### 3.8 $$\int \frac{e^{\coth ^{-1}(a x)}}{x^4} \, dx$$

Optimal. Leaf size=75 $-\frac{1}{3} a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+a^3 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 x}-\frac{1}{2} a^3 \csc ^{-1}(a x)$

[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*ArcCsc[
a*x])/2

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Rubi [A]  time = 0.0618059, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, 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[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*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 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 = 0.0783049, size = 51, normalized size = 0.68 $\frac{1}{6} a \left (\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left (4 a^2 x^2+3 a x+2\right )}{x^2}-3 a^2 \sin ^{-1}\left (\frac{1}{a x}\right )\right )$

Warning: Unable to verify antiderivative.

[In]

Integrate[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))/x^2 - 3*a^2*ArcSin[1/(a*x)]))/6

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Maple [B]  time = 0.135, size = 284, normalized size = 3.8 \begin{align*} -{\frac{ax-1}{6\,{x}^{3}} \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{{\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^4,x)

[Out]

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

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Maxima [B]  time = 1.55237, size = 184, normalized size = 2.45 \begin{align*} \frac{1}{3} \,{\left (3 \, a^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \frac{3 \, 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}} + 9 \, 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(1/((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))) + (3*a^2*((a*x - 1)/(a*x + 1))^(5/2) + 4*a^2*((a*x - 1)/(a*x + 1)
)^(3/2) + 9*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 = 1.55917, size = 161, normalized size = 2.15 \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} + 7 \, a^{2} x^{2} + 5 \, 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(1/((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 + 7*a^2*x^2 + 5*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{1}{x^{4} \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**4,x)

[Out]

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

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Giac [B]  time = 1.17312, size = 176, normalized size = 2.35 \begin{align*} \frac{1}{3} \,{\left (3 \, a^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \frac{\frac{4 \,{\left (a x - 1\right )} a^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + \frac{3 \,{\left (a x - 1\right )}^{2} a^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 9 \, a^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{3}}\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^4,x, algorithm="giac")

[Out]

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