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]

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

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Rubi [A]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.05, 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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fricas [A]  time = 0.63, size = 60, normalized size = 1.58 \[ \frac {2 \, a^{2} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (2 \, a^{2} x^{2} + 3 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, x^{2}} \]

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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giac [B]  time = 0.14, size = 87, normalized size = 2.29 \[ {\left (a \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {\frac {{\left (a x - 1\right )} a \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + 3 \, a \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \]

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

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maple [B]  time = 0.05, size = 260, normalized size = 6.84 \[ \frac {\left (a x -1\right ) \left (2 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}+2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}-2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a -\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}-a^{2} x^{2} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+2 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{2 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{2} \sqrt {a^{2}}} \]

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*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^2*a^2-2*(a^2*x^2-
1)^(3/2)*(a^2)^(1/2)*x*a-(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^2*a^2-a^2*x^2*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))
-2*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3+2*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(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 = 0.41, size = 91, normalized size = 2.39 \[ {\left (a \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 3 \, a \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )}}{a x + 1} + \frac {{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}\right )} a \]

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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mupad [B]  time = 1.20, size = 81, normalized size = 2.13 \[ a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,x^2}+a^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+\frac {3\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]

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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