Integrand size = 17, antiderivative size = 157 \[ \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3 \, dx=-x \text {arctanh}(a x)+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}+\frac {2 \text {arctanh}(a x)^3}{3 a}+\frac {2}{3} x \text {arctanh}(a x)^3+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3-\frac {2 \text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-\frac {2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a} \]
-x*arctanh(a*x)+1/2*(-a^2*x^2+1)*arctanh(a*x)^2/a+2/3*arctanh(a*x)^3/a+2/3 *x*arctanh(a*x)^3+1/3*x*(-a^2*x^2+1)*arctanh(a*x)^3-2*arctanh(a*x)^2*ln(2/ (-a*x+1))/a-1/2*ln(-a^2*x^2+1)/a-2*arctanh(a*x)*polylog(2,1-2/(-a*x+1))/a+ polylog(3,1-2/(-a*x+1))/a
Time = 0.23 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.85 \[ \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3 \, dx=-\frac {6 a x \text {arctanh}(a x)-3 \text {arctanh}(a x)^2+3 a^2 x^2 \text {arctanh}(a x)^2+4 \text {arctanh}(a x)^3-6 a x \text {arctanh}(a x)^3+2 a^3 x^3 \text {arctanh}(a x)^3+12 \text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+3 \log \left (1-a^2 x^2\right )-12 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )-6 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )}{6 a} \]
-1/6*(6*a*x*ArcTanh[a*x] - 3*ArcTanh[a*x]^2 + 3*a^2*x^2*ArcTanh[a*x]^2 + 4 *ArcTanh[a*x]^3 - 6*a*x*ArcTanh[a*x]^3 + 2*a^3*x^3*ArcTanh[a*x]^3 + 12*Arc Tanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] + 3*Log[1 - a^2*x^2] - 12*ArcTanh [a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] - 6*PolyLog[3, -E^(-2*ArcTanh[a*x]) ])/a
Time = 0.92 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6506, 6436, 240, 6546, 6470, 6620, 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.
| \(\displaystyle \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3 \, dx\) |
| \(\Big \downarrow \) 6506 |
| \(\displaystyle -\int \text {arctanh}(a x)dx+\frac {2}{3} \int \text {arctanh}(a x)^3dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle \frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx\right )+a \int \frac {x}{1-a^2 x^2}dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-x \text {arctanh}(a x)\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \left (\frac {\int \frac {\text {arctanh}(a x)^2}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \int \frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle \frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{4 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\) |
-(x*ArcTanh[a*x]) + ((1 - a^2*x^2)*ArcTanh[a*x]^2)/(2*a) + (x*(1 - a^2*x^2 )*ArcTanh[a*x]^3)/3 - Log[1 - a^2*x^2]/(2*a) + (2*(x*ArcTanh[a*x]^3 - 3*a* (-1/3*ArcTanh[a*x]^3/a^2 + ((ArcTanh[a*x]^2*Log[2/(1 - a*x)])/a - 2*(-1/2* (ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/a + PolyLog[3, 1 - 2/(1 - a*x)] /(4*a)))/a)))/3
3.2.83.3.1 Defintions of rubi rules used
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTanh[c*x^n]) ^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[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_)*((d_) + (e_.)*(x_)^2)^(q_.), x _Symbol] :> Simp[b*p*(d + e*x^2)^q*((a + b*ArcTanh[c*x])^(p - 1)/(2*c*q*(2* q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTanh[c*x])^p/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1))) Int[(d + e*x^2)^(q - 1)* (a + b*ArcTanh[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c ^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[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[(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 = 8.66 (sec) , antiderivative size = 749, normalized size of antiderivative = 4.77
| method | result | size |
| derivativedivides | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{3} a^{3} x^{3}}{3}+\operatorname {arctanh}\left (a x \right )^{3} a x -\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\operatorname {arctanh}\left (a x \right ) \left (3 i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )+6 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {arctanh}\left (a x \right )+3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )-3 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )-3 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {arctanh}\left (a x \right )+3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )+3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {arctanh}\left (a x \right )+6 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{3} \operatorname {arctanh}\left (a x \right )-6 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2} \operatorname {arctanh}\left (a x \right )+6 i \pi \,\operatorname {arctanh}\left (a x \right )+12 \ln \left (2\right ) \operatorname {arctanh}\left (a x \right )-4 \operatorname {arctanh}\left (a x \right )^{2}-3 \,\operatorname {arctanh}\left (a x \right )+6 a x +6\right )}{6}+\ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a}\) | \(749\) |
