\(\int \frac {\coth ^{-1}(x)}{\sqrt {a-a x^2}} \, dx\) [9]

Optimal result

Integrand size = 15, antiderivative size = 144 \[ \int \frac {\coth ^{-1}(x)}{\sqrt {a-a x^2}} \, dx=-\frac {2 \sqrt {1-x^2} \coth ^{-1}(x) \arctan \left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right )}{\sqrt {a-a x^2}}-\frac {i \sqrt {1-x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{\sqrt {a-a x^2}}+\frac {i \sqrt {1-x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{\sqrt {a-a x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.53 \[ \int \frac {\coth ^{-1}(x)}{\sqrt {a-a x^2}} \, dx=\frac {\sqrt {a-a x^2} \left (\coth ^{-1}(x) \left (\log \left (1-e^{-\coth ^{-1}(x)}\right )-\log \left (1+e^{-\coth ^{-1}(x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-\coth ^{-1}(x)}\right )-\operatorname {PolyLog}\left (2,e^{-\coth ^{-1}(x)}\right )\right )}{a \sqrt {1-\frac {1}{x^2}} x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.69, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6517, 6513}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6517

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

\(\Big \downarrow \) 6513

\(\displaystyle \frac {\sqrt {1-x^2} \left (-2 \arctan \left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \coth ^{-1}(x)-i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )+i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )\right )}{\sqrt {a-a x^2}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 6513

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*S 
qrt[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 6517

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[Sqrt[1 - c^2*x^2]/Sqrt[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]
 

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.32

method	result	size
default	\(-\frac {\ln \left (\frac {1}{\sqrt {\frac {x -1}{x +1}}}+1\right ) \operatorname {arccoth}\left (x \right ) \sqrt {\frac {x -1}{x +1}}\, \sqrt {-\left (x -1\right ) \left (x +1\right ) a}}{a \left (x -1\right )}-\frac {\operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {x -1}{x +1}}}\right ) \sqrt {\frac {x -1}{x +1}}\, \sqrt {-\left (x -1\right ) \left (x +1\right ) a}}{a \left (x -1\right )}+\frac {\ln \left (1-\frac {1}{\sqrt {\frac {x -1}{x +1}}}\right ) \operatorname {arccoth}\left (x \right ) \sqrt {\frac {x -1}{x +1}}\, \sqrt {-\left (x -1\right ) \left (x +1\right ) a}}{a \left (x -1\right )}+\frac {\operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {x -1}{x +1}}}\right ) \sqrt {\frac {x -1}{x +1}}\, \sqrt {-\left (x -1\right ) \left (x +1\right ) a}}{a \left (x -1\right )}\)	\(190\)

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\coth ^{-1}(x)}{\sqrt {a-a x^2}} \, dx=\int { \frac {\operatorname {arcoth}\left (x\right )}{\sqrt {-a x^{2} + a}} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {\coth ^{-1}(x)}{\sqrt {a-a x^2}} \, dx=\int \frac {\operatorname {acoth}{\left (x \right )}}{\sqrt {- a \left (x - 1\right ) \left (x + 1\right )}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\coth ^{-1}(x)}{\sqrt {a-a x^2}} \, dx=\int { \frac {\operatorname {arcoth}\left (x\right )}{\sqrt {-a x^{2} + a}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\coth ^{-1}(x)}{\sqrt {a-a x^2}} \, dx=\int { \frac {\operatorname {arcoth}\left (x\right )}{\sqrt {-a x^{2} + a}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(x)}{\sqrt {a-a x^2}} \, dx=\int \frac {\mathrm {acoth}\left (x\right )}{\sqrt {a-a\,x^2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\coth ^{-1}(x)}{\sqrt {a-a x^2}} \, dx=\frac {\int \frac {\mathit {acoth} \left (x \right )}{\sqrt {-x^{2}+1}}d x}{\sqrt {a}} \] Input:

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

Output:

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