### 3.9 $$\int \frac{e^{\coth ^{-1}(a x)}}{x^5} \, dx$$

Optimal. Leaf size=88 $\frac{1}{24} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a+\frac{9}{x}\right )+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 x^2}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}-\frac{3}{8} a^4 \csc ^{-1}(a x)$

[Out]

(a^3*Sqrt[1 - 1/(a^2*x^2)]*(16*a + 9/x))/24 + (a*Sqrt[1 - 1/(a^2*x^2)])/(4*x^3) + (a^2*Sqrt[1 - 1/(a^2*x^2)])/
(3*x^2) - (3*a^4*ArcCsc[a*x])/8

________________________________________________________________________________________

Rubi [A]  time = 0.0828003, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {6169, 833, 780, 216} $\frac{1}{24} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a+\frac{9}{x}\right )+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 x^2}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}-\frac{3}{8} a^4 \csc ^{-1}(a x)$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCoth[a*x]/x^5,x]

[Out]

(a^3*Sqrt[1 - 1/(a^2*x^2)]*(16*a + 9/x))/24 + (a*Sqrt[1 - 1/(a^2*x^2)])/(4*x^3) + (a^2*Sqrt[1 - 1/(a^2*x^2)])/
(3*x^2) - (3*a^4*ArcCsc[a*x])/8

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 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

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^5} \, dx &=-\operatorname{Subst}\left (\int \frac{x^3 \left (1+\frac{x}{a}\right )}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}+\frac{1}{4} a^2 \operatorname{Subst}\left (\int \frac{x^2 \left (-\frac{3}{a}-\frac{4 x}{a^2}\right )}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 x^2}-\frac{1}{12} a^4 \operatorname{Subst}\left (\int \frac{x \left (\frac{8}{a^2}+\frac{9 x}{a^3}\right )}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{24} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a+\frac{9}{x}\right )+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 x^2}-\frac{1}{8} \left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{24} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a+\frac{9}{x}\right )+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 x^2}-\frac{3}{8} a^4 \csc ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0942163, size = 59, normalized size = 0.67 $\frac{1}{24} a \left (\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left (16 a^3 x^3+9 a^2 x^2+8 a x+6\right )}{x^3}-9 a^3 \sin ^{-1}\left (\frac{1}{a x}\right )\right )$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCoth[a*x]/x^5,x]

[Out]

(a*((Sqrt[1 - 1/(a^2*x^2)]*(6 + 8*a*x + 9*a^2*x^2 + 16*a^3*x^3))/x^3 - 9*a^3*ArcSin[1/(a*x)]))/24

________________________________________________________________________________________

Maple [B]  time = 0.138, size = 308, normalized size = 3.5 \begin{align*} -{\frac{ax-1}{24\,{x}^{4}} \left ( -24\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{5}{a}^{5}+24\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+9\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{4}{a}^{4}+9\,{a}^{4}{x}^{4}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{4}{a}^{5}-24\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{4}{a}^{4}-24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{4}{a}^{5}+15\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+8\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+6\, \left ({a}^{2}{x}^{2}-1 \right ) ^{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^5,x)

[Out]

-1/24*(a*x-1)*(-24*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^5*a^5+24*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^3*a^3+9*(a^2*x^2-1
)^(1/2)*(a^2)^(1/2)*x^4*a^4+9*a^4*x^4*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))+24*ln((a^2*x+(a^2*x^2-1)^(1/2)*(
a^2)^(1/2))/(a^2)^(1/2))*x^4*a^5-24*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4-24*ln((a^2*x+(a^2)^(1/2)*((a*x
-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^4*a^5+15*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^2*a^2+8*(a^2)^(1/2)*(a^2*x^2-1)^(3
/2)*x*a+6*(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^4/(a^2)^(1/2)

________________________________________________________________________________________

Maxima [B]  time = 1.56124, size = 232, normalized size = 2.64 \begin{align*} \frac{1}{12} \,{\left (9 \, a^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \frac{9 \, a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 49 \, a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 31 \, a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 39 \, a^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{4 \,{\left (a x - 1\right )}}{a x + 1} + \frac{6 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{4 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac{{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 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^5,x, algorithm="maxima")

[Out]

1/12*(9*a^3*arctan(sqrt((a*x - 1)/(a*x + 1))) + (9*a^3*((a*x - 1)/(a*x + 1))^(7/2) + 49*a^3*((a*x - 1)/(a*x +
1))^(5/2) + 31*a^3*((a*x - 1)/(a*x + 1))^(3/2) + 39*a^3*sqrt((a*x - 1)/(a*x + 1)))/(4*(a*x - 1)/(a*x + 1) + 6*
(a*x - 1)^2/(a*x + 1)^2 + 4*(a*x - 1)^3/(a*x + 1)^3 + (a*x - 1)^4/(a*x + 1)^4 + 1))*a

________________________________________________________________________________________

Fricas [A]  time = 1.96609, size = 185, normalized size = 2.1 \begin{align*} \frac{18 \, a^{4} x^{4} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) +{\left (16 \, a^{4} x^{4} + 25 \, a^{3} x^{3} + 17 \, a^{2} x^{2} + 14 \, a x + 6\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{24 \, x^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/x^5,x, algorithm="fricas")

[Out]

1/24*(18*a^4*x^4*arctan(sqrt((a*x - 1)/(a*x + 1))) + (16*a^4*x^4 + 25*a^3*x^3 + 17*a^2*x^2 + 14*a*x + 6)*sqrt(
(a*x - 1)/(a*x + 1)))/x^4

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(1/2)/x**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.15896, size = 221, normalized size = 2.51 \begin{align*} \frac{1}{12} \,{\left (9 \, a^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \frac{\frac{31 \,{\left (a x - 1\right )} a^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + \frac{49 \,{\left (a x - 1\right )}^{2} a^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + \frac{9 \,{\left (a x - 1\right )}^{3} a^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + 39 \, a^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{4}}\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^5,x, algorithm="giac")

[Out]

1/12*(9*a^3*arctan(sqrt((a*x - 1)/(a*x + 1))) + (31*(a*x - 1)*a^3*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1) + 49*(a*
x - 1)^2*a^3*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^2 + 9*(a*x - 1)^3*a^3*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^3 +
39*a^3*sqrt((a*x - 1)/(a*x + 1)))/((a*x - 1)/(a*x + 1) + 1)^4)*a