Integrand size = 16, antiderivative size = 264 \[ \int x^2 \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=-\frac {1}{6} x^3 \log \left (1-a-b f^{c+d x}\right )+\frac {1}{6} x^3 \log \left (1+a+b f^{c+d x}\right )+\frac {1}{6} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )-\frac {1}{6} x^3 \log \left (1+\frac {b f^{c+d x}}{1+a}\right )+\frac {x^2 \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac {x \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{1-a}\right )}{d^2 \log ^2(f)}+\frac {x \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{1+a}\right )}{d^2 \log ^2(f)}+\frac {\operatorname {PolyLog}\left (4,\frac {b f^{c+d x}}{1-a}\right )}{d^3 \log ^3(f)}-\frac {\operatorname {PolyLog}\left (4,-\frac {b f^{c+d x}}{1+a}\right )}{d^3 \log ^3(f)} \]
-1/6*x^3*ln(1-a-b*f^(d*x+c))+1/6*x^3*ln(1+a+b*f^(d*x+c))+1/6*x^3*ln(1-b*f^ (d*x+c)/(1-a))-1/6*x^3*ln(1+b*f^(d*x+c)/(1+a))+1/2*x^2*polylog(2,b*f^(d*x+ c)/(1-a))/d/ln(f)-1/2*x^2*polylog(2,-b*f^(d*x+c)/(1+a))/d/ln(f)-x*polylog( 3,b*f^(d*x+c)/(1-a))/d^2/ln(f)^2+x*polylog(3,-b*f^(d*x+c)/(1+a))/d^2/ln(f) ^2+polylog(4,b*f^(d*x+c)/(1-a))/d^3/ln(f)^3-polylog(4,-b*f^(d*x+c)/(1+a))/ d^3/ln(f)^3
Time = 0.05 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.89 \[ \int x^2 \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=\frac {2 d^3 x^3 \text {arctanh}\left (a+b f^{c+d x}\right ) \log ^3(f)+d^3 x^3 \log ^3(f) \log \left (1+\frac {b f^{c+d x}}{-1+a}\right )-d^3 x^3 \log ^3(f) \log \left (1+\frac {b f^{c+d x}}{1+a}\right )+3 d^2 x^2 \log ^2(f) \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{-1+a}\right )-3 d^2 x^2 \log ^2(f) \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right )-6 d x \log (f) \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{-1+a}\right )+6 d x \log (f) \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{1+a}\right )+6 \operatorname {PolyLog}\left (4,-\frac {b f^{c+d x}}{-1+a}\right )-6 \operatorname {PolyLog}\left (4,-\frac {b f^{c+d x}}{1+a}\right )}{6 d^3 \log ^3(f)} \]
(2*d^3*x^3*ArcTanh[a + b*f^(c + d*x)]*Log[f]^3 + d^3*x^3*Log[f]^3*Log[1 + (b*f^(c + d*x))/(-1 + a)] - d^3*x^3*Log[f]^3*Log[1 + (b*f^(c + d*x))/(1 + a)] + 3*d^2*x^2*Log[f]^2*PolyLog[2, -((b*f^(c + d*x))/(-1 + a))] - 3*d^2*x ^2*Log[f]^2*PolyLog[2, -((b*f^(c + d*x))/(1 + a))] - 6*d*x*Log[f]*PolyLog[ 3, -((b*f^(c + d*x))/(-1 + a))] + 6*d*x*Log[f]*PolyLog[3, -((b*f^(c + d*x) )/(1 + a))] + 6*PolyLog[4, -((b*f^(c + d*x))/(-1 + a))] - 6*PolyLog[4, -(( b*f^(c + d*x))/(1 + a))])/(6*d^3*Log[f]^3)
Time = 0.99 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6767, 3012, 3011, 7163, 2720, 7143}
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.
