Integrand size = 18, antiderivative size = 48 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x} \, dx=-\frac {a x}{2}+\frac {1}{2} \text {arctanh}(a x)-\frac {1}{2} a^2 x^2 \text {arctanh}(a x)-\frac {\operatorname {PolyLog}(2,-a x)}{2}+\frac {\operatorname {PolyLog}(2,a x)}{2} \]
Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.25 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x} \, dx=-\frac {a x}{2}-\frac {1}{2} a^2 x^2 \text {arctanh}(a x)-\frac {1}{4} \log (1-a x)+\frac {1}{4} \log (1+a x)+\frac {1}{2} (-\operatorname {PolyLog}(2,-a x)+\operatorname {PolyLog}(2,a x)) \]
-1/2*(a*x) - (a^2*x^2*ArcTanh[a*x])/2 - Log[1 - a*x]/4 + Log[1 + a*x]/4 + (-PolyLog[2, -(a*x)] + PolyLog[2, a*x])/2
Time = 0.33 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6576, 6446, 6452, 262, 219}
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.
\(\displaystyle \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x} \, dx\) |
\(\Big \downarrow \) 6576 |
\(\displaystyle \int \frac {\text {arctanh}(a x)}{x}dx-a^2 \int x \text {arctanh}(a x)dx\) |
\(\Big \downarrow \) 6446 |
\(\displaystyle a^2 (-\int x \text {arctanh}(a x)dx)-\frac {\operatorname {PolyLog}(2,-a x)}{2}+\frac {\operatorname {PolyLog}(2,a x)}{2}\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle -\left (a^2 \left (\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \int \frac {x^2}{1-a^2 x^2}dx\right )\right )-\frac {\operatorname {PolyLog}(2,-a x)}{2}+\frac {\operatorname {PolyLog}(2,a x)}{2}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\left (a^2 \left (\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\int \frac {1}{1-a^2 x^2}dx}{a^2}-\frac {x}{a^2}\right )\right )\right )-\frac {\operatorname {PolyLog}(2,-a x)}{2}+\frac {\operatorname {PolyLog}(2,a x)}{2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\left (a^2 \left (\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )\right )\right )-\frac {\operatorname {PolyLog}(2,-a x)}{2}+\frac {\operatorname {PolyLog}(2,a x)}{2}\) |
-(a^2*((x^2*ArcTanh[a*x])/2 - (a*(-(x/a^2) + ArcTanh[a*x]/a^3))/2)) - Poly Log[2, -(a*x)]/2 + PolyLog[2, a*x]/2
3.2.66.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x ] + (-Simp[(b/2)*PolyLog[2, (-c)*x], x] + Simp[(b/2)*PolyLog[2, c*x], x]) / ; FreeQ[{a, b, c}, x]
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_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x ^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ [p, 1] && IntegerQ[q]))
Time = 0.15 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.44
method | result | size |
derivativedivides | \(-\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{2}+\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {a x}{2}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\operatorname {dilog}\left (a x \right )}{2}\) | \(69\) |
default | \(-\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{2}+\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {a x}{2}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\operatorname {dilog}\left (a x \right )}{2}\) | \(69\) |
risch | \(\frac {\left (-a x +1\right )^{2} \ln \left (-a x +1\right )}{4}-\frac {a x}{2}-\frac {\left (-a x +1\right ) \ln \left (-a x +1\right )}{2}+\frac {\operatorname {dilog}\left (-a x +1\right )}{2}-\frac {\left (a x +1\right )^{2} \ln \left (a x +1\right )}{4}+\frac {\left (a x +1\right ) \ln \left (a x +1\right )}{2}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}\) | \(83\) |
meijerg | \(-\frac {i \left (\frac {2 i a x \operatorname {polylog}\left (2, \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-\frac {2 i a x \operatorname {polylog}\left (2, -\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}\right )}{4}-\frac {i \left (-2 i x a +2 i \left (-a x +1\right ) \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )\right )}{4}\) | \(85\) |
parts | \(-\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{2}+\operatorname {arctanh}\left (a x \right ) \ln \left (x \right )-\frac {a \left (x +\frac {\ln \left (a x -1\right )}{2 a}-\frac {\ln \left (a x +1\right )}{2 a}-\frac {\left (\ln \left (x \right )-\ln \left (a x \right )\right ) \ln \left (-a x +1\right )}{a}+\frac {\operatorname {dilog}\left (a x \right )}{a}+\frac {\operatorname {dilog}\left (a x +1\right )}{a}+\frac {\ln \left (x \right ) \ln \left (a x +1\right )}{a}\right )}{2}\) | \(99\) |
-1/2*a^2*x^2*arctanh(a*x)+arctanh(a*x)*ln(a*x)-1/2*a*x-1/4*ln(a*x-1)+1/4*l n(a*x+1)-1/2*dilog(a*x+1)-1/2*ln(a*x)*ln(a*x+1)-1/2*dilog(a*x)
\[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )}{x} \,d x } \]
\[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x} \, dx=- \int \left (- \frac {\operatorname {atanh}{\left (a x \right )}}{x}\right )\, dx - \int a^{2} x \operatorname {atanh}{\left (a x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (36) = 72\).
Time = 0.17 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.85 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x} \, dx=-\frac {1}{4} \, a {\left (2 \, x + \frac {2 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} - \frac {2 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a} - \frac {\log \left (a x + 1\right )}{a} + \frac {\log \left (a x - 1\right )}{a}\right )} - \frac {1}{2} \, {\left (a^{2} x^{2} - \log \left (x^{2}\right )\right )} \operatorname {artanh}\left (a x\right ) \]
-1/4*a*(2*x + 2*(log(a*x + 1)*log(x) + dilog(-a*x))/a - 2*(log(-a*x + 1)*l og(x) + dilog(a*x))/a - log(a*x + 1)/a + log(a*x - 1)/a) - 1/2*(a^2*x^2 - log(x^2))*arctanh(a*x)
\[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x} \, dx=-\int \frac {\mathrm {atanh}\left (a\,x\right )\,\left (a^2\,x^2-1\right )}{x} \,d x \]