Integrand size = 20, antiderivative size = 41 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx=-\frac {\text {arctanh}(a x)}{x}+\frac {1}{2} a \text {arctanh}(a x)^2+a \log (x)-\frac {1}{2} a \log \left (1-a^2 x^2\right ) \]
Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx=-\frac {\text {arctanh}(a x)}{x}+\frac {1}{2} a \text {arctanh}(a x)^2+a \log (x)-\frac {1}{2} a \log \left (1-a^2 x^2\right ) \]
Time = 0.38 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6544, 6452, 243, 47, 14, 16, 6510}
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.
\(\displaystyle \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx\) |
\(\Big \downarrow \) 6544 |
\(\displaystyle a^2 \int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx+\int \frac {\text {arctanh}(a x)}{x^2}dx\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle a^2 \int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx+a \int \frac {1}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{x}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle a^2 \int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx+\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )}dx^2-\frac {\text {arctanh}(a x)}{x}\) |
\(\Big \downarrow \) 47 |
\(\displaystyle a^2 \int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx+\frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx^2+\int \frac {1}{x^2}dx^2\right )-\frac {\text {arctanh}(a x)}{x}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle a^2 \int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx+\frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx^2+\log \left (x^2\right )\right )-\frac {\text {arctanh}(a x)}{x}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle a^2 \int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )-\frac {\text {arctanh}(a x)}{x}\) |
\(\Big \downarrow \) 6510 |
\(\displaystyle \frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )+\frac {1}{2} a \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)}{x}\) |
3.3.32.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x ], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d + e*x ^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12
method | result | size |
parallelrisch | \(\frac {\operatorname {arctanh}\left (a x \right )^{2} a x +2 a \ln \left (x \right ) x -2 \ln \left (a x -1\right ) a x -2 a x \,\operatorname {arctanh}\left (a x \right )-2 \,\operatorname {arctanh}\left (a x \right )}{2 x}\) | \(46\) |
risch | \(\frac {a \ln \left (a x +1\right )^{2}}{8}-\frac {\left (a x \ln \left (-a x +1\right )+2\right ) \ln \left (a x +1\right )}{4 x}+\frac {a \ln \left (-a x +1\right )^{2} x +8 a \ln \left (x \right ) x -4 a \ln \left (a^{2} x^{2}-1\right ) x +4 \ln \left (-a x +1\right )}{8 x}\) | \(83\) |
parts | \(\frac {a \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {a \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )}{x}-\frac {a \left (-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right )^{2}}{4}+\ln \left (a x -1\right )+\ln \left (a x +1\right )-2 \ln \left (x \right )-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x +1\right )^{2}}{4}\right )}{2}\) | \(117\) |
derivativedivides | \(a \left (-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )}{a x}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}+\ln \left (a x \right )-\frac {\ln \left (a x -1\right )}{2}-\frac {\ln \left (a x +1\right )}{2}-\frac {\ln \left (a x +1\right )^{2}}{8}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}\right )\) | \(120\) |
default | \(a \left (-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )}{a x}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}+\ln \left (a x \right )-\frac {\ln \left (a x -1\right )}{2}-\frac {\ln \left (a x +1\right )}{2}-\frac {\ln \left (a x +1\right )^{2}}{8}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}\right )\) | \(120\) |
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.54 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx=\frac {a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 4 \, a x \log \left (a^{2} x^{2} - 1\right ) + 8 \, a x \log \left (x\right ) - 4 \, \log \left (-\frac {a x + 1}{a x - 1}\right )}{8 \, x} \]
1/8*(a*x*log(-(a*x + 1)/(a*x - 1))^2 - 4*a*x*log(a^2*x^2 - 1) + 8*a*x*log( x) - 4*log(-(a*x + 1)/(a*x - 1)))/x
Time = 0.43 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx=\begin {cases} a \log {\left (x \right )} - a \log {\left (x - \frac {1}{a} \right )} + \frac {a \operatorname {atanh}^{2}{\left (a x \right )}}{2} - a \operatorname {atanh}{\left (a x \right )} - \frac {\operatorname {atanh}{\left (a x \right )}}{x} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((a*log(x) - a*log(x - 1/a) + a*atanh(a*x)**2/2 - a*atanh(a*x) - atanh(a*x)/x, Ne(a, 0)), (0, True))
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (37) = 74\).
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.00 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx=\frac {1}{8} \, {\left (2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right ) + 8 \, \log \left (x\right )\right )} a + \frac {1}{2} \, {\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} \operatorname {artanh}\left (a x\right ) \]
1/8*(2*(log(a*x - 1) - 2)*log(a*x + 1) - log(a*x + 1)^2 - log(a*x - 1)^2 - 4*log(a*x - 1) + 8*log(x))*a + 1/2*(a*log(a*x + 1) - a*log(a*x - 1) - 2/x )*arctanh(a*x)
\[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )}{{\left (a^{2} x^{2} - 1\right )} x^{2}} \,d x } \]
Time = 3.61 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.95 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx=\frac {a\,{\ln \left (a\,x+1\right )}^2}{8}+\frac {a\,{\ln \left (1-a\,x\right )}^2}{8}-\frac {\ln \left (a\,x+1\right )}{2\,x}+\frac {\ln \left (1-a\,x\right )}{2\,x}-\frac {a\,\ln \left (a^2\,x^2-1\right )}{2}+a\,\ln \left (x\right )-\frac {a\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{4} \]