Integrand size = 22, antiderivative size = 136 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^3} \, dx=-\frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}+a^2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {1}{2} a^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {1}{2} a^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \]
a^2*arctanh(a*x)*arctanh((-a*x+1)^(1/2)/(a*x+1)^(1/2))-1/2*a^2*polylog(2,- (-a*x+1)^(1/2)/(a*x+1)^(1/2))+1/2*a^2*polylog(2,(-a*x+1)^(1/2)/(a*x+1)^(1/ 2))-1/2*a*(-a^2*x^2+1)^(1/2)/x-1/2*arctanh(a*x)*(-a^2*x^2+1)^(1/2)/x^2
Time = 0.54 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^3} \, dx=\frac {1}{8} a^2 \left (-2 \coth \left (\frac {1}{2} \text {arctanh}(a x)\right )-\text {arctanh}(a x) \text {csch}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )-4 \text {arctanh}(a x) \log \left (1-e^{-\text {arctanh}(a x)}\right )+4 \text {arctanh}(a x) \log \left (1+e^{-\text {arctanh}(a x)}\right )-4 \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )+4 \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \text {sech}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )+2 \tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right ) \]
(a^2*(-2*Coth[ArcTanh[a*x]/2] - ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^2 - 4*Ar cTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] + 4*ArcTanh[a*x]*Log[1 + E^(-ArcTanh [a*x])] - 4*PolyLog[2, -E^(-ArcTanh[a*x])] + 4*PolyLog[2, E^(-ArcTanh[a*x] )] - ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^2 + 2*Tanh[ArcTanh[a*x]/2]))/8
Time = 0.53 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6572, 242, 6588, 242, 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.
\(\displaystyle \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^3} \, dx\) |
\(\Big \downarrow \) 6572 |
\(\displaystyle -\int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx+a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^2}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^2}-\frac {a \sqrt {1-a^2 x^2}}{x}\) |
\(\Big \downarrow \) 6588 |
\(\displaystyle -\frac {1}{2} a^2 \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{x}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {1}{2} a^2 \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{2 x}\) |
\(\Big \downarrow \) 6580 |
\(\displaystyle -\frac {1}{2} a^2 \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{2 x}\) |
-1/2*(a*Sqrt[1 - a^2*x^2])/x - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(2*x^2) - (a^2*(-2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + PolyLog[2, -( Sqrt[1 - a*x]/Sqrt[1 + a*x])] - PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]))/ 2
3.5.33.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
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]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*A rcTanh[c*x])^p/(d*f*(m + 1))), x] + (-Simp[b*c*(p/(f*(m + 1))) Int[(f*x)^ (m + 1)*((a + b*ArcTanh[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] + Simp[c^2*( (m + 2)/(f^2*(m + 1))) Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/Sqrt[d + e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && G tQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
Time = 0.17 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.02
method | result | size |
default | \(-\frac {\left (a x +\operatorname {arctanh}\left (a x \right )\right ) \sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {a^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {a^{2} \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {a^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\) | \(139\) |
-1/2*(a*x+arctanh(a*x))*(-a^2*x^2+1)^(1/2)/x^2-1/2*a^2*arctanh(a*x)*ln(1-( a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*a^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+1 /2*a^2*arctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*a^2*polylog(2,-(a *x+1)/(-a^2*x^2+1)^(1/2))
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^3} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )}{x^{3}} \,d x } \]
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^3} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}{\left (a x \right )}}{x^{3}}\, dx \]
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^3} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )}{x^{3}} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^3} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^3} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}}{x^3} \,d x \]