Integrand size = 19, antiderivative size = 95 \[ \int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a} \] Output:
-2*arctan((-a*x+1)^(1/2)/(a*x+1)^(1/2))*arctanh(a*x)/a-I*polylog(2,-I*(-a* x+1)^(1/2)/(a*x+1)^(1/2))/a+I*polylog(2,I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.80 \[ \int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {i \left (\text {arctanh}(a x) \left (\log \left (1-i e^{-\text {arctanh}(a x)}\right )-\log \left (1+i e^{-\text {arctanh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )\right )}{a} \] Input:
Integrate[ArcTanh[a*x]/Sqrt[1 - a^2*x^2],x]
Output:
((-I)*(ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]] ) + PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/a
Time = 0.24 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {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.
\(\displaystyle \int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 6512 |
\(\displaystyle -\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\) |
Input:
Int[ArcTanh[a*x]/Sqrt[1 - a^2*x^2],x]
Output:
(-2*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a - (I*PolyLog[2, (( -I)*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqr t[1 + a*x]])/a
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]
Time = 0.71 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.19
method | result | size |
default | \(-\frac {i \left (\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{a}\) | \(113\) |
Input:
int(arctanh(a*x)/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-I/a*(arctanh(a*x)*ln(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)*ln(1-I* (a*x+1)/(-a^2*x^2+1)^(1/2))-dilog(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))+dilog(1+ I*(a*x+1)/(-a^2*x^2+1)^(1/2)))
\[ \int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(arctanh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-a^2*x^2 + 1)*arctanh(a*x)/(a^2*x^2 - 1), x)
\[ \int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {atanh}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:
integrate(atanh(a*x)/(-a**2*x**2+1)**(1/2),x)
Output:
Integral(atanh(a*x)/sqrt(-(a*x - 1)*(a*x + 1)), x)
\[ \int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(arctanh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
integrate(arctanh(a*x)/sqrt(-a^2*x^2 + 1), x)
\[ \int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(arctanh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
integrate(arctanh(a*x)/sqrt(-a^2*x^2 + 1), x)
Timed out. \[ \int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \] Input:
int(atanh(a*x)/(1 - a^2*x^2)^(1/2),x)
Output:
int(atanh(a*x)/(1 - a^2*x^2)^(1/2), x)
\[ \int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {atanh} \left (a x \right )}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:
int(atanh(a*x)/(-a^2*x^2+1)^(1/2),x)
Output:
int(atanh(a*x)/sqrt( - a**2*x**2 + 1),x)