3.1.22 \(\int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx\) [22]

Optimal. Leaf size=51 \[ -3 a \sqrt {1-\frac {1}{a^2 x^2}}-\frac {2 \left (a+\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 a \csc ^{-1}(a x) \]

[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 = 51, 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[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 = 0.06, size = 41, normalized size = 0.80 \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[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. \(592\) vs. \(2(47)=94\).
time = 0.10, size = 593, normalized size = 11.63

method	result	size
risch	\(-\frac {a x -1}{x \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (-\frac {4 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}+3 a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(116\)
default	\(-\frac {\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}+\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-5 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-3 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+2 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a x -2 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+2 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +7 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+6 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{2} x^{2}+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x -3 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a x -\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x}{x \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\)	\(593\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((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+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)*a^3*x^3+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))-ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/
2))*a^4*x^3+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)^(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*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2
)^(1/2))/(a^2)^(1/2))*a^3*x^2+((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a*x+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)*arct
an(1/(a^2*x^2-1)^(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)/x/(a^2)^(1/2)/((a*x+1
)*(a*x-1))^(1/2)/(a*x+1)/((a*x-1)/(a*x+1))^(3/2)

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Maxima [A]
time = 0.46, size = 72, normalized size = 1.41 \begin {gather*} -2 \, a {\left (\frac {\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + 2}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + \sqrt {\frac {a x - 1}{a x + 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(1/((a*x-1)/(a*x+1))^(3/2)/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 74, normalized size = 1.45 \begin {gather*} -\frac {6 \, {\left (a^{2} x^{2} - a x\right )} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x^{2} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(6*(a^2*x^2 - a*x)*arctan(sqrt((a*x - 1)/(a*x + 1))) + (5*a^2*x^2 + 4*a*x - 1)*sqrt((a*x - 1)/(a*x + 1)))/(a*
x^2 - x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**2*((a*x - 1)/(a*x + 1))**(3/2)), 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(1/((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 = 57, normalized size = 1.12 \begin {gather*} \frac {1}{x\,\sqrt {\frac {a\,x-1}{a\,x+1}}}-6\,a\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-\frac {5\,a}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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