Integrand size = 14, antiderivative size = 162 \[ \int x^4 (a+b \text {arctanh}(c x))^2 \, dx=\frac {3 b^2 x}{10 c^4}+\frac {b^2 x^3}{30 c^2}-\frac {3 b^2 \text {arctanh}(c x)}{10 c^5}+\frac {b x^2 (a+b \text {arctanh}(c x))}{5 c^3}+\frac {b x^4 (a+b \text {arctanh}(c x))}{10 c}+\frac {(a+b \text {arctanh}(c x))^2}{5 c^5}+\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2 b (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{5 c^5}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{5 c^5} \]
3/10*b^2*x/c^4+1/30*b^2*x^3/c^2-3/10*b^2*arctanh(c*x)/c^5+1/5*b*x^2*(a+b*a rctanh(c*x))/c^3+1/10*b*x^4*(a+b*arctanh(c*x))/c+1/5*(a+b*arctanh(c*x))^2/ c^5+1/5*x^5*(a+b*arctanh(c*x))^2-2/5*b*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c ^5-1/5*b^2*polylog(2,1-2/(-c*x+1))/c^5
Time = 0.34 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.99 \[ \int x^4 (a+b \text {arctanh}(c x))^2 \, dx=\frac {-9 a b+9 b^2 c x+6 a b c^2 x^2+b^2 c^3 x^3+3 a b c^4 x^4+6 a^2 c^5 x^5+6 b^2 \left (-1+c^5 x^5\right ) \text {arctanh}(c x)^2+3 b \text {arctanh}(c x) \left (4 a c^5 x^5+b \left (-3+2 c^2 x^2+c^4 x^4\right )-4 b \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+6 a b \log \left (-1+c^2 x^2\right )+6 b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )}{30 c^5} \]
(-9*a*b + 9*b^2*c*x + 6*a*b*c^2*x^2 + b^2*c^3*x^3 + 3*a*b*c^4*x^4 + 6*a^2* c^5*x^5 + 6*b^2*(-1 + c^5*x^5)*ArcTanh[c*x]^2 + 3*b*ArcTanh[c*x]*(4*a*c^5* x^5 + b*(-3 + 2*c^2*x^2 + c^4*x^4) - 4*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 6 *a*b*Log[-1 + c^2*x^2] + 6*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(30*c^5)
Time = 1.31 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.23, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6452, 6542, 6452, 254, 2009, 6542, 6452, 262, 219, 6546, 6470, 2849, 2752}
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.
| \(\displaystyle \int x^4 (a+b \text {arctanh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\int \frac {x^3 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\int x^3 (a+b \text {arctanh}(c x))dx}{c^2}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\int \frac {x^3 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \int \frac {x^4}{1-c^2 x^2}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\int \frac {x^3 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \int \left (-\frac {x^2}{c^2}+\frac {1}{c^4 \left (1-c^2 x^2\right )}-\frac {1}{c^4}\right )dx}{c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\int \frac {x^3 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\int x (a+b \text {arctanh}(c x))dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \int \frac {x^2}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{1-c^2 x^2}dx}{c^2}-\frac {x}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\frac {\int \frac {a+b \text {arctanh}(c x)}{1-c x}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}-b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-\frac {2}{1-c x}}d\frac {1}{1-c x}}{c}+\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )\) |
(x^5*(a + b*ArcTanh[c*x])^2)/5 - (2*b*c*(-(((x^4*(a + b*ArcTanh[c*x]))/4 - (b*c*(-(x/c^4) - x^3/(3*c^2) + ArcTanh[c*x]/c^5))/4)/c^2) + (-(((x^2*(a + b*ArcTanh[c*x]))/2 - (b*c*(-(x/c^2) + ArcTanh[c*x]/c^3))/2)/c^2) + (-1/2* (a + b*ArcTanh[c*x])^2/(b*c^2) + (((a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/ c + (b*PolyLog[2, 1 - 2/(1 - c*x)])/(2*c))/c)/c^2)/c^2))/5
3.1.14.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -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_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* x])^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcTanh[c*x])^p/ (d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 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]
Time = 1.09 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.41
| method | result | size |
