Integrand size = 16, antiderivative size = 148 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^n\right )\right )^2}{x} \, dx=\frac {2 \left (a+b \text {arctanh}\left (c x^n\right )\right )^2 \text {arctanh}\left (1-\frac {2}{1-c x^n}\right )}{n}-\frac {b \left (a+b \text {arctanh}\left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^n}\right )}{n}+\frac {b \left (a+b \text {arctanh}\left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x^n}\right )}{n}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^n}\right )}{2 n}-\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x^n}\right )}{2 n} \]
-2*(a+b*arctanh(c*x^n))^2*arctanh(-1+2/(1-c*x^n))/n-b*(a+b*arctanh(c*x^n)) *polylog(2,1-2/(1-c*x^n))/n+b*(a+b*arctanh(c*x^n))*polylog(2,-1+2/(1-c*x^n ))/n+1/2*b^2*polylog(3,1-2/(1-c*x^n))/n-1/2*b^2*polylog(3,-1+2/(1-c*x^n))/ n
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^n\right )\right )^2}{x} \, dx=a^2 \log (x)+\frac {a b \left (-\operatorname {PolyLog}\left (2,-c x^n\right )+\operatorname {PolyLog}\left (2,c x^n\right )\right )}{n}+\frac {b^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}\left (c x^n\right )^3-\text {arctanh}\left (c x^n\right )^2 \log \left (1+e^{-2 \text {arctanh}\left (c x^n\right )}\right )+\text {arctanh}\left (c x^n\right )^2 \log \left (1-e^{2 \text {arctanh}\left (c x^n\right )}\right )+\text {arctanh}\left (c x^n\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^n\right )}\right )+\text {arctanh}\left (c x^n\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (c x^n\right )}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (c x^n\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (c x^n\right )}\right )\right )}{n} \]
a^2*Log[x] + (a*b*(-PolyLog[2, -(c*x^n)] + PolyLog[2, c*x^n]))/n + (b^2*(( I/24)*Pi^3 - (2*ArcTanh[c*x^n]^3)/3 - ArcTanh[c*x^n]^2*Log[1 + E^(-2*ArcTa nh[c*x^n])] + ArcTanh[c*x^n]^2*Log[1 - E^(2*ArcTanh[c*x^n])] + ArcTanh[c*x ^n]*PolyLog[2, -E^(-2*ArcTanh[c*x^n])] + ArcTanh[c*x^n]*PolyLog[2, E^(2*Ar cTanh[c*x^n])] + PolyLog[3, -E^(-2*ArcTanh[c*x^n])]/2 - PolyLog[3, E^(2*Ar cTanh[c*x^n])]/2))/n
Time = 0.82 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6450, 6448, 6614, 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.
| \(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (c x^n\right )\right )^2}{x} \, dx\) |
| \(\Big \downarrow \) 6450 |
| \(\displaystyle \frac {\int x^{-n} \left (a+b \text {arctanh}\left (c x^n\right )\right )^2dx^n}{n}\) |
| \(\Big \downarrow \) 6448 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{1-c x^n}\right ) \left (a+b \text {arctanh}\left (c x^n\right )\right )^2-4 b c \int \frac {\left (a+b \text {arctanh}\left (c x^n\right )\right ) \text {arctanh}\left (1-\frac {2}{1-c x^n}\right )}{1-c^2 x^{2 n}}dx^n}{n}\) |
| \(\Big \downarrow \) 6614 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{1-c x^n}\right ) \left (a+b \text {arctanh}\left (c x^n\right )\right )^2-4 b c \left (\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (c x^n\right )\right ) \log \left (2-\frac {2}{1-c x^n}\right )}{1-c^2 x^{2 n}}dx^n-\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (c x^n\right )\right ) \log \left (\frac {2}{1-c x^n}\right )}{1-c^2 x^{2 n}}dx^n\right )}{n}\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{1-c x^n}\right ) \left (a+b \text {arctanh}\left (c x^n\right )\right )^2-4 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^n}\right ) \left (a+b \text {arctanh}\left (c x^n\right )\right )}{2 c}-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^n}\right )}{1-c^2 x^{2 n}}dx^n\right )+\frac {1}{2} \left (\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c x^n}-1\right )}{1-c^2 x^{2 n}}dx^n-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c x^n}-1\right ) \left (a+b \text {arctanh}\left (c x^n\right )\right )}{2 c}\right )\right )}{n}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{1-c x^n}\right ) \left (a+b \text {arctanh}\left (c x^n\right )\right )^2-4 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^n}\right ) \left (a+b \text {arctanh}\left (c x^n\right )\right )}{2 c}-\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^n}\right )}{4 c}\right )+\frac {1}{2} \left (\frac {b \operatorname {PolyLog}\left (3,\frac {2}{1-c x^n}-1\right )}{4 c}-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c x^n}-1\right ) \left (a+b \text {arctanh}\left (c x^n\right )\right )}{2 c}\right )\right )}{n}\) |
