### 3.7 $$\int \frac{e^{\coth ^{-1}(a x)}}{x^3} \, dx$$

Optimal. Leaf size=38 $\frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a+\frac{1}{x}\right )-\frac{1}{2} a^2 \csc ^{-1}(a x)$

[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 = 0.0308692, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.3, 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[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 = 0.0419218, size = 42, normalized size = 1.11 $\frac{a \left (\sqrt{1-\frac{1}{a^2 x^2}} (2 a x+1)-a x \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{2 x}$

Warning: Unable to verify antiderivative.

[In]

Integrate[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 = 0.126, size = 257, normalized size = 6.8 \begin{align*} -{\frac{ax-1}{2\,{x}^{2}} \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{{\frac{ax-1}{ax+1}}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.52503, size = 123, normalized size = 3.24 \begin{align*}{\left (a \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \frac{a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 3 \, 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(1/((a*x-1)/(a*x+1))^(1/2)/x^3,x, algorithm="maxima")

[Out]

(a*arctan(sqrt((a*x - 1)/(a*x + 1))) + (a*((a*x - 1)/(a*x + 1))^(3/2) + 3*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 = 1.72788, size = 144, normalized size = 3.79 \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} + 3 \, 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(1/((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 + 3*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{1}{x^{3} \sqrt{\frac{a x - 1}{a x + 1}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.13562, size = 117, normalized size = 3.08 \begin{align*}{\left (a \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \frac{\frac{{\left (a x - 1\right )} a \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + 3 \, a \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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