3.1.55 \(\int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx\) [55]

Optimal. Leaf size=53 \[ 3 a \sqrt {1-\frac {1}{a^2 x^2}}+\frac {2 \left (a-\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 a \csc ^{-1}(a x) \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6304, 867, 683, 655, 222} \begin {gather*} \frac {2 \left (a-\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 a \sqrt {1-\frac {1}{a^2 x^2}}+3 a \csc ^{-1}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcCoth[a*x])*x^2),x]

[Out]

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

Rule 222

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 655

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 683

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

Rule 867

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

Rule 6304

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^{-3 \coth ^{-1}(a x)}}{x^2} \, dx &=-\text {Subst}\left (\int \frac {\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 )\\ &=-\text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 \left (a-\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 \text {Subst}\left (\int \frac {1-\frac {x}{a}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=3 a \sqrt {1-\frac {1}{a^2 x^2}}+\frac {2 \left (a-\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=3 a \sqrt {1-\frac {1}{a^2 x^2}}+\frac {2 \left (a-\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 a \csc ^{-1}(a x)\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 41, normalized size = 0.77 \begin {gather*} \frac {a \sqrt {1-\frac {1}{a^2 x^2}} (1+5 a x)}{1+a x}+3 a \text {ArcSin}\left (\frac {1}{a x}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*x^2),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(593\) vs. \(2(49)=98\).
time = 0.10, size = 594, normalized size = 11.21

method	result	size
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{x}+\frac {\left (\frac {4 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}+3 a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a x -1}\)	\(109\)
default	\(\frac {\left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+5 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+3 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}-\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-2 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +7 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+6 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{2} x^{2}-2 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a x -2 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+3 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +3 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a x -\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x -\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x \right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{x \sqrt {a^{2}}\, \left (a x -1\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}}\)	\(594\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a^2*x^2-1)^(1/2)*(a^2)^(1/2)*a^4*x^4+ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^4*x^3-(a^
2*x^2-1)^(3/2)*(a^2)^(1/2)*a^2*x^2+5*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*a^3*x^3+3*a^3*x^3*(a^2)^(1/2)*arctan(1/(a^2
*x^2-1)^(1/2))-(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)*a^3*x^3-ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2
))*a^4*x^3+2*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^3*x^2-2*(a^2)^(1/2)*(a^2*x^2-1)^(3/
2)*a*x+7*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*a^2*x^2+6*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))*a^2*x^2-2*(a^2)^(1/2)
*((a*x+1)*(a*x-1))^(3/2)*a*x-2*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a^2*x^2-2*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)
^(1/2))/(a^2)^(1/2))*a^3*x^2+ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^2*x-(a^2*x^2-1)^(3/
2)*(a^2)^(1/2)+3*(a^2)^(1/2)*(a^2*x^2-1)^(1/2)*a*x+3*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))*a*x-((a*x+1)*(a*x
-1))^(1/2)*(a^2)^(1/2)*a*x-ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^2*x)*((a*x-1)/(a*x+1))^(3/2
)/x/(a^2)^(1/2)/(a*x-1)/((a*x+1)*(a*x-1))^(1/2)

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Maxima [A]
time = 0.46, size = 72, normalized size = 1.36 \begin {gather*} 2 \, a {\left (2 \, \sqrt {\frac {a x - 1}{a x + 1}} + \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\frac {a x - 1}{a x + 1} + 1} - 3 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 49, normalized size = 0.92 \begin {gather*} -\frac {6 \, a x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (5 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(6*a*x*arctan(sqrt((a*x - 1)/(a*x + 1))) - (5*a*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef

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Mupad [B]
time = 0.05, size = 59, normalized size = 1.11 \begin {gather*} \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}+5\,a\,x\,\sqrt {\frac {a\,x-1}{a\,x+1}}-6\,a\,x\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((a*x - 1)/(a*x + 1))^(1/2) + 5*a*x*((a*x - 1)/(a*x + 1))^(1/2) - 6*a*x*atan(((a*x - 1)/(a*x + 1))^(1/2)))/x

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