### 3.56 $$\int \frac{e^{-3 \coth ^{-1}(a x)}}{x^3} \, dx$$

Optimal. Leaf size=87 $-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (a+\frac{1}{x}\right )^3}-\frac{3 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 \left (a+\frac{1}{x}\right )}-\frac{9}{2} a^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{9}{2} a^2 \csc ^{-1}(a x)$

[Out]

(-9*a^2*Sqrt[1 - 1/(a^2*x^2)])/2 - (a^5*(1 - 1/(a^2*x^2))^(5/2))/(a + x^(-1))^3 - (3*a^3*(1 - 1/(a^2*x^2))^(3/
2))/(2*(a + x^(-1))) - (9*a^2*ArcCsc[a*x])/2

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Rubi [A]  time = 0.444977, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.583, Rules used = {6169, 1633, 1593, 12, 793, 665, 216} $-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (a+\frac{1}{x}\right )^3}-\frac{3 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 \left (a+\frac{1}{x}\right )}-\frac{9}{2} a^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{9}{2} a^2 \csc ^{-1}(a x)$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^(3*ArcCoth[a*x])*x^3),x]

[Out]

(-9*a^2*Sqrt[1 - 1/(a^2*x^2)])/2 - (a^5*(1 - 1/(a^2*x^2))^(5/2))/(a + x^(-1))^3 - (3*a^3*(1 - 1/(a^2*x^2))^(3/
2))/(2*(a + x^(-1))) - (9*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 1633

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d*e, Int[(d + e*x)^(m - 1)*
PolynomialQuotient[Pq, a*e + c*d*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && EqQ[c*d^2 + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + c*d*x, x], 0]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 793

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

Rule 665

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 2*p + 1)), x] - Dist[(2*c*d*p)/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[
m + 2*p + 1, 0] && IntegerQ[2*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^{-3 \coth ^{-1}(a x)}}{x^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x \left (1-\frac{x}{a}\right )^2}{\left (1+\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a x-x^2\right ) \sqrt{1-\frac{x^2}{a^2}}}{\left (1+\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a-x) x \sqrt{1-\frac{x^2}{a^2}}}{\left (1+\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{a^2 x \left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (1+\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )}{a^2}\\ &=-\operatorname{Subst}\left (\int \frac{x \left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (1+\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (a+\frac{1}{x}\right )^3}-(3 a) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (1+\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (a+\frac{1}{x}\right )^3}-\frac{3 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 \left (a+\frac{1}{x}\right )}-\frac{1}{2} (9 a) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{1+\frac{x}{a}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{9}{2} a^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (a+\frac{1}{x}\right )^3}-\frac{3 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 \left (a+\frac{1}{x}\right )}-\frac{1}{2} (9 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{9}{2} a^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (a+\frac{1}{x}\right )^3}-\frac{3 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 \left (a+\frac{1}{x}\right )}-\frac{9}{2} a^2 \csc ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.136651, size = 56, normalized size = 0.64 $\frac{1}{2} a \left (\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left (-14 a^2 x^2-5 a x+1\right )}{x (a x+1)}-9 a \sin ^{-1}\left (\frac{1}{a x}\right )\right )$

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*x^3),x]

[Out]

(a*((Sqrt[1 - 1/(a^2*x^2)]*(1 - 5*a*x - 14*a^2*x^2))/(x*(1 + a*x)) - 9*a*ArcSin[1/(a*x)]))/2

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Maple [B]  time = 0.131, size = 642, normalized size = 7.4 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(-6*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^5*a^5+6*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^3*a^3-21*(a^2*x^2-1)^(1/2)*(a^
2)^(1/2)*x^4*a^4+6*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^4*a^5-9*a^4*x^4*(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^4*a^4-6*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(
1/2))/(a^2)^(1/2))*x^4*a^5+11*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^2*a^2-24*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^3*a^3+1
2*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^3*a^4-18*a^3*x^3*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1
/2))+4*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2+12*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3-12*ln((a^2*x
+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^3*a^4+4*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*x*a-9*(a^2*x^2-1)^(
1/2)*(a^2)^(1/2)*x^2*a^2+6*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3-9*a^2*x^2*(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^2*a^2-6*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))^(3/2)/(a^2)^(1/2)/x^2/(a*x
-1)/((a*x-1)*(a*x+1))^(1/2)

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Maxima [A]  time = 1.49877, size = 151, normalized size = 1.74 \begin{align*}{\left (9 \, a \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - 4 \, a \sqrt{\frac{a x - 1}{a x + 1}} - \frac{7 \, a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 5 \, 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))^(3/2)/x^3,x, algorithm="maxima")

[Out]

(9*a*arctan(sqrt((a*x - 1)/(a*x + 1))) - 4*a*sqrt((a*x - 1)/(a*x + 1)) - (7*a*((a*x - 1)/(a*x + 1))^(3/2) + 5*
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 = 1.59415, size = 147, normalized size = 1.69 \begin{align*} \frac{18 \, a^{2} x^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) -{\left (14 \, a^{2} x^{2} + 5 \, 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))^(3/2)/x^3,x, algorithm="fricas")

[Out]

1/2*(18*a^2*x^2*arctan(sqrt((a*x - 1)/(a*x + 1))) - (14*a^2*x^2 + 5*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{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{3}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef