Integrand size = 15, antiderivative size = 186 \[ \int \sqrt {a-a x^2} \text {arctanh}(x) \, dx=\frac {1}{2} \sqrt {a-a x^2}+\frac {1}{2} x \sqrt {a-a x^2} \text {arctanh}(x)-\frac {a \sqrt {1-x^2} \arctan \left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \text {arctanh}(x)}{\sqrt {a-a x^2}}-\frac {i a \sqrt {1-x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{2 \sqrt {a-a x^2}}+\frac {i a \sqrt {1-x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{2 \sqrt {a-a x^2}} \]
-a*arctan((1-x)^(1/2)/(1+x)^(1/2))*arctanh(x)*(-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*arctanh(x)*(-a*x^2+a)^(1/2)
Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.52 \[ \int \sqrt {a-a x^2} \text {arctanh}(x) \, dx=\frac {1}{2} \sqrt {a \left (1-x^2\right )} \left (1+x \text {arctanh}(x)-\frac {i \left (\text {arctanh}(x) \left (\log \left (1-i e^{-\text {arctanh}(x)}\right )-\log \left (1+i e^{-\text {arctanh}(x)}\right )\right )+\operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(x)}\right )\right )}{\sqrt {1-x^2}}\right ) \]
(Sqrt[a*(1 - x^2)]*(1 + x*ArcTanh[x] - (I*(ArcTanh[x]*(Log[1 - I/E^ArcTanh [x]] - Log[1 + I/E^ArcTanh[x]]) + PolyLog[2, (-I)/E^ArcTanh[x]] - PolyLog[ 2, I/E^ArcTanh[x]]))/Sqrt[1 - x^2]))/2
Time = 0.41 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6504, 6516, 6512}
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.
\(\displaystyle \int \sqrt {a-a x^2} \text {arctanh}(x) \, dx\) |
\(\Big \downarrow \) 6504 |
\(\displaystyle \frac {1}{2} a \int \frac {\text {arctanh}(x)}{\sqrt {a-a x^2}}dx+\frac {1}{2} x \sqrt {a-a x^2} \text {arctanh}(x)+\frac {1}{2} \sqrt {a-a x^2}\) |
\(\Big \downarrow \) 6516 |
\(\displaystyle \frac {a \sqrt {1-x^2} \int \frac {\text {arctanh}(x)}{\sqrt {1-x^2}}dx}{2 \sqrt {a-a x^2}}+\frac {1}{2} x \sqrt {a-a x^2} \text {arctanh}(x)+\frac {1}{2} \sqrt {a-a x^2}\) |
\(\Big \downarrow \) 6512 |
\(\displaystyle \frac {a \sqrt {1-x^2} \left (-2 \arctan \left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \text {arctanh}(x)-i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )+i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )\right )}{2 \sqrt {a-a x^2}}+\frac {1}{2} x \sqrt {a-a x^2} \text {arctanh}(x)+\frac {1}{2} \sqrt {a-a x^2}\) |
Sqrt[a - a*x^2]/2 + (x*Sqrt[a - a*x^2]*ArcTanh[x])/2 + (a*Sqrt[1 - x^2]*(- 2*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]]*ArcTanh[x] - I*PolyLog[2, ((-I)*Sqrt[1 - x])/Sqrt[1 + x]] + I*PolyLog[2, (I*Sqrt[1 - x])/Sqrt[1 + x]]))/(2*Sqrt[a - a*x^2])
3.6.17.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symb ol] :> Simp[b*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2)^q *((a + b*ArcTanh[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e *x^2)^(q - 1)*(a + b*ArcTanh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol ] :> Simp[-2*(a + b*ArcTanh[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]
Int[((a_.) + ArcTanh[(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*ArcTanh[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]
Time = 0.26 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.23
method | result | size |
default | \(\frac {\left (\operatorname {arctanh}\left (x \right ) x +1\right ) \sqrt {-\left (x -1\right ) \left (1+x \right ) a}}{2}+\frac {i \sqrt {-\left (x -1\right ) \left (1+x \right ) a}\, \sqrt {-x^{2}+1}\, \operatorname {arctanh}\left (x \right ) \ln \left (1+\frac {i \left (1+x \right )}{\sqrt {-x^{2}+1}}\right )}{2 \left (1+x \right ) \left (x -1\right )}-\frac {i \sqrt {-\left (x -1\right ) \left (1+x \right ) a}\, \sqrt {-x^{2}+1}\, \operatorname {arctanh}\left (x \right ) \ln \left (1-\frac {i \left (1+x \right )}{\sqrt {-x^{2}+1}}\right )}{2 \left (1+x \right ) \left (x -1\right )}+\frac {i \sqrt {-\left (x -1\right ) \left (1+x \right ) a}\, \sqrt {-x^{2}+1}\, \operatorname {dilog}\left (1+\frac {i \left (1+x \right )}{\sqrt {-x^{2}+1}}\right )}{2 \left (1+x \right ) \left (x -1\right )}-\frac {i \sqrt {-\left (x -1\right ) \left (1+x \right ) a}\, \sqrt {-x^{2}+1}\, \operatorname {dilog}\left (1-\frac {i \left (1+x \right )}{\sqrt {-x^{2}+1}}\right )}{2 \left (1+x \right ) \left (x -1\right )}\) | \(229\) |
1/2*(arctanh(x)*x+1)*(-(x-1)*(1+x)*a)^(1/2)+1/2*I*(-(x-1)*(1+x)*a)^(1/2)/( 1+x)*(-x^2+1)^(1/2)/(x-1)*arctanh(x)*ln(1+I*(1+x)/(-x^2+1)^(1/2))-1/2*I*(- (x-1)*(1+x)*a)^(1/2)/(1+x)*(-x^2+1)^(1/2)/(x-1)*arctanh(x)*ln(1-I*(1+x)/(- x^2+1)^(1/2))+1/2*I*(-(x-1)*(1+x)*a)^(1/2)/(1+x)*(-x^2+1)^(1/2)/(x-1)*dilo g(1+I*(1+x)/(-x^2+1)^(1/2))-1/2*I*(-(x-1)*(1+x)*a)^(1/2)/(1+x)*(-x^2+1)^(1 /2)/(x-1)*dilog(1-I*(1+x)/(-x^2+1)^(1/2))
\[ \int \sqrt {a-a x^2} \text {arctanh}(x) \, dx=\int { \sqrt {-a x^{2} + a} \operatorname {artanh}\left (x\right ) \,d x } \]
\[ \int \sqrt {a-a x^2} \text {arctanh}(x) \, dx=\int \sqrt {- a \left (x - 1\right ) \left (x + 1\right )} \operatorname {atanh}{\left (x \right )}\, dx \]
\[ \int \sqrt {a-a x^2} \text {arctanh}(x) \, dx=\int { \sqrt {-a x^{2} + a} \operatorname {artanh}\left (x\right ) \,d x } \]
\[ \int \sqrt {a-a x^2} \text {arctanh}(x) \, dx=\int { \sqrt {-a x^{2} + a} \operatorname {artanh}\left (x\right ) \,d x } \]
Timed out. \[ \int \sqrt {a-a x^2} \text {arctanh}(x) \, dx=\int \mathrm {atanh}\left (x\right )\,\sqrt {a-a\,x^2} \,d x \]