Integrand size = 22, antiderivative size = 91 \[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,-1+\frac {2}{1+a x}\right ) \]
1/4*arctanh(a*x)^4+arctanh(a*x)^3*ln(2-2/(a*x+1))-3/2*arctanh(a*x)^2*polyl og(2,-1+2/(a*x+1))-3/2*arctanh(a*x)*polylog(3,-1+2/(a*x+1))-3/4*polylog(4, -1+2/(a*x+1))
Time = 0.38 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.91 \[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=-\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (1-e^{2 \text {arctanh}(a x)}\right )+\frac {3}{2} \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )-\frac {3}{2} \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )+\frac {3}{4} \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(a x)}\right ) \]
-1/4*ArcTanh[a*x]^4 + ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] + (3*ArcT anh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])])/2 - (3*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])])/2 + (3*PolyLog[4, E^(2*ArcTanh[a*x])])/4
Time = 0.73 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6550, 6494, 6618, 6622, 7164}
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.
\(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx\) |
\(\Big \downarrow \) 6550 |
\(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x (a x+1)}dx+\frac {1}{4} \text {arctanh}(a x)^4\) |
\(\Big \downarrow \) 6494 |
\(\displaystyle -3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\) |
\(\Big \downarrow \) 6618 |
\(\displaystyle -3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\) |
\(\Big \downarrow \) 6622 |
\(\displaystyle -3 a \left (-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle -3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,\frac {2}{a x+1}-1\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\) |
ArcTanh[a*x]^4/4 + ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - 3*a*((ArcTanh[a*x ]^2*PolyLog[2, -1 + 2/(1 + a*x)])/(2*a) + (ArcTanh[a*x]*PolyLog[3, -1 + 2/ (1 + a*x)])/(2*a) + PolyLog[4, -1 + 2/(1 + a*x)]/(4*a))
3.3.45.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))] /(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c ^2*d^2 - e^2, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ d Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x ] - Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_ .)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(PolyLog[k + 1, u]/ (2*c*d)), x] + Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[k + 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] & & EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.43 (sec) , antiderivative size = 1165, normalized size of antiderivative = 12.80
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1165\) |
default | \(\text {Expression too large to display}\) | \(1165\) |
parts | \(\text {Expression too large to display}\) | \(1558\) |
arctanh(a*x)^3*ln(a*x)-1/2*arctanh(a*x)^3*ln(a*x-1)-1/2*arctanh(a*x)^3*ln( a*x+1)+arctanh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*arctanh(a*x)^4-ar ctanh(a*x)^3*ln((a*x+1)^2/(-a^2*x^2+1)-1)+arctanh(a*x)^3*ln(1-(a*x+1)/(-a^ 2*x^2+1)^(1/2))+3*arctanh(a*x)^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-6*a rctanh(a*x)*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+6*polylog(4,(a*x+1)/(-a^ 2*x^2+1)^(1/2))+arctanh(a*x)^3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh( a*x)^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6*arctanh(a*x)*polylog(3,-(a *x+1)/(-a^2*x^2+1)^(1/2))+6*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*(2* I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2+2* I*Pi*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^3+I*Pi*c sgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*csgn(I/(1-(a*x+1) ^2/(a^2*x^2-1)))+2*I*Pi*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))^3-2*I*Pi*csgn(I* (-(a*x+1)^2/(a^2*x^2-1)-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*csgn(I/(1-(a*x+1)^ 2/(a^2*x^2-1)))+I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3+I*Pi*csgn(I*(a*x+1)^2 /(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^3-2*I*Pi*csgn(I/(1-(a*x+1)^2/(a^2* x^2-1)))^2-I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2* csgn(I*(a*x+1)^2/(a^2*x^2-1))-2*I*Pi*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1))*cs gn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2+2*I*Pi-I*Pi*c sgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I/(1-(a*x+1)^2 /(a^2*x^2-1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))+2*I*Pi*csgn(I*(-(a*x+1)^2/...
\[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )} x} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=- \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a^{2} x^{3} - x}\, dx \]
\[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )} x} \,d x } \]
1/16*log(a*x + 1)*log(-a*x + 1)^3 + 1/64*log(-a*x + 1)^4 - 1/8*integrate(1 /2*(3*(a^2*x^2 + a*x + 2)*log(a*x + 1)*log(-a*x + 1)^2 + 2*log(a*x + 1)^3 - 6*log(a*x + 1)^2*log(-a*x + 1))/(a^2*x^3 - x), x)
\[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )} x} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=-\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x\,\left (a^2\,x^2-1\right )} \,d x \]