(2*(a + b*ArcTanh[c*x^n])^2*ArcTanh[1 - 2/(1 - c*x^n)] - 4*b*c*((((a + b*A rcTanh[c*x^n])*PolyLog[2, 1 - 2/(1 - c*x^n)])/(2*c) - (b*PolyLog[3, 1 - 2/ (1 - c*x^n)])/(4*c))/2 + (-1/2*((a + b*ArcTanh[c*x^n])*PolyLog[2, -1 + 2/( 1 - c*x^n)])/c + (b*PolyLog[3, -1 + 2/(1 - c*x^n)])/(4*c))/2))/n
3.3.33.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcTanh[c*x])^p*ArcTanh[1 - 2/(1 - c*x)], x] - Simp[2*b*c*p Int[(a + b *ArcTanh[c*x])^(p - 1)*(ArcTanh[1 - 2/(1 - c*x)]/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[ 1/n Subst[Int[(a + b*ArcTanh[c*x])^p/x, x], x, x^n], x] /; FreeQ[{a, b, c , n}, x] && IGtQ[p, 0]
Int[(ArcTanh[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*( x_)^2), x_Symbol] :> Simp[1/2 Int[Log[1 + u]*((a + b*ArcTanh[c*x])^p/(d + e*x^2)), x], x] - Simp[1/2 Int[Log[1 - u]*((a + b*ArcTanh[c*x])^p/(d + e *x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 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 = 23.72 (sec) , antiderivative size = 750, normalized size of antiderivative = 5.07
| method | result | size |
| parts | \(a^{2} \ln \left (x \right )+\frac {b^{2} \left (\ln \left (c \,x^{n}\right ) \operatorname {arctanh}\left (c \,x^{n}\right )^{2}-\operatorname {arctanh}\left (c \,x^{n}\right ) \operatorname {polylog}\left (2, -\frac {\left (c \,x^{n}+1\right )^{2}}{-c^{2} x^{2 n}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (c \,x^{n}+1\right )^{2}}{-c^{2} x^{2 n}+1}\right )}{2}-\operatorname {arctanh}\left (c \,x^{n}\right )^{2} \ln \left (\frac {\left (c \,x^{n}+1\right )^{2}}{-c^{2} x^{2 n}+1}-1\right )+\operatorname {arctanh}\left (c \,x^{n}\right )^{2} \ln \left (1+\frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{n}\right ) \operatorname {polylog}\left (2, -\frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )-2 \operatorname {polylog}\left (3, -\frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )+\operatorname {arctanh}\left (c \,x^{n}\right )^{2} \ln \left (1-\frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{n}\right ) \operatorname {polylog}\left (2, \frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right )}^{2}\right ) \operatorname {arctanh}\left (c \,x^{n}\right )^{2}}{2}\right )}{n}+\frac {2 a b \left (\ln \left (c \,x^{n}\right ) \operatorname {arctanh}\left (c \,x^{n}\right )-\frac {\operatorname {dilog}\left (c \,x^{n}+1\right )}{2}-\frac {\ln \left (c \,x^{n}\right ) \ln \left (c \,x^{n}+1\right )}{2}-\frac {\operatorname {dilog}\left (c \,x^{n}\right )}{2}\right )}{n}\) | \(750\) |
| derivativedivides | \(\frac {a^{2} \ln \left (c \,x^{n}\right )+b^{2} \left (\ln \left (c \,x^{n}\right ) \operatorname {arctanh}\left (c \,x^{n}\right )^{2}-\operatorname {arctanh}\left (c \,x^{n}\right ) \operatorname {polylog}\left (2, -\frac {\left (c \,x^{n}+1\right )^{2}}{-c^{2} x^{2 n}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (c \,x^{n}+1\right )^{2}}{-c^{2} x^{2 n}+1}\right )}{2}-\operatorname {arctanh}\left (c \,x^{n}\right )^{2} \ln \left (\frac {\left (c \,x^{n}+1\right )^{2}}{-c^{2} x^{2 n}+1}-1\right )+\operatorname {arctanh}\left (c \,x^{n}\right )^{2} \ln \left (1+\frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{n}\right ) \operatorname {polylog}\left (2, -\frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )-2 \operatorname {polylog}\left (3, -\frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )+\operatorname {arctanh}\left (c \,x^{n}\right )^{2} \ln \left (1-\frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{n}\right ) \operatorname {polylog}\left (2, \frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right )}^{2}\right ) \operatorname {arctanh}\left (c \,x^{n}\right )^{2}}{2}\right )+2 a b \left (\ln \left (c \,x^{n}\right ) \operatorname {arctanh}\left (c \,x^{n}\right )-\frac {\operatorname {dilog}\left (c \,x^{n}+1\right )}{2}-\frac {\ln \left (c \,x^{n}\right ) \ln \left (c \,x^{n}+1\right )}{2}-\frac {\operatorname {dilog}\left (c \,x^{n}\right )}{2}\right )}{n}\) | \(752\) |
