### 3.40 $$\int \frac{e^{-\coth ^{-1}(a x)}}{x^4} \, dx$$

Optimal. Leaf size=76 $\frac{1}{3} a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-a^3 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 x}-\frac{1}{2} a^3 \csc ^{-1}(a x)$

[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

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Rubi [A]  time = 0.0656624, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.417, Rules used = {6169, 797, 641, 195, 216} $\frac{1}{3} a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-a^3 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 x}-\frac{1}{2} a^3 \csc ^{-1}(a x)$

Antiderivative was successfully veriﬁed.

[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 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 797

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 641

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 195

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 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^4} \, dx &=-\operatorname{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 \operatorname{Subst}\left (\int \frac{1-\frac{x}{a}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\right )+a^2 \operatorname{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 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )+a^2 \operatorname{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 \operatorname{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]  time = 0.0786569, size = 52, normalized size = 0.68 $-\frac{a \sqrt{1-\frac{1}{a^2 x^2}} \left (4 a^2 x^2-3 a x+2\right )}{6 x^2}-\frac{1}{2} a^3 \sin ^{-1}\left (\frac{1}{a x}\right )$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.13, size = 284, normalized size = 3.7 \begin{align*}{\frac{ax+1}{6\,{x}^{3}}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -6\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{4}{a}^{4}+6\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-3\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+6\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}-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 ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}-6\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}-3\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+2\, \left ({a}^{2}{x}^{2}-1 \right ) ^{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^4,x)

[Out]

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

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Maxima [B]  time = 1.57039, size = 185, normalized size = 2.43 \begin{align*} \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 \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.85405, size = 155, normalized size = 2.04 \begin{align*} \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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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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^{4}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.16876, size = 219, normalized size = 2.88 \begin{align*} a^{3} \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right ) - \frac{3 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{5} a^{3} \mathrm{sgn}\left (a x + 1\right ) + 12 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} a^{2}{\left | a \right |} \mathrm{sgn}\left (a x + 1\right ) - 3 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )} a^{3} \mathrm{sgn}\left (a x + 1\right ) + 4 \, a^{2}{\left | a \right |} \mathrm{sgn}\left (a x + 1\right )}{3 \,{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1) - 1/3*(3*(x*abs(a) - sqrt(a^2*x^2 - 1))^5*a^3*sgn(a*x +
1) + 12*(x*abs(a) - sqrt(a^2*x^2 - 1))^2*a^2*abs(a)*sgn(a*x + 1) - 3*(x*abs(a) - sqrt(a^2*x^2 - 1))*a^3*sgn(a
*x + 1) + 4*a^2*abs(a)*sgn(a*x + 1))/((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^3