 12, antiderivative size = 76 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=-a^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a^3 \csc ^{-1}(a x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6304, 811, 655, 201, 222} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=-\frac {1}{2} a^3 \csc ^{-1}(a x)+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-a^3 \sqrt {1-\frac {1}{a^2 x^2}} \]

[In]

Int[1/(E^ArcCoth[a*x]*x^4),x]

[Out]

-(a^3*Sqrt[1 - 1/(a^2*x^2)]) + (a^3*(1 - 1/(a^2*x^2))^(3/2))/3 + (a^2*Sqrt[1 - 1/(a^2*x^2)])/(2*x) - (a^3*ArcC
sc[a*x])/2

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 655

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

Rule 811

Int[(x_)^2*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/c, Int[(f + g*x)*(a + c*x^2)^(p
+ 1), x], x] - Dist[a/c, Int[(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, f, g, p}, x] && EqQ[a*g^2 + f^2*
c, 0]

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^2 \left (1-\frac {x}{a}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\left (a^2 \text {Subst}\left (\int \frac {1-\frac {x}{a}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\right )+a^2 \text {Subst}\left (\int \left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}} \, dx,x,\frac {1}{x}\right ) \\ & = -a^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-a^2 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )+a^2 \text {Subst}\left (\int \sqrt {1-\frac {x^2}{a^2}} \, dx,x,\frac {1}{x}\right ) \\ & = -a^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-a^3 \csc ^{-1}(a x)+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -a^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a^3 \csc ^{-1}(a x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=-\frac {a \sqrt {1-\frac {1}{a^2 x^2}} \left (2-3 a x+4 a^2 x^2\right )}{6 x^2}-\frac {1}{2} a^3 \arcsin \left (\frac {1}{a x}\right ) \]

[In]

Integrate[1/(E^ArcCoth[a*x]*x^4),x]

[Out]

-1/6*(a*Sqrt[1 - 1/(a^2*x^2)]*(2 - 3*a*x + 4*a^2*x^2))/x^2 - (a^3*ArcSin[1/(a*x)])/2

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.22

method	result	size
risch	\(-\frac {\left (a x +1\right ) \left (4 a^{2} x^{2}-3 a x +2\right ) \sqrt {\frac {a x -1}{a x +1}}}{6 x^{3}}-\frac {a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 \left (a x -1\right )}\)	\(93\)
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-6 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-3 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-6 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-3 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} \sqrt {a^{2}}}\)	\(284\)

[In]

int(((a*x-1)/(a*x+1))^(1/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

-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)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=\frac {6 \, a^{3} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (4 \, a^{3} x^{3} + a^{2} x^{2} - a x + 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, x^{3}} \]

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/x^4,x, algorithm="fricas")

[Out]

1/6*(6*a^3*x^3*arctan(sqrt((a*x - 1)/(a*x + 1))) - (4*a^3*x^3 + a^2*x^2 - a*x + 2)*sqrt((a*x - 1)/(a*x + 1)))/
x^3

Sympy [F]

\[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=\int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{x^{4}}\, dx \]

[In]

integrate(((a*x-1)/(a*x+1))**(1/2)/x**4,x)

[Out]

Integral(sqrt((a*x - 1)/(a*x + 1))/x**4, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (64) = 128\).

Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.80 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{3} \, {\left (3 \, a^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - \frac {9 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 4 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 3 \, a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}\right )} a \]

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/x^4,x, algorithm="maxima")

[Out]

1/3*(3*a^2*arctan(sqrt((a*x - 1)/(a*x + 1))) - (9*a^2*((a*x - 1)/(a*x + 1))^(5/2) + 4*a^2*((a*x - 1)/(a*x + 1)
)^(3/2) + 3*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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/x^4,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 4.38 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.38 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=a^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,x^3}-\frac {2\,a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3}-\frac {a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{6\,x}+\frac {a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{6\,x^2} \]

[In]

int(((a*x - 1)/(a*x + 1))^(1/2)/x^4,x)

[Out]

a^3*atan(((a*x - 1)/(a*x + 1))^(1/2)) - ((a*x - 1)/(a*x + 1))^(1/2)/(3*x^3) - (2*a^3*((a*x - 1)/(a*x + 1))^(1/
2))/3 - (a^2*((a*x - 1)/(a*x + 1))^(1/2))/(6*x) + (a*((a*x - 1)/(a*x + 1))^(1/2))/(6*x^2)