| default | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{3} a^{3} x^{3}}{3}+\operatorname {arctanh}\left (a x \right )^{3} a x -\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\operatorname {arctanh}\left (a x \right ) \left (3 i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )+6 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {arctanh}\left (a x \right )+3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )-3 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )-3 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {arctanh}\left (a x \right )+3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )+3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {arctanh}\left (a x \right )+6 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{3} \operatorname {arctanh}\left (a x \right )-6 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2} \operatorname {arctanh}\left (a x \right )+6 i \pi \,\operatorname {arctanh}\left (a x \right )+12 \ln \left (2\right ) \operatorname {arctanh}\left (a x \right )-4 \operatorname {arctanh}\left (a x \right )^{2}-3 \,\operatorname {arctanh}\left (a x \right )+6 a x +6\right )}{6}+\ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a}\) | \(749\) |
| parts | \(-\frac {\operatorname {arctanh}\left (a x \right )^{3} a^{2} x^{3}}{3}+x \operatorname {arctanh}\left (a x \right )^{3}-\frac {a \operatorname {arctanh}\left (a x \right )^{2} x^{2}}{2}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{a}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{a}+\frac {-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\operatorname {arctanh}\left (a x \right ) \left (3 i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )+6 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {arctanh}\left (a x \right )+3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )-3 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )-3 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {arctanh}\left (a x \right )+3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )+3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {arctanh}\left (a x \right )+6 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{3} \operatorname {arctanh}\left (a x \right )-6 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2} \operatorname {arctanh}\left (a x \right )+6 i \pi \,\operatorname {arctanh}\left (a x \right )+12 \ln \left (2\right ) \operatorname {arctanh}\left (a x \right )-4 \operatorname {arctanh}\left (a x \right )^{2}-3 \,\operatorname {arctanh}\left (a x \right )+6 a x +6\right )}{6}+\ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a}\) | \(753\) |
1/a*(-1/3*arctanh(a*x)^3*a^3*x^3+arctanh(a*x)^3*a*x-1/2*a^2*x^2*arctanh(a* x)^2+arctanh(a*x)^2*ln(a*x-1)+arctanh(a*x)^2*ln(a*x+1)-2*arctanh(a*x)^2*ln ((a*x+1)/(-a^2*x^2+1)^(1/2))-1/6*arctanh(a*x)*(3*I*Pi*csgn(I*(a*x+1)/(-a^2 *x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*arctanh(a*x)+6*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*arctanh(a*x)+3 *I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)-3*I*Pi*csgn(I*(a*x+1)^2 /(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*ar ctanh(a*x)-3*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(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)))*arctanh(a* x)+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*arctan h(a*x)+3*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*cs gn(I/(1-(a*x+1)^2/(a^2*x^2-1)))*arctanh(a*x)+6*I*Pi*csgn(I/(1-(a*x+1)^2/(a ^2*x^2-1)))^3*arctanh(a*x)-6*I*Pi*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))^2*arct anh(a*x)+6*I*Pi*arctanh(a*x)+12*ln(2)*arctanh(a*x)-4*arctanh(a*x)^2-3*arct anh(a*x)+6*a*x+6)+ln(1+(a*x+1)^2/(-a^2*x^2+1))-2*arctanh(a*x)*polylog(2,-( a*x+1)^2/(-a^2*x^2+1))+polylog(3,-(a*x+1)^2/(-a^2*x^2+1)))
\[ \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3 \, dx=\int { -{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{3} \,d x } \]
\[ \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3 \, dx=- \int a^{2} x^{2} \operatorname {atanh}^{3}{\left (a x \right )}\, dx - \int \left (- \operatorname {atanh}^{3}{\left (a x \right )}\right )\, dx \]
\[ \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3 \, dx=\int { -{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{3} \,d x } \]
1/48*(2*a^3*x^3 - 3*a^2*x^2 - 12*a*x - 6*(a^3*x^3 - 3*a*x - 2)*log(a*x + 1 ))*log(-a*x + 1)^2/a - 1/8*(log(-a*x + 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a *x + 1) - 6)*(a*x - 1)/a + 1/864*(4*(9*log(-a*x + 1)^3 - 9*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 27*(4*log(-a*x + 1)^3 - 6*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 3)*(a*x - 1)^2 + 108*(log(-a*x + 1)^3 - 3*log(-a *x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1))/a + 1/8*integrate(-1/3*(3*(a^3 *x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1)^3 + (2*a^3*x^3 - 3*a^2*x^2 - 9*(a^3 *x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1)^2 - 12*a*x - 6*(a^3*x^3 - 3*a*x - 2 )*log(a*x + 1))*log(-a*x + 1))/(a*x - 1), x)
\[ \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3 \, dx=\int { -{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{3} \,d x } \]
Timed out. \[ \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3 \, dx=-\int {\mathrm {atanh}\left (a\,x\right )}^3\,\left (a^2\,x^2-1\right ) \,d x \]