### 3.39 $$\int \frac{e^{-\coth ^{-1}(a x)}}{x^3} \, dx$$

Optimal. Leaf size=40 $\frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a-\frac{1}{x}\right )+\frac{1}{2} a^2 \csc ^{-1}(a x)$

[Out]

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

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Rubi [A]  time = 0.036613, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {6169, 780, 216} $\frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a-\frac{1}{x}\right )+\frac{1}{2} a^2 \csc ^{-1}(a x)$

Antiderivative was successfully veriﬁed.

[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 6169

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]

Rule 780

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 216

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]

Rubi steps

\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{x^3} \, dx &=-\operatorname{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 \operatorname{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]  time = 0.0464496, size = 41, normalized size = 1.02 $\frac{a \left (\sqrt{1-\frac{1}{a^2 x^2}} (2 a x-1)+a x \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{2 x}$

Warning: Unable to verify antiderivative.

[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)

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Maple [B]  time = 0.138, size = 260, normalized size = 6.5 \begin{align*} -{\frac{ax+1}{2\,{x}^{2}}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -2\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+2\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+2\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-{a}^{2}{x}^{2}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +2\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}-2\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}- \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.57412, size = 126, normalized size = 3.15 \begin{align*} -{\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 \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.85432, size = 143, normalized size = 3.58 \begin{align*} -\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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{x^{3}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.1941, size = 212, normalized size = 5.3 \begin{align*} -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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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