| \(\displaystyle \int x^2 \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx\) |
| \(\Big \downarrow \) 6767 |
| \(\displaystyle \frac {1}{2} \int x^2 \log \left (b f^{c+d x}+a+1\right )dx-\frac {1}{2} \int x^2 \log \left (-b f^{c+d x}-a+1\right )dx\) |
| \(\Big \downarrow \) 3012 |
| \(\displaystyle \frac {1}{2} \left (-\int x^2 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )dx-\frac {1}{3} x^3 \log \left (-a-b f^{c+d x}+1\right )+\frac {1}{3} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )\right )+\frac {1}{2} \left (\int x^2 \log \left (\frac {b f^{c+d x}}{a+1}+1\right )dx+\frac {1}{3} x^3 \log \left (a+b f^{c+d x}+1\right )-\frac {1}{3} x^3 \log \left (\frac {b f^{c+d x}}{a+1}+1\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 \int x \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{1-a}\right )dx}{d \log (f)}+\frac {x^2 \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{1-a}\right )}{d \log (f)}-\frac {1}{3} x^3 \log \left (-a-b f^{c+d x}+1\right )+\frac {1}{3} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )\right )+\frac {1}{2} \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{a+1}\right )dx}{d \log (f)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{a+1}\right )}{d \log (f)}+\frac {1}{3} x^3 \log \left (a+b f^{c+d x}+1\right )-\frac {1}{3} x^3 \log \left (\frac {b f^{c+d x}}{a+1}+1\right )\right )\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{1-a}\right )}{d \log (f)}-\frac {\int \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{1-a}\right )dx}{d \log (f)}\right )}{d \log (f)}+\frac {x^2 \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{1-a}\right )}{d \log (f)}-\frac {1}{3} x^3 \log \left (-a-b f^{c+d x}+1\right )+\frac {1}{3} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )\right )+\frac {1}{2} \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{a+1}\right )}{d \log (f)}-\frac {\int \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{a+1}\right )dx}{d \log (f)}\right )}{d \log (f)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{a+1}\right )}{d \log (f)}+\frac {1}{3} x^3 \log \left (a+b f^{c+d x}+1\right )-\frac {1}{3} x^3 \log \left (\frac {b f^{c+d x}}{a+1}+1\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{1-a}\right )}{d \log (f)}-\frac {\int f^{-c-d x} \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{1-a}\right )df^{c+d x}}{d^2 \log ^2(f)}\right )}{d \log (f)}+\frac {x^2 \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{1-a}\right )}{d \log (f)}-\frac {1}{3} x^3 \log \left (-a-b f^{c+d x}+1\right )+\frac {1}{3} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )\right )+\frac {1}{2} \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{a+1}\right )}{d \log (f)}-\frac {\int f^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{a+1}\right )df^{c+d x}}{d^2 \log ^2(f)}\right )}{d \log (f)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{a+1}\right )}{d \log (f)}+\frac {1}{3} x^3 \log \left (a+b f^{c+d x}+1\right )-\frac {1}{3} x^3 \log \left (\frac {b f^{c+d x}}{a+1}+1\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{1-a}\right )}{d \log (f)}-\frac {\operatorname {PolyLog}\left (4,\frac {b f^{c+d x}}{1-a}\right )}{d^2 \log ^2(f)}\right )}{d \log (f)}+\frac {x^2 \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{1-a}\right )}{d \log (f)}-\frac {1}{3} x^3 \log \left (-a-b f^{c+d x}+1\right )+\frac {1}{3} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )\right )+\frac {1}{2} \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{a+1}\right )}{d \log (f)}-\frac {\operatorname {PolyLog}\left (4,-\frac {b f^{c+d x}}{a+1}\right )}{d^2 \log ^2(f)}\right )}{d \log (f)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{a+1}\right )}{d \log (f)}+\frac {1}{3} x^3 \log \left (a+b f^{c+d x}+1\right )-\frac {1}{3} x^3 \log \left (\frac {b f^{c+d x}}{a+1}+1\right )\right )\) |
(-1/3*(x^3*Log[1 - a - b*f^(c + d*x)]) + (x^3*Log[1 - (b*f^(c + d*x))/(1 - a)])/3 + (x^2*PolyLog[2, (b*f^(c + d*x))/(1 - a)])/(d*Log[f]) - (2*((x*Po lyLog[3, (b*f^(c + d*x))/(1 - a)])/(d*Log[f]) - PolyLog[4, (b*f^(c + d*x)) /(1 - a)]/(d^2*Log[f]^2)))/(d*Log[f]))/2 + ((x^3*Log[1 + a + b*f^(c + d*x) ])/3 - (x^3*Log[1 + (b*f^(c + d*x))/(1 + a)])/3 - (x^2*PolyLog[2, -((b*f^( c + d*x))/(1 + a))])/(d*Log[f]) + (2*((x*PolyLog[3, -((b*f^(c + d*x))/(1 + a))])/(d*Log[f]) - PolyLog[4, -((b*f^(c + d*x))/(1 + a))]/(d^2*Log[f]^2)) )/(d*Log[f]))/2
3.4.54.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[Log[(d_) + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g _.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*(Log[d + e*(F^(c*(a + b*x)))^n]/(g*(m + 1))), x] + (Int[(f + g*x)^m*Log[1 + (e/d)*(F^(c*(a + b*x) ))^n], x] - Simp[(f + g*x)^(m + 1)*(Log[1 + (e/d)*(F^(c*(a + b*x)))^n]/(g*( m + 1))), x]) /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && NeQ[ d, 1]
Int[ArcTanh[(a_.) + (b_.)*(f_)^((c_.) + (d_.)*(x_))]*(x_)^(m_.), x_Symbol] :> Simp[1/2 Int[x^m*Log[1 + a + b*f^(c + d*x)], x], x] - Simp[1/2 Int[x ^m*Log[1 - a - b*f^(c + d*x)], x], x] /; FreeQ[{a, b, c, d, f}, x] && IGtQ[ m, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(671\) vs. \(2(252)=504\).
Time = 1.14 (sec) , antiderivative size = 672, normalized size of antiderivative = 2.55
| method | result | size |
| risch | \(\frac {x^{3} \ln \left (1+a +b \,f^{d x +c}\right )}{6}-\frac {x^{3} \ln \left (1-a -b \,f^{d x +c}\right )}{6}+\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{1-a}\right ) x^{3}}{6}-\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{1-a}\right ) x \,c^{2}}{2 d^{2}}-\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{1-a}\right ) c^{3}}{3 d^{3}}+\frac {\operatorname {polylog}\left (2, \frac {b \,f^{d x} f^{c}}{1-a}\right ) x^{2}}{2 \ln \left (f \right ) d}-\frac {\operatorname {polylog}\left (2, \frac {b \,f^{d x} f^{c}}{1-a}\right ) c^{2}}{2 \ln \left (f \right ) d^{3}}-\frac {\operatorname {polylog}\left (3, \frac {b \,f^{d x} f^{c}}{1-a}\right ) x}{\ln \left (f \right )^{2} d^{2}}+\frac {\operatorname {polylog}\left (4, \frac {b \,f^{d x} f^{c}}{1-a}\right )}{\ln \left (f \right )^{3} d^{3}}-\frac {c^{3} \ln \left (1-a -f^{d x} f^{c} b \right )}{6 d^{3}}+\frac {c^{2} \operatorname {dilog}\left (\frac {f^{d x} f^{c} b +a -1}{-1+a}\right )}{2 \ln \left (f \right ) d^{3}}+\frac {c^{2} \ln \left (\frac {f^{d x} f^{c} b +a -1}{-1+a}\right ) x}{2 d^{2}}+\frac {c^{3} \ln \left (\frac {f^{d x} f^{c} b +a -1}{-1+a}\right )}{2 d^{3}}-\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{-1-a}\right ) x^{3}}{6}+\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{-1-a}\right ) x \,c^{2}}{2 d^{2}}+\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{-1-a}\right ) c^{3}}{3 d^{3}}-\frac {\operatorname {polylog}\left (2, \frac {b \,f^{d x} f^{c}}{-1-a}\right ) x^{2}}{2 \ln \left (f \right ) d}+\frac {\operatorname {polylog}\left (2, \frac {b \,f^{d x} f^{c}}{-1-a}\right ) c^{2}}{2 \ln \left (f \right ) d^{3}}+\frac {\operatorname {polylog}\left (3, \frac {b \,f^{d x} f^{c}}{-1-a}\right ) x}{\ln \left (f \right )^{2} d^{2}}-\frac {\operatorname {polylog}\left (4, \frac {b \,f^{d x} f^{c}}{-1-a}\right )}{\ln \left (f \right )^{3} d^{3}}+\frac {c^{3} \ln \left (1+a +f^{d x} f^{c} b \right )}{6 d^{3}}-\frac {c^{2} \operatorname {dilog}\left (\frac {1+a +f^{d x} f^{c} b}{1+a}\right )}{2 \ln \left (f \right ) d^{3}}-\frac {c^{2} \ln \left (\frac {1+a +f^{d x} f^{c} b}{1+a}\right ) x}{2 d^{2}}-\frac {c^{3} \ln \left (\frac {1+a +f^{d x} f^{c} b}{1+a}\right )}{2 d^{3}}\) | \(672\) |
1/6*x^3*ln(1+a+b*f^(d*x+c))-1/6*x^3*ln(1-a-b*f^(d*x+c))+1/6*ln(1-b*f^(d*x) *f^c/(1-a))*x^3-1/2/d^2*ln(1-b*f^(d*x)*f^c/(1-a))*x*c^2-1/3/d^3*ln(1-b*f^( d*x)*f^c/(1-a))*c^3+1/2/ln(f)/d*polylog(2,b*f^(d*x)*f^c/(1-a))*x^2-1/2/ln( f)/d^3*polylog(2,b*f^(d*x)*f^c/(1-a))*c^2-1/ln(f)^2/d^2*polylog(3,b*f^(d*x )*f^c/(1-a))*x+1/ln(f)^3/d^3*polylog(4,b*f^(d*x)*f^c/(1-a))-1/6/d^3*c^3*ln (1-a-f^(d*x)*f^c*b)+1/2/ln(f)/d^3*c^2*dilog((f^(d*x)*f^c*b+a-1)/(-1+a))+1/ 2/d^2*c^2*ln((f^(d*x)*f^c*b+a-1)/(-1+a))*x+1/2/d^3*c^3*ln((f^(d*x)*f^c*b+a -1)/(-1+a))-1/6*ln(1-b*f^(d*x)*f^c/(-1-a))*x^3+1/2/d^2*ln(1-b*f^(d*x)*f^c/ (-1-a))*x*c^2+1/3/d^3*ln(1-b*f^(d*x)*f^c/(-1-a))*c^3-1/2/ln(f)/d*polylog(2 ,b*f^(d*x)*f^c/(-1-a))*x^2+1/2/ln(f)/d^3*polylog(2,b*f^(d*x)*f^c/(-1-a))*c ^2+1/ln(f)^2/d^2*polylog(3,b*f^(d*x)*f^c/(-1-a))*x-1/ln(f)^3/d^3*polylog(4 ,b*f^(d*x)*f^c/(-1-a))+1/6/d^3*c^3*ln(1+a+f^(d*x)*f^c*b)-1/2/ln(f)/d^3*c^2 *dilog((1+a+f^(d*x)*f^c*b)/(1+a))-1/2/d^2*c^2*ln((1+a+f^(d*x)*f^c*b)/(1+a) )*x-1/2/d^3*c^3*ln((1+a+f^(d*x)*f^c*b)/(1+a))
Time = 0.27 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.82 \[ \int x^2 \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=\frac {d^{3} x^{3} \log \left (f\right )^{3} \log \left (-\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1}{b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1}\right ) - 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1}{a + 1} + 1\right ) \log \left (f\right )^{2} + 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1}{a - 1} + 1\right ) \log \left (f\right )^{2} + c^{3} \log \left (b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1\right ) \log \left (f\right )^{3} - c^{3} \log \left (b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1\right ) \log \left (f\right )^{3} - {\left (d^{3} x^{3} + c^{3}\right )} \log \left (f\right )^{3} \log \left (\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1}{a + 1}\right ) + {\left (d^{3} x^{3} + c^{3}\right )} \log \left (f\right )^{3} \log \left (\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1}{a - 1}\right ) + 6 \, d x \log \left (f\right ) {\rm polylog}\left (3, -\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right )}{a + 1}\right ) - 6 \, d x \log \left (f\right ) {\rm polylog}\left (3, -\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right )}{a - 1}\right ) - 6 \, {\rm polylog}\left (4, -\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right )}{a + 1}\right ) + 6 \, {\rm polylog}\left (4, -\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right )}{a - 1}\right )}{6 \, d^{3} \log \left (f\right )^{3}} \]
