Integrand size = 19, antiderivative size = 143 \[ \int \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx=\frac {\sqrt {1-a^2 x^2}}{2 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {\arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a} \] Output:
1/2*(-a^2*x^2+1)^(1/2)/a+1/2*x*(-a^2*x^2+1)^(1/2)*arctanh(a*x)-arctan((-a* x+1)^(1/2)/(a*x+1)^(1/2))*arctanh(a*x)/a-1/2*I*polylog(2,-I*(-a*x+1)^(1/2) /(a*x+1)^(1/2))/a+1/2*I*polylog(2,I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a
Time = 0.22 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.82 \[ \int \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx=\frac {\sqrt {1-a^2 x^2} \left (1+a x \text {arctanh}(a x)-\frac {i \left (\text {arctanh}(a x) \left (\log \left (1-i e^{-\text {arctanh}(a x)}\right )-\log \left (1+i e^{-\text {arctanh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{2 a} \] Input:
Integrate[Sqrt[1 - a^2*x^2]*ArcTanh[a*x],x]
Output:
(Sqrt[1 - a^2*x^2]*(1 + a*x*ArcTanh[a*x] - (I*(ArcTanh[a*x]*(Log[1 - I/E^A rcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]]) + PolyLog[2, (-I)/E^ArcTanh[a*x] ] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(2*a)
Time = 0.34 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6504, 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.
\(\displaystyle \int \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx\) |
\(\Big \downarrow \) 6504 |
\(\displaystyle \frac {1}{2} \int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2} \text {arctanh}(a x)+\frac {\sqrt {1-a^2 x^2}}{2 a}\) |
\(\Big \downarrow \) 6512 |
\(\displaystyle \frac {1}{2} x \sqrt {1-a^2 x^2} \text {arctanh}(a x)+\frac {\sqrt {1-a^2 x^2}}{2 a}+\frac {1}{2} \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )\) |
Input:
Int[Sqrt[1 - a^2*x^2]*ArcTanh[a*x],x]
Output:
Sqrt[1 - a^2*x^2]/(2*a) + (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/2 + ((-2*ArcT an[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])/Sqrt[1 + a* x]])/a)/2
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]
Time = 0.69 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.06
method | result | size |
default | \(\frac {\left (a x \,\operatorname {arctanh}\left (a x \right )+1\right ) \sqrt {-a^{2} x^{2}+1}}{2 a}-\frac {i \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a}+\frac {i \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a}-\frac {i \operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a}+\frac {i \operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a}\) | \(152\) |
Input:
int((-a^2*x^2+1)^(1/2)*arctanh(a*x),x,method=_RETURNVERBOSE)
Output:
1/2*(a*x*arctanh(a*x)+1)*(-a^2*x^2+1)^(1/2)/a-1/2*I/a*arctanh(a*x)*ln(1+I* (a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I/a*arctanh(a*x)*ln(1-I*(a*x+1)/(-a^2*x^2+ 1)^(1/2))-1/2*I/a*dilog(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I/a*dilog(1-I* (a*x+1)/(-a^2*x^2+1)^(1/2))
\[ \int \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx=\int { \sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right ) \,d x } \] Input:
integrate((-a^2*x^2+1)^(1/2)*arctanh(a*x),x, algorithm="fricas")
Output:
integral(sqrt(-a^2*x^2 + 1)*arctanh(a*x), x)
\[ \int \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx=\int \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}{\left (a x \right )}\, dx \] Input:
integrate((-a**2*x**2+1)**(1/2)*atanh(a*x),x)
Output:
Integral(sqrt(-(a*x - 1)*(a*x + 1))*atanh(a*x), x)
\[ \int \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx=\int { \sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right ) \,d x } \] Input:
integrate((-a^2*x^2+1)^(1/2)*arctanh(a*x),x, algorithm="maxima")
Output:
integrate(sqrt(-a^2*x^2 + 1)*arctanh(a*x), x)
Exception generated. \[ \int \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a^2*x^2+1)^(1/2)*arctanh(a*x),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx=\int \mathrm {atanh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2} \,d x \] 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)
\[ \int \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx=\int \sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )d x \] Input:
int((-a^2*x^2+1)^(1/2)*atanh(a*x),x)
Output:
int(sqrt( - a**2*x**2 + 1)*atanh(a*x),x)