Integrand size = 12, antiderivative size = 263 \[ \int x^3 \text {arctanh}(a+b x)^2 \, dx=-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {a \text {arctanh}(a+b x)}{b^4}+\frac {\left (1+6 a^2\right ) (a+b x) \text {arctanh}(a+b x)}{2 b^4}-\frac {a (a+b x)^2 \text {arctanh}(a+b x)}{b^4}+\frac {(a+b x)^3 \text {arctanh}(a+b x)}{6 b^4}-\frac {a \left (1+a^2\right ) \text {arctanh}(a+b x)^2}{b^4}-\frac {\left (1+6 a^2+a^4\right ) \text {arctanh}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \text {arctanh}(a+b x)^2+\frac {2 a \left (1+a^2\right ) \text {arctanh}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b^4}+\frac {\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac {\left (1+6 a^2\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}+\frac {a \left (1+a^2\right ) \operatorname {PolyLog}\left (2,-\frac {1+a+b x}{1-a-b x}\right )}{b^4} \] Output:
-a*x/b^3+1/12*(b*x+a)^2/b^4+a*arctanh(b*x+a)/b^4+1/2*(6*a^2+1)*(b*x+a)*arc tanh(b*x+a)/b^4-a*(b*x+a)^2*arctanh(b*x+a)/b^4+1/6*(b*x+a)^3*arctanh(b*x+a )/b^4-a*(a^2+1)*arctanh(b*x+a)^2/b^4-1/4*(a^4+6*a^2+1)*arctanh(b*x+a)^2/b^ 4+1/4*x^4*arctanh(b*x+a)^2+2*a*(a^2+1)*arctanh(b*x+a)*ln(2/(-b*x-a+1))/b^4 +1/12*ln(1-(b*x+a)^2)/b^4+1/4*(6*a^2+1)*ln(1-(b*x+a)^2)/b^4+a*(a^2+1)*poly log(2,-(b*x+a+1)/(-b*x-a+1))/b^4
Time = 0.97 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.71 \[ \int x^3 \text {arctanh}(a+b x)^2 \, dx=-\frac {1+11 a^2+10 a b x-b^2 x^2+3 \left (1-4 a+6 a^2-4 a^3+a^4-b^4 x^4\right ) \text {arctanh}(a+b x)^2-2 \text {arctanh}(a+b x) \left (9 a+13 a^3+3 b x+9 a^2 b x-3 a b^2 x^2+b^3 x^3+12 \left (a+a^3\right ) \log \left (1+e^{-2 \text {arctanh}(a+b x)}\right )\right )+8 \log \left (\frac {1}{\sqrt {1-(a+b x)^2}}\right )+36 a^2 \log \left (\frac {1}{\sqrt {1-(a+b x)^2}}\right )+12 \left (a+a^3\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a+b x)}\right )}{12 b^4} \] Input:
Integrate[x^3*ArcTanh[a + b*x]^2,x]
Output:
-1/12*(1 + 11*a^2 + 10*a*b*x - b^2*x^2 + 3*(1 - 4*a + 6*a^2 - 4*a^3 + a^4 - b^4*x^4)*ArcTanh[a + b*x]^2 - 2*ArcTanh[a + b*x]*(9*a + 13*a^3 + 3*b*x + 9*a^2*b*x - 3*a*b^2*x^2 + b^3*x^3 + 12*(a + a^3)*Log[1 + E^(-2*ArcTanh[a + b*x])]) + 8*Log[1/Sqrt[1 - (a + b*x)^2]] + 36*a^2*Log[1/Sqrt[1 - (a + b* x)^2]] + 12*(a + a^3)*PolyLog[2, -E^(-2*ArcTanh[a + b*x])])/b^4
Time = 0.54 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6661, 25, 27, 6480, 2009}
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.