| default | \(\frac {a^{2} \ln \left (c \,x^{n}\right )+b^{2} \left (\ln \left (c \,x^{n}\right ) \operatorname {arctanh}\left (c \,x^{n}\right )^{2}-\operatorname {arctanh}\left (c \,x^{n}\right ) \operatorname {polylog}\left (2, -\frac {\left (c \,x^{n}+1\right )^{2}}{-c^{2} x^{2 n}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (c \,x^{n}+1\right )^{2}}{-c^{2} x^{2 n}+1}\right )}{2}-\operatorname {arctanh}\left (c \,x^{n}\right )^{2} \ln \left (\frac {\left (c \,x^{n}+1\right )^{2}}{-c^{2} x^{2 n}+1}-1\right )+\operatorname {arctanh}\left (c \,x^{n}\right )^{2} \ln \left (1+\frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{n}\right ) \operatorname {polylog}\left (2, -\frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )-2 \operatorname {polylog}\left (3, -\frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )+\operatorname {arctanh}\left (c \,x^{n}\right )^{2} \ln \left (1-\frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{n}\right ) \operatorname {polylog}\left (2, \frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {c \,x^{n}+1}{\sqrt {-c^{2} x^{2 n}+1}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}-1\right )}{1-\frac {\left (c \,x^{n}+1\right )^{2}}{c^{2} x^{2 n}-1}}\right )}^{2}\right ) \operatorname {arctanh}\left (c \,x^{n}\right )^{2}}{2}\right )+2 a b \left (\ln \left (c \,x^{n}\right ) \operatorname {arctanh}\left (c \,x^{n}\right )-\frac {\operatorname {dilog}\left (c \,x^{n}+1\right )}{2}-\frac {\ln \left (c \,x^{n}\right ) \ln \left (c \,x^{n}+1\right )}{2}-\frac {\operatorname {dilog}\left (c \,x^{n}\right )}{2}\right )}{n}\) | \(752\) |
a^2*ln(x)+b^2/n*(ln(c*x^n)*arctanh(c*x^n)^2-arctanh(c*x^n)*polylog(2,-(c*x ^n+1)^2/(-c^2*(x^n)^2+1))+1/2*polylog(3,-(c*x^n+1)^2/(-c^2*(x^n)^2+1))-arc tanh(c*x^n)^2*ln((c*x^n+1)^2/(-c^2*(x^n)^2+1)-1)+arctanh(c*x^n)^2*ln(1+(c* x^n+1)/(-c^2*(x^n)^2+1)^(1/2))+2*arctanh(c*x^n)*polylog(2,-(c*x^n+1)/(-c^2 *(x^n)^2+1)^(1/2))-2*polylog(3,-(c*x^n+1)/(-c^2*(x^n)^2+1)^(1/2))+arctanh( c*x^n)^2*ln(1-(c*x^n+1)/(-c^2*(x^n)^2+1)^(1/2))+2*arctanh(c*x^n)*polylog(2 ,(c*x^n+1)/(-c^2*(x^n)^2+1)^(1/2))-2*polylog(3,(c*x^n+1)/(-c^2*(x^n)^2+1)^ (1/2))+1/2*I*Pi*csgn(I*(-(c*x^n+1)^2/(c^2*(x^n)^2-1)-1)/(1-(c*x^n+1)^2/(c^ 2*(x^n)^2-1)))*(csgn(I*(-(c*x^n+1)^2/(c^2*(x^n)^2-1)-1))*csgn(I/(1-(c*x^n+ 1)^2/(c^2*(x^n)^2-1)))-csgn(I*(-(c*x^n+1)^2/(c^2*(x^n)^2-1)-1))*csgn(I*(-( c*x^n+1)^2/(c^2*(x^n)^2-1)-1)/(1-(c*x^n+1)^2/(c^2*(x^n)^2-1)))-csgn(I*(-(c *x^n+1)^2/(c^2*(x^n)^2-1)-1)/(1-(c*x^n+1)^2/(c^2*(x^n)^2-1)))*csgn(I/(1-(c *x^n+1)^2/(c^2*(x^n)^2-1)))+csgn(I*(-(c*x^n+1)^2/(c^2*(x^n)^2-1)-1)/(1-(c* x^n+1)^2/(c^2*(x^n)^2-1)))^2)*arctanh(c*x^n)^2)+2*a*b/n*(ln(c*x^n)*arctanh (c*x^n)-1/2*dilog(c*x^n+1)-1/2*ln(c*x^n)*ln(c*x^n+1)-1/2*dilog(c*x^n))
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{n}\right ) + a\right )}^{2}}{x} \,d x } \]
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^n\right )\right )^2}{x} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x^{n} \right )}\right )^{2}}{x}\, dx \]
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{n}\right ) + a\right )}^{2}}{x} \,d x } \]
1/4*b^2*log(-c*x^n + 1)^2*log(x) + a^2*log(x) - integrate(-1/4*((b^2*c*x^n - b^2)*log(c*x^n + 1)^2 + 4*(a*b*c*x^n - a*b)*log(c*x^n + 1) + 2*(2*a*b - (b^2*c*n*log(x) + 2*a*b*c)*x^n - (b^2*c*x^n - b^2)*log(c*x^n + 1))*log(-c *x^n + 1))/(c*x*x^n - x), x)
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{n}\right ) + a\right )}^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c x^n\right )\right )^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^n\right )\right )}^2}{x} \,d x \]