3.1.48 \(\int \sqrt {a-a x^2} \coth ^{-1}(x) \, dx\) [48]

Optimal. Leaf size=186 \[ \frac {1}{2} \sqrt {a-a x^2}+\frac {1}{2} x \sqrt {a-a x^2} \coth ^{-1}(x)-\frac {a \sqrt {1-x^2} \coth ^{-1}(x) \text {ArcTan}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right )}{\sqrt {a-a x^2}}-\frac {i a \sqrt {1-x^2} \text {PolyLog}\left (2,-\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{2 \sqrt {a-a x^2}}+\frac {i a \sqrt {1-x^2} \text {PolyLog}\left (2,\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{2 \sqrt {a-a x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6090, 6102, 6098} \begin {gather*} -\frac {a \sqrt {1-x^2} \text {ArcTan}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \coth ^{-1}(x)}{\sqrt {a-a x^2}}-\frac {i a \sqrt {1-x^2} \text {Li}_2\left (-\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )}{2 \sqrt {a-a x^2}}+\frac {i a \sqrt {1-x^2} \text {Li}_2\left (\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )}{2 \sqrt {a-a x^2}}+\frac {1}{2} \sqrt {a-a x^2}+\frac {1}{2} x \sqrt {a-a x^2} \coth ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a - a*x^2]*ArcCoth[x],x]

[Out]

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

Rule 6090

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

Rule 6098

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 6102

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

________________________________________________________________________________________

Mathematica [A]
time = 0.88, size = 125, normalized size = 0.67 \begin {gather*} -\frac {\sqrt {a-a x^2} \left (-2 \coth \left (\frac {1}{2} \coth ^{-1}(x)\right )-\coth ^{-1}(x) \text {csch}^2\left (\frac {1}{2} \coth ^{-1}(x)\right )-4 \coth ^{-1}(x) \log \left (1-e^{-\coth ^{-1}(x)}\right )+4 \coth ^{-1}(x) \log \left (1+e^{-\coth ^{-1}(x)}\right )-4 \text {PolyLog}\left (2,-e^{-\coth ^{-1}(x)}\right )+4 \text {PolyLog}\left (2,e^{-\coth ^{-1}(x)}\right )-\coth ^{-1}(x) \text {sech}^2\left (\frac {1}{2} \coth ^{-1}(x)\right )+2 \tanh \left (\frac {1}{2} \coth ^{-1}(x)\right )\right )}{8 \sqrt {1-\frac {1}{x^2}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(Sqrt[a - a*x^2]*(-2*Coth[ArcCoth[x]/2] - ArcCoth[x]*Csch[ArcCoth[x]/2]^2 - 4*ArcCoth[x]*Log[1 - E^(-ArcC
oth[x])] + 4*ArcCoth[x]*Log[1 + E^(-ArcCoth[x])] - 4*PolyLog[2, -E^(-ArcCoth[x])] + 4*PolyLog[2, E^(-ArcCoth[x
])] - ArcCoth[x]*Sech[ArcCoth[x]/2]^2 + 2*Tanh[ArcCoth[x]/2]))/(Sqrt[1 - x^(-2)]*x)

________________________________________________________________________________________

Maple [A]
time = 0.45, size = 199, normalized size = 1.07

method	result	size
default	\(\frac {\left (\mathrm {arccoth}\left (x \right ) x +1\right ) \sqrt {-a \left (1+x \right ) \left (-1+x \right )}}{2}-\frac {\sqrt {-a \left (1+x \right ) \left (-1+x \right )}\, \sqrt {\frac {-1+x}{1+x}}\, \mathrm {arccoth}\left (x \right ) \ln \left (\frac {1}{\sqrt {\frac {-1+x}{1+x}}}+1\right )}{2 \left (-1+x \right )}-\frac {\sqrt {-a \left (1+x \right ) \left (-1+x \right )}\, \sqrt {\frac {-1+x}{1+x}}\, \polylog \left (2, -\frac {1}{\sqrt {\frac {-1+x}{1+x}}}\right )}{2 \left (-1+x \right )}+\frac {\sqrt {-a \left (1+x \right ) \left (-1+x \right )}\, \sqrt {\frac {-1+x}{1+x}}\, \mathrm {arccoth}\left (x \right ) \ln \left (1-\frac {1}{\sqrt {\frac {-1+x}{1+x}}}\right )}{-2+2 x}+\frac {\sqrt {-a \left (1+x \right ) \left (-1+x \right )}\, \sqrt {\frac {-1+x}{1+x}}\, \polylog \left (2, \frac {1}{\sqrt {\frac {-1+x}{1+x}}}\right )}{-2+2 x}\)	\(199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {acoth}\left (x\right )\,\sqrt {a-a\,x^2} \,d x \end {gather*}

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)

________________________________________________________________________________________