Integrand size = 22, antiderivative size = 100 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x} \, dx=-\arcsin (a x)+\sqrt {1-a^2 x^2} \text {arctanh}(a x)-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \] Output:
-arcsin(a*x)+(-a^2*x^2+1)^(1/2)*arctanh(a*x)-2*arctanh(a*x)*arctanh((-a*x+ 1)^(1/2)/(a*x+1)^(1/2))+polylog(2,-(-a*x+1)^(1/2)/(a*x+1)^(1/2))-polylog(2 ,(-a*x+1)^(1/2)/(a*x+1)^(1/2))
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x} \, dx=-2 \arctan \left (\tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )+\sqrt {1-a^2 x^2} \text {arctanh}(a x)+\text {arctanh}(a x) \log \left (1-e^{-\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \log \left (1+e^{-\text {arctanh}(a x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right ) \] Input:
Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x,x]
Output:
-2*ArcTan[Tanh[ArcTanh[a*x]/2]] + Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + ArcTanh [a*x]*Log[1 - E^(-ArcTanh[a*x])] - ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] + PolyLog[2, -E^(-ArcTanh[a*x])] - PolyLog[2, E^(-ArcTanh[a*x])]
Time = 0.40 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6572, 223, 6580}
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.
\(\displaystyle \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x} \, dx\) |
\(\Big \downarrow \) 6572 |
\(\displaystyle \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx-a \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\sqrt {1-a^2 x^2} \text {arctanh}(a x)\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx+\sqrt {1-a^2 x^2} \text {arctanh}(a x)-\arcsin (a x)\) |
\(\Big \downarrow \) 6580 |
\(\displaystyle \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\arcsin (a x)-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\) |
Input:
Int[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x,x]
Output:
-ArcSin[a*x] + Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - 2*ArcTanh[a*x]*ArcTanh[Sqr t[1 - a*x]/Sqrt[1 + a*x]] + PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - P olyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcTanh[c *x])/(f*(m + 2))), x] + (Simp[d/(m + 2) Int[(f*x)^m*((a + b*ArcTanh[c*x]) /Sqrt[d + e*x^2]), x], x] - Simp[b*c*(d/(f*(m + 2))) Int[(f*x)^(m + 1)/Sq rt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && NeQ[m, -2]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x _Symbol] :> Simp[(-2/Sqrt[d])*(a + b*ArcTanh[c*x])*ArcTanh[Sqrt[1 - c*x]/Sq rt[1 + c*x]], x] + (Simp[(b/Sqrt[d])*PolyLog[2, -Sqrt[1 - c*x]/Sqrt[1 + c*x ]], x] - Simp[(b/Sqrt[d])*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]], x]) /; F reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
Time = 0.97 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.11
method | result | size |
default | \(\sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right )-2 \arctan \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(111\) |
Input:
int((-a^2*x^2+1)^(1/2)*arctanh(a*x)/x,x,method=_RETURNVERBOSE)
Output:
(-a^2*x^2+1)^(1/2)*arctanh(a*x)-2*arctan((a*x+1)/(-a^2*x^2+1)^(1/2))-dilog (1+(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2 ))-dilog((a*x+1)/(-a^2*x^2+1)^(1/2))
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )}{x} \,d x } \] Input:
integrate((-a^2*x^2+1)^(1/2)*arctanh(a*x)/x,x, algorithm="fricas")
Output:
integral(sqrt(-a^2*x^2 + 1)*arctanh(a*x)/x, x)
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}{\left (a x \right )}}{x}\, dx \] Input:
integrate((-a**2*x**2+1)**(1/2)*atanh(a*x)/x,x)
Output:
Integral(sqrt(-(a*x - 1)*(a*x + 1))*atanh(a*x)/x, x)
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )}{x} \,d x } \] Input:
integrate((-a^2*x^2+1)^(1/2)*arctanh(a*x)/x,x, algorithm="maxima")
Output:
integrate(sqrt(-a^2*x^2 + 1)*arctanh(a*x)/x, x)
Exception generated. \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a^2*x^2+1)^(1/2)*arctanh(a*x)/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. \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}}{x} \,d x \] Input:
int((atanh(a*x)*(1 - a^2*x^2)^(1/2))/x,x)
Output:
int((atanh(a*x)*(1 - a^2*x^2)^(1/2))/x, x)
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x} \, dx=\int \frac {\sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )}{x}d x \] Input:
int((-a^2*x^2+1)^(1/2)*atanh(a*x)/x,x)
Output:
int((sqrt( - a**2*x**2 + 1)*atanh(a*x))/x,x)