Integrand size = 22, antiderivative size = 112 \[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {a x}{\sqrt {1-a^2 x^2}}+\frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \]
-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))-a*x/(-a^2*x^2 +1)^(1/2)+arctanh(a*x)/(-a^2*x^2+1)^(1/2)
Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.87 \[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {a x}{\sqrt {1-a^2 x^2}}+\frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}+\text {arctanh}(a x) \log \left (1-e^{-\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \log \left (1+e^{-\text {arctanh}(a x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right ) \]
-((a*x)/Sqrt[1 - a^2*x^2]) + ArcTanh[a*x]/Sqrt[1 - a^2*x^2] + ArcTanh[a*x] *Log[1 - E^(-ArcTanh[a*x])] - ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] + Po lyLog[2, -E^(-ArcTanh[a*x])] - PolyLog[2, E^(-ArcTanh[a*x])]
Time = 0.57 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6592, 6556, 208, 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.
\(\displaystyle \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6592 |
\(\displaystyle a^2 \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 6556 |
\(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx}{a}\right )+\int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx+a^2 \left (\frac {\text {arctanh}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {x}{a \sqrt {1-a^2 x^2}}\right )\) |
\(\Big \downarrow \) 6580 |
\(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {x}{a \sqrt {1-a^2 x^2}}\right )-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\) |
a^2*(-(x/(a*Sqrt[1 - a^2*x^2])) + ArcTanh[a*x]/(a^2*Sqrt[1 - a^2*x^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]]
3.4.92.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q _.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(2*e*(q + 1))), x] + Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTanh[c* x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]
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_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[1/d Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh [c*x])^p, x], x] - Simp[e/d Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh[c* x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Integers Q[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
Time = 0.17 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.40
method | result | size |
default | \(-\frac {\left (\operatorname {arctanh}\left (a x \right )-1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 \left (a x -1\right )}+\frac {\left (1+\operatorname {arctanh}\left (a x \right )\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a x +2}+\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(157\) |
-1/2*(arctanh(a*x)-1)*(-(a*x-1)*(a*x+1))^(1/2)/(a*x-1)+1/2*(1+arctanh(a*x) )*(-(a*x-1)*(a*x+1))^(1/2)/(a*x+1)+arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^ (1/2))+polylog(2,(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))-polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))
\[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {atanh}{\left (a x \right )}}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )}{x\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]