| \(\displaystyle \int x^3 \text {arctanh}(a+b x)^2 \, dx\) |
| \(\Big \downarrow \) 6661 |
| \(\displaystyle \frac {\int x^3 \text {arctanh}(a+b x)^2d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -x^3 \text {arctanh}(a+b x)^2d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -b^3 x^3 \text {arctanh}(a+b x)^2d(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle -\frac {\frac {1}{2} \int \left (-\text {arctanh}(a+b x) (a+b x)^2+4 a \text {arctanh}(a+b x) (a+b x)-\left (6 a^2+1\right ) \text {arctanh}(a+b x)+\frac {\left (a^4+6 a^2-4 \left (a^2+1\right ) (a+b x) a+1\right ) \text {arctanh}(a+b x)}{1-(a+b x)^2}\right )d(a+b x)-\frac {1}{4} b^4 x^4 \text {arctanh}(a+b x)^2}{b^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\left (6 a^2+1\right ) (a+b x) \text {arctanh}(a+b x)+2 a \left (a^2+1\right ) \text {arctanh}(a+b x)^2-4 a \left (a^2+1\right ) \text {arctanh}(a+b x) \log \left (\frac {2}{-a-b x+1}\right )-2 a \left (a^2+1\right ) \operatorname {PolyLog}\left (2,-\frac {a+b x+1}{-a-b x+1}\right )-\frac {1}{2} \left (6 a^2+1\right ) \log \left (1-(a+b x)^2\right )+\frac {1}{2} \left (a^4+6 a^2+1\right ) \text {arctanh}(a+b x)^2-\frac {1}{3} (a+b x)^3 \text {arctanh}(a+b x)+2 a (a+b x)^2 \text {arctanh}(a+b x)-2 a \text {arctanh}(a+b x)-\frac {1}{6} (a+b x)^2+2 a (a+b x)-\frac {1}{6} \log \left (1-(a+b x)^2\right )\right )-\frac {1}{4} b^4 x^4 \text {arctanh}(a+b x)^2}{b^4}\) |
Input:
Int[x^3*ArcTanh[a + b*x]^2,x]
Output:
-((-1/4*(b^4*x^4*ArcTanh[a + b*x]^2) + (2*a*(a + b*x) - (a + b*x)^2/6 - 2* a*ArcTanh[a + b*x] - (1 + 6*a^2)*(a + b*x)*ArcTanh[a + b*x] + 2*a*(a + b*x )^2*ArcTanh[a + b*x] - ((a + b*x)^3*ArcTanh[a + b*x])/3 + 2*a*(1 + a^2)*Ar cTanh[a + b*x]^2 + ((1 + 6*a^2 + a^4)*ArcTanh[a + b*x]^2)/2 - 4*a*(1 + a^2 )*ArcTanh[a + b*x]*Log[2/(1 - a - b*x)] - Log[1 - (a + b*x)^2]/6 - ((1 + 6 *a^2)*Log[1 - (a + b*x)^2])/2 - 2*a*(1 + a^2)*PolyLog[2, -((1 + a + b*x)/( 1 - a - b*x))])/2)/b^4)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_S ymbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])^p/(e*(q + 1))), x] - Simp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1 ), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IG tQ[p, 0]
Time = 1.29 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.71
| method | result | size |
| parts | \(\frac {x^{4} \operatorname {arctanh}\left (b x +a \right )^{2}}{4}-\frac {-6 \,\operatorname {arctanh}\left (b x +a \right ) a^{2} \left (b x +a \right )+2 \,\operatorname {arctanh}\left (b x +a \right ) \left (b x +a \right )^{2} a -\frac {\operatorname {arctanh}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}-\operatorname {arctanh}\left (b x +a \right ) \left (b x +a \right )-\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{4}}{2}+2 \,\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{3}-3 \,\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{2}+2 \,\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right ) a -\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right )}{2}+\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{4}}{2}+2 \,\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{3}+3 \,\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{2}+2 \,\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right ) a +\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2}+2 \left (b x +a \right ) a -\frac {\left (b x +a \right )^{2}}{6}-\frac {\left (18 a^{2}-6 a +4\right ) \ln \left (b x +a -1\right )}{6}+\frac {\left (-18 a^{2}-6 a -4\right ) \ln \left (b x +a +1\right )}{6}-\frac {\left (3 a^{4}-12 a^{3}+18 a^{2}-12 a +3\right ) \left (\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{6}-\frac {\left (-3 a^{4}-12 a^{3}-18 a^{2}-12 a -3\right ) \left (-\frac {\ln \left (b x +a +1\right )^{2}}{4}+\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{6}}{2 b^{4}}\) | \(449\) |
| derivativedivides | \(\frac {-\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right ) a +\frac {3 \,\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{2}}{2}+\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{4}}{4}-\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{3}+3 \,\operatorname {arctanh}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arctanh}\left (b x +a \right ) \left (b x +a \right )^{2} a +\frac {\left (b x +a \right )^{2}}{12}-\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{4}}{4}-\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{3}-\frac {3 \,\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{2}}{2}-\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right ) a -\operatorname {arctanh}\left (b x +a \right )^{2} a^{3} \left (b x +a \right )+\frac {3 \operatorname {arctanh}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arctanh}\left (b x +a \right )^{2} a \left (b x +a \right )^{3}-\frac {\left (-18 a^{2}-6 a -4\right ) \ln \left (b x +a +1\right )}{12}+\frac {\left (18 a^{2}-6 a +4\right ) \ln \left (b x +a -1\right )}{12}+\frac {\operatorname {arctanh}\left (b x +a \right ) \left (b x +a \right )^{3}}{6}+\frac {\operatorname {arctanh}\left (b x +a \right ) \left (b x +a \right )}{2}+\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right )}{4}-\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right )}{4}-\left (b x +a \right ) a +\frac {\operatorname {arctanh}\left (b x +a \right )^{2} a^{4}}{4}+\frac {\operatorname {arctanh}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}+\frac {\left (-3 a^{4}-12 a^{3}-18 a^{2}-12 a -3\right ) \left (-\frac {\ln \left (b x +a +1\right )^{2}}{4}+\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{12}+\frac {\left (3 a^{4}-12 a^{3}+18 a^{2}-12 a +3\right ) \left (\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{12}}{b^{4}}\) | \(520\) |