1/6*(d^3*x^3*log(f)^3*log(-(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*lo g(f)) + a + 1)/(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a - 1)) - 3*d^2*x^2*dilog(-(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f) ) + a + 1)/(a + 1) + 1)*log(f)^2 + 3*d^2*x^2*dilog(-(b*cosh((d*x + c)*log( f)) + b*sinh((d*x + c)*log(f)) + a - 1)/(a - 1) + 1)*log(f)^2 + c^3*log(b* cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a + 1)*log(f)^3 - c^3* log(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a - 1)*log(f)^3 - (d^3*x^3 + c^3)*log(f)^3*log((b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c )*log(f)) + a + 1)/(a + 1)) + (d^3*x^3 + c^3)*log(f)^3*log((b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a - 1)/(a - 1)) + 6*d*x*log(f)*pol ylog(3, -(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)))/(a + 1)) - 6*d*x*log(f)*polylog(3, -(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log( f)))/(a - 1)) - 6*polylog(4, -(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c) *log(f)))/(a + 1)) + 6*polylog(4, -(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)))/(a - 1)))/(d^3*log(f)^3)
\[ \int x^2 \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=\int x^{2} \operatorname {atanh}{\left (a + b f^{c + d x} \right )}\, dx \]
Time = 0.23 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.96 \[ \int x^2 \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=\frac {1}{3} \, x^{3} \operatorname {artanh}\left (b f^{d x + c} + a\right ) - \frac {1}{6} \, b d {\left (\frac {d^{3} x^{3} \log \left (\frac {b f^{d x} f^{c}}{a + 1} + 1\right ) \log \left (f\right )^{3} + 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b f^{d x} f^{c}}{a + 1}\right ) \log \left (f\right )^{2} - 6 \, d x \log \left (f\right ) {\rm Li}_{3}(-\frac {b f^{d x} f^{c}}{a + 1}) + 6 \, {\rm Li}_{4}(-\frac {b f^{d x} f^{c}}{a + 1})}{b d^{4} \log \left (f\right )^{4}} - \frac {d^{3} x^{3} \log \left (\frac {b f^{d x} f^{c}}{a - 1} + 1\right ) \log \left (f\right )^{3} + 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b f^{d x} f^{c}}{a - 1}\right ) \log \left (f\right )^{2} - 6 \, d x \log \left (f\right ) {\rm Li}_{3}(-\frac {b f^{d x} f^{c}}{a - 1}) + 6 \, {\rm Li}_{4}(-\frac {b f^{d x} f^{c}}{a - 1})}{b d^{4} \log \left (f\right )^{4}}\right )} \log \left (f\right ) \]
1/3*x^3*arctanh(b*f^(d*x + c) + a) - 1/6*b*d*((d^3*x^3*log(b*f^(d*x)*f^c/( a + 1) + 1)*log(f)^3 + 3*d^2*x^2*dilog(-b*f^(d*x)*f^c/(a + 1))*log(f)^2 - 6*d*x*log(f)*polylog(3, -b*f^(d*x)*f^c/(a + 1)) + 6*polylog(4, -b*f^(d*x)* f^c/(a + 1)))/(b*d^4*log(f)^4) - (d^3*x^3*log(b*f^(d*x)*f^c/(a - 1) + 1)*l og(f)^3 + 3*d^2*x^2*dilog(-b*f^(d*x)*f^c/(a - 1))*log(f)^2 - 6*d*x*log(f)* polylog(3, -b*f^(d*x)*f^c/(a - 1)) + 6*polylog(4, -b*f^(d*x)*f^c/(a - 1))) /(b*d^4*log(f)^4))*log(f)
\[ \int x^2 \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=\int { x^{2} \operatorname {artanh}\left (b f^{d x + c} + a\right ) \,d x } \]
Timed out. \[ \int x^2 \text {arctanh}\left (a+b f^{c+d x}\right ) \, dx=\int x^2\,\mathrm {atanh}\left (a+b\,f^{c+d\,x}\right ) \,d x \]