3.49 \(\int \frac {\coth ^{-1}(x)}{\sqrt {a-a x^2}} \, dx\)

Optimal. Leaf size=144 \[ -\frac {i \sqrt {1-x^2} \text {Li}_2\left (-\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )}{\sqrt {a-a x^2}}+\frac {i \sqrt {1-x^2} \text {Li}_2\left (\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )}{\sqrt {a-a x^2}}-\frac {2 \sqrt {1-x^2} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \coth ^{-1}(x)}{\sqrt {a-a x^2}} \]

[Out]

-2*arccoth(x)*arctan((1-x)^(1/2)/(1+x)^(1/2))*(-x^2+1)^(1/2)/(-a*x^2+a)^(1/2)-I*polylog(2,-I*(1-x)^(1/2)/(1+x)
^(1/2))*(-x^2+1)^(1/2)/(-a*x^2+a)^(1/2)+I*polylog(2,I*(1-x)^(1/2)/(1+x)^(1/2))*(-x^2+1)^(1/2)/(-a*x^2+a)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5955, 5951} \[ -\frac {i \sqrt {1-x^2} \text {PolyLog}\left (2,-\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )}{\sqrt {a-a x^2}}+\frac {i \sqrt {1-x^2} \text {PolyLog}\left (2,\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )}{\sqrt {a-a x^2}}-\frac {2 \sqrt {1-x^2} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \coth ^{-1}(x)}{\sqrt {a-a x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[x]/Sqrt[a - a*x^2],x]

[Out]

(-2*Sqrt[1 - x^2]*ArcCoth[x]*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]])/Sqrt[a - a*x^2] - (I*Sqrt[1 - x^2]*PolyLog[2, ((
-I)*Sqrt[1 - x])/Sqrt[1 + x]])/Sqrt[a - a*x^2] + (I*Sqrt[1 - x^2]*PolyLog[2, (I*Sqrt[1 - x])/Sqrt[1 + x]])/Sqr
t[a - a*x^2]

Rule 5951

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-2*(a + b*ArcCoth[c*x])*
ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(c*Sqrt[d]), x] + (-Simp[(I*b*PolyLog[2, -((I*Sqrt[1 - c*x])/Sqrt[1 + c*x
])])/(c*Sqrt[d]), x] + Simp[(I*b*PolyLog[2, (I*Sqrt[1 - c*x])/Sqrt[1 + c*x]])/(c*Sqrt[d]), x]) /; FreeQ[{a, b,
c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]

Rule 5955

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 - c^2*x^2]/S
qrt[d + e*x^2], Int[(a + b*ArcCoth[c*x])^p/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d
+ e, 0] && IGtQ[p, 0] &&  !GtQ[d, 0]

Rubi steps

\begin {align*} \int \frac {\coth ^{-1}(x)}{\sqrt {a-a x^2}} \, dx &=\frac {\sqrt {1-x^2} \int \frac {\coth ^{-1}(x)}{\sqrt {1-x^2}} \, dx}{\sqrt {a-a x^2}}\\ &=-\frac {2 \sqrt {1-x^2} \coth ^{-1}(x) \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right )}{\sqrt {a-a x^2}}-\frac {i \sqrt {1-x^2} \text {Li}_2\left (-\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{\sqrt {a-a x^2}}+\frac {i \sqrt {1-x^2} \text {Li}_2\left (\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{\sqrt {a-a x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 77, normalized size = 0.53 \[ \frac {\sqrt {a-a x^2} \left (\text {Li}_2\left (-e^{-\coth ^{-1}(x)}\right )-\text {Li}_2\left (e^{-\coth ^{-1}(x)}\right )+\coth ^{-1}(x) \left (\log \left (1-e^{-\coth ^{-1}(x)}\right )-\log \left (e^{-\coth ^{-1}(x)}+1\right )\right )\right )}{a \sqrt {1-\frac {1}{x^2}} x} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCoth[x]/Sqrt[a - a*x^2],x]

[Out]

(Sqrt[a - a*x^2]*(ArcCoth[x]*(Log[1 - E^(-ArcCoth[x])] - Log[1 + E^(-ArcCoth[x])]) + PolyLog[2, -E^(-ArcCoth[x
])] - PolyLog[2, E^(-ArcCoth[x])]))/(a*Sqrt[1 - x^(-2)]*x)

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a x^{2} + a} \operatorname {arcoth}\relax (x)}{a x^{2} - a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-a*x^2 + a)*arccoth(x)/(a*x^2 - a), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\relax (x)}{\sqrt {-a x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccoth(x)/sqrt(-a*x^2 + a), x)

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maple [A]  time = 0.56, size = 190, normalized size = 1.32 \[ \frac {\ln \left (1-\frac {1}{\sqrt {\frac {-1+x}{1+x}}}\right ) \mathrm {arccoth}\relax (x ) \sqrt {\frac {-1+x}{1+x}}\, \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{\left (-1+x \right ) a}+\frac {\polylog \left (2, \frac {1}{\sqrt {\frac {-1+x}{1+x}}}\right ) \sqrt {\frac {-1+x}{1+x}}\, \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{\left (-1+x \right ) a}-\frac {\ln \left (1+\frac {1}{\sqrt {\frac {-1+x}{1+x}}}\right ) \mathrm {arccoth}\relax (x ) \sqrt {\frac {-1+x}{1+x}}\, \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{\left (-1+x \right ) a}-\frac {\polylog \left (2, -\frac {1}{\sqrt {\frac {-1+x}{1+x}}}\right ) \sqrt {\frac {-1+x}{1+x}}\, \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{\left (-1+x \right ) a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(1-1/((-1+x)/(1+x))^(1/2))*arccoth(x)*((-1+x)/(1+x))^(1/2)*(-(-1+x)*(1+x)*a)^(1/2)/(-1+x)/a+polylog(2,1/((-1
+x)/(1+x))^(1/2))*((-1+x)/(1+x))^(1/2)*(-(-1+x)*(1+x)*a)^(1/2)/(-1+x)/a-ln(1+1/((-1+x)/(1+x))^(1/2))*arccoth(x
)*((-1+x)/(1+x))^(1/2)*(-(-1+x)*(1+x)*a)^(1/2)/(-1+x)/a-polylog(2,-1/((-1+x)/(1+x))^(1/2))*((-1+x)/(1+x))^(1/2
)*(-(-1+x)*(1+x)*a)^(1/2)/(-1+x)/a

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\relax (x)}{\sqrt {-a x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccoth(x)/sqrt(-a*x^2 + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acoth}\relax (x)}{\sqrt {a-a\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(x)/(a - a*x^2)^(1/2),x)

[Out]

int(acoth(x)/(a - a*x^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acoth}{\relax (x )}}{\sqrt {- a \left (x - 1\right ) \left (x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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