| default | \(\frac {-\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right ) a +\frac {3 \,\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{2}}{2}+\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{4}}{4}-\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{3}+3 \,\operatorname {arctanh}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arctanh}\left (b x +a \right ) \left (b x +a \right )^{2} a +\frac {\left (b x +a \right )^{2}}{12}-\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{4}}{4}-\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{3}-\frac {3 \,\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{2}}{2}-\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right ) a -\operatorname {arctanh}\left (b x +a \right )^{2} a^{3} \left (b x +a \right )+\frac {3 \operatorname {arctanh}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arctanh}\left (b x +a \right )^{2} a \left (b x +a \right )^{3}-\frac {\left (-18 a^{2}-6 a -4\right ) \ln \left (b x +a +1\right )}{12}+\frac {\left (18 a^{2}-6 a +4\right ) \ln \left (b x +a -1\right )}{12}+\frac {\operatorname {arctanh}\left (b x +a \right ) \left (b x +a \right )^{3}}{6}+\frac {\operatorname {arctanh}\left (b x +a \right ) \left (b x +a \right )}{2}+\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a -1\right )}{4}-\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (b x +a +1\right )}{4}-\left (b x +a \right ) a +\frac {\operatorname {arctanh}\left (b x +a \right )^{2} a^{4}}{4}+\frac {\operatorname {arctanh}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}+\frac {\left (-3 a^{4}-12 a^{3}-18 a^{2}-12 a -3\right ) \left (-\frac {\ln \left (b x +a +1\right )^{2}}{4}+\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{12}+\frac {\left (3 a^{4}-12 a^{3}+18 a^{2}-12 a +3\right ) \left (\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{12}}{b^{4}}\) | \(520\) |
| risch | \(-\frac {1}{12 b^{4}}-\frac {5 a x}{6 b^{3}}-\frac {\ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a^{3}}{b^{4}}+\frac {\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) \ln \left (-b x -a +1\right ) a^{3}}{b^{4}}-\frac {\ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a}{b^{4}}+\frac {\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) \ln \left (-b x -a +1\right ) a}{b^{4}}+\frac {\ln \left (-b x -a -1\right )}{3 b^{4}}+\left (-\frac {x^{4} \ln \left (-b x -a +1\right )}{8}-\frac {-2 b^{3} x^{3}-3 \ln \left (-b x -a +1\right ) a^{4}+6 a \,b^{2} x^{2}+12 \ln \left (-b x -a +1\right ) a^{3}-18 a^{2} b x -18 \ln \left (-b x -a +1\right ) a^{2}+12 \ln \left (-b x -a +1\right ) a -6 b x -3 \ln \left (-b x -a +1\right )}{24 b^{4}}\right ) \ln \left (b x +a +1\right )-\frac {\operatorname {dilog}\left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) a^{3}}{b^{4}}-\frac {\operatorname {dilog}\left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) a}{b^{4}}+\frac {13 \ln \left (-b x -a -1\right ) a^{3}}{12 b^{4}}+\frac {3 \ln \left (-b x -a -1\right ) a^{2}}{2 b^{4}}+\frac {3 \ln \left (-b x -a -1\right ) a}{4 b^{4}}-\frac {\left (-b^{4} x^{4}+a^{4}+4 a^{3}+6 a^{2}+4 a +1\right ) \ln \left (b x +a +1\right )^{2}}{16 b^{4}}+\frac {\ln \left (-b x -a +1\right )^{2} x^{4}}{16}+\frac {a}{b^{4}}-\frac {11 a^{2}}{12 b^{4}}+\frac {x^{2}}{12 b^{2}}+\frac {\ln \left (-b x -a +1\right )}{3 b^{4}}-\frac {\ln \left (-b x -a +1\right )^{2}}{16 b^{4}}-\frac {\ln \left (-b x -a +1\right ) x}{4 b^{3}}+\frac {\ln \left (-b x -a +1\right )^{2} a^{3}}{4 b^{4}}-\frac {3 \ln \left (-b x -a +1\right )^{2} a^{2}}{8 b^{4}}+\frac {\ln \left (-b x -a +1\right )^{2} a}{4 b^{4}}-\frac {13 \ln \left (-b x -a +1\right ) a^{3}}{12 b^{4}}+\frac {3 \ln \left (-b x -a +1\right ) a^{2}}{2 b^{4}}-\frac {3 \ln \left (-b x -a +1\right ) a}{4 b^{4}}-\frac {\ln \left (-b x -a +1\right )^{2} a^{4}}{16 b^{4}}-\frac {\ln \left (-b x -a +1\right ) x^{3}}{12 b}+\frac {\ln \left (-b x -a +1\right ) x^{2} a}{4 b^{2}}-\frac {3 \ln \left (-b x -a +1\right ) x \,a^{2}}{4 b^{3}}\) | \(661\) |
Input:
int(x^3*arctanh(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/4*x^4*arctanh(b*x+a)^2-1/2/b^4*(-6*arctanh(b*x+a)*a^2*(b*x+a)+2*arctanh( b*x+a)*(b*x+a)^2*a-1/3*arctanh(b*x+a)*(b*x+a)^3-arctanh(b*x+a)*(b*x+a)-1/2 *arctanh(b*x+a)*ln(b*x+a-1)*a^4+2*arctanh(b*x+a)*ln(b*x+a-1)*a^3-3*arctanh (b*x+a)*ln(b*x+a-1)*a^2+2*arctanh(b*x+a)*ln(b*x+a-1)*a-1/2*arctanh(b*x+a)* ln(b*x+a-1)+1/2*arctanh(b*x+a)*ln(b*x+a+1)*a^4+2*arctanh(b*x+a)*ln(b*x+a+1 )*a^3+3*arctanh(b*x+a)*ln(b*x+a+1)*a^2+2*arctanh(b*x+a)*ln(b*x+a+1)*a+1/2* arctanh(b*x+a)*ln(b*x+a+1)+2*(b*x+a)*a-1/6*(b*x+a)^2-1/6*(18*a^2-6*a+4)*ln (b*x+a-1)+1/6*(-18*a^2-6*a-4)*ln(b*x+a+1)-1/6*(3*a^4-12*a^3+18*a^2-12*a+3) *(1/4*ln(b*x+a-1)^2-1/2*dilog(1/2*b*x+1/2*a+1/2)-1/2*ln(b*x+a-1)*ln(1/2*b* x+1/2*a+1/2))-1/6*(-3*a^4-12*a^3-18*a^2-12*a-3)*(-1/4*ln(b*x+a+1)^2+1/2*(l n(b*x+a+1)-ln(1/2*b*x+1/2*a+1/2))*ln(-1/2*b*x-1/2*a+1/2)-1/2*dilog(1/2*b*x +1/2*a+1/2)))
\[ \int x^3 \text {arctanh}(a+b x)^2 \, dx=\int { x^{3} \operatorname {artanh}\left (b x + a\right )^{2} \,d x } \] Input:
integrate(x^3*arctanh(b*x+a)^2,x, algorithm="fricas")
Output:
integral(x^3*arctanh(b*x + a)^2, x)
\[ \int x^3 \text {arctanh}(a+b x)^2 \, dx=\int x^{3} \operatorname {atanh}^{2}{\left (a + b x \right )}\, dx \] Input:
integrate(x**3*atanh(b*x+a)**2,x)
Output:
Integral(x**3*atanh(a + b*x)**2, x)
Time = 0.03 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.22 \[ \int x^3 \text {arctanh}(a+b x)^2 \, dx=\frac {1}{4} \, x^{4} \operatorname {artanh}\left (b x + a\right )^{2} + \frac {1}{48} \, b^{2} {\left (\frac {48 \, {\left (a^{3} + a\right )} {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )}}{b^{6}} + \frac {4 \, {\left (13 \, a^{3} + 18 \, a^{2} + 9 \, a + 4\right )} \log \left (b x + a + 1\right )}{b^{6}} + \frac {4 \, b^{2} x^{2} - 40 \, a b x + 3 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 6 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + 3 \, {\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )^{2} - 4 \, {\left (13 \, a^{3} - 18 \, a^{2} + 9 \, a - 4\right )} \log \left (b x + a - 1\right )}{b^{6}}\right )} + \frac {1}{12} \, b {\left (\frac {2 \, {\left (b^{2} x^{3} - 3 \, a b x^{2} + 3 \, {\left (3 \, a^{2} + 1\right )} x\right )}}{b^{4}} - \frac {3 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{5}} + \frac {3 \, {\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{5}}\right )} \operatorname {artanh}\left (b x + a\right ) \] Input:
integrate(x^3*arctanh(b*x+a)^2,x, algorithm="maxima")
Output:
1/4*x^4*arctanh(b*x + a)^2 + 1/48*b^2*(48*(a^3 + a)*(log(b*x + a - 1)*log( 1/2*b*x + 1/2*a + 1/2) + dilog(-1/2*b*x - 1/2*a + 1/2))/b^6 + 4*(13*a^3 + 18*a^2 + 9*a + 4)*log(b*x + a + 1)/b^6 + (4*b^2*x^2 - 40*a*b*x + 3*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1)^2 - 6*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1)*log(b*x + a - 1) + 3*(a^4 - 4*a^3 + 6*a^2 - 4*a + 1 )*log(b*x + a - 1)^2 - 4*(13*a^3 - 18*a^2 + 9*a - 4)*log(b*x + a - 1))/b^6 ) + 1/12*b*(2*(b^2*x^3 - 3*a*b*x^2 + 3*(3*a^2 + 1)*x)/b^4 - 3*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1)/b^5 + 3*(a^4 - 4*a^3 + 6*a^2 - 4*a + 1)*log(b*x + a - 1)/b^5)*arctanh(b*x + a)
\[ \int x^3 \text {arctanh}(a+b x)^2 \, dx=\int { x^{3} \operatorname {artanh}\left (b x + a\right )^{2} \,d x } \] Input:
integrate(x^3*arctanh(b*x+a)^2,x, algorithm="giac")
Output:
integrate(x^3*arctanh(b*x + a)^2, x)
Timed out. \[ \int x^3 \text {arctanh}(a+b x)^2 \, dx=\int x^3\,{\mathrm {atanh}\left (a+b\,x\right )}^2 \,d x \] Input:
int(x^3*atanh(a + b*x)^2,x)
Output:
int(x^3*atanh(a + b*x)^2, x)
\[ \int x^3 \text {arctanh}(a+b x)^2 \, dx=\frac {3 \mathit {atanh} \left (b x +a \right )^{2} a^{4}-6 \mathit {atanh} \left (b x +a \right )^{2} a^{2}+3 \mathit {atanh} \left (b x +a \right )^{2} b^{4} x^{4}+3 \mathit {atanh} \left (b x +a \right )^{2}+14 \mathit {atanh} \left (b x +a \right ) a^{3}+6 \mathit {atanh} \left (b x +a \right ) a^{2} b x +24 \mathit {atanh} \left (b x +a \right ) a^{2}-6 \mathit {atanh} \left (b x +a \right ) a \,b^{2} x^{2}+6 \mathit {atanh} \left (b x +a \right ) a +2 \mathit {atanh} \left (b x +a \right ) b^{3} x^{3}-6 \mathit {atanh} \left (b x +a \right ) b x -4 \mathit {atanh} \left (b x +a \right )+12 \left (\int \frac {\mathit {atanh} \left (b x +a \right ) x^{2}}{b^{2} x^{2}+2 a b x +a^{2}-1}d x \right ) a^{2} b^{3}+12 \left (\int \frac {\mathit {atanh} \left (b x +a \right ) x^{2}}{b^{2} x^{2}+2 a b x +a^{2}-1}d x \right ) b^{3}+24 \,\mathrm {log}\left (b x +a -1\right ) a^{2}-4 \,\mathrm {log}\left (b x +a -1\right )-10 a b x +b^{2} x^{2}}{12 b^{4}} \] Input:
int(x^3*atanh(b*x+a)^2,x)
Output:
(3*atanh(a + b*x)**2*a**4 - 6*atanh(a + b*x)**2*a**2 + 3*atanh(a + b*x)**2 *b**4*x**4 + 3*atanh(a + b*x)**2 + 14*atanh(a + b*x)*a**3 + 6*atanh(a + b* x)*a**2*b*x + 24*atanh(a + b*x)*a**2 - 6*atanh(a + b*x)*a*b**2*x**2 + 6*at anh(a + b*x)*a + 2*atanh(a + b*x)*b**3*x**3 - 6*atanh(a + b*x)*b*x - 4*ata nh(a + b*x) + 12*int((atanh(a + b*x)*x**2)/(a**2 + 2*a*b*x + b**2*x**2 - 1 ),x)*a**2*b**3 + 12*int((atanh(a + b*x)*x**2)/(a**2 + 2*a*b*x + b**2*x**2 - 1),x)*b**3 + 24*log(a + b*x - 1)*a**2 - 4*log(a + b*x - 1) - 10*a*b*x + b**2*x**2)/(12*b**4)