| parts | \(\frac {a^{2} x^{5}}{5}+\frac {b^{2} \left (\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )^{2}}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{10}+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{5}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{5}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{5}+\frac {c^{3} x^{3}}{30}+\frac {3 c x}{10}+\frac {3 \ln \left (c x -1\right )}{20}-\frac {3 \ln \left (c x +1\right )}{20}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{10}+\frac {\ln \left (c x -1\right )^{2}}{20}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{10}-\frac {\ln \left (c x +1\right )^{2}}{20}\right )}{c^{5}}+\frac {2 a b \left (\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {c^{4} x^{4}}{20}+\frac {c^{2} x^{2}}{10}+\frac {\ln \left (c x -1\right )}{10}+\frac {\ln \left (c x +1\right )}{10}\right )}{c^{5}}\) | \(229\) |
| derivativedivides | \(\frac {\frac {c^{5} x^{5} a^{2}}{5}+b^{2} \left (\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )^{2}}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{10}+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{5}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{5}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{5}+\frac {c^{3} x^{3}}{30}+\frac {3 c x}{10}+\frac {3 \ln \left (c x -1\right )}{20}-\frac {3 \ln \left (c x +1\right )}{20}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{10}+\frac {\ln \left (c x -1\right )^{2}}{20}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{10}-\frac {\ln \left (c x +1\right )^{2}}{20}\right )+2 a b \left (\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {c^{4} x^{4}}{20}+\frac {c^{2} x^{2}}{10}+\frac {\ln \left (c x -1\right )}{10}+\frac {\ln \left (c x +1\right )}{10}\right )}{c^{5}}\) | \(230\) |
| default | \(\frac {\frac {c^{5} x^{5} a^{2}}{5}+b^{2} \left (\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )^{2}}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{10}+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{5}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{5}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{5}+\frac {c^{3} x^{3}}{30}+\frac {3 c x}{10}+\frac {3 \ln \left (c x -1\right )}{20}-\frac {3 \ln \left (c x +1\right )}{20}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{10}+\frac {\ln \left (c x -1\right )^{2}}{20}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{10}-\frac {\ln \left (c x +1\right )^{2}}{20}\right )+2 a b \left (\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {c^{4} x^{4}}{20}+\frac {c^{2} x^{2}}{10}+\frac {\ln \left (c x -1\right )}{10}+\frac {\ln \left (c x +1\right )}{10}\right )}{c^{5}}\) | \(230\) |
| risch | \(\frac {b^{2} \ln \left (c x +1\right ) x^{2}}{10 c^{3}}-\frac {a^{2}}{5 c^{5}}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{5 c^{5}}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{5 c^{5}}+\frac {b a \ln \left (c x +1\right )}{5 c^{5}}+\frac {b a \ln \left (c x +1\right ) x^{5}}{5}-\frac {b^{2} \ln \left (-c x +1\right ) \ln \left (c x +1\right ) x^{5}}{10}-\frac {b^{2} \ln \left (-c x +1\right ) \ln \left (c x +1\right )}{10 c^{5}}+\frac {b^{2} \ln \left (c x +1\right ) x^{4}}{20 c}-\frac {a b \ln \left (-c x +1\right ) x^{5}}{5}-\frac {b^{2} \ln \left (-c x +1\right )^{2}}{20 c^{5}}+\frac {137 b^{2} \ln \left (-c x +1\right )}{300 c^{5}}+\frac {b^{2} \ln \left (-c x +1\right )^{2} x^{5}}{20}-\frac {23 b^{2} \ln \left (c x -1\right )}{75 c^{5}}-\frac {b^{2} \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{5 c^{5}}+\frac {a b \ln \left (-c x +1\right )}{5 c^{5}}+\frac {a b \,x^{4}}{10 c}-\frac {b^{2} \ln \left (-c x +1\right ) x^{4}}{20 c}-\frac {b^{2} \ln \left (-c x +1\right ) x^{2}}{10 c^{3}}+\frac {a b \,x^{2}}{5 c^{3}}+\frac {3 b^{2} x}{10 c^{4}}+\frac {b^{2} x^{3}}{30 c^{2}}-\frac {3 b^{2} \ln \left (c x +1\right )}{20 c^{5}}+\frac {b^{2} \ln \left (c x +1\right )^{2}}{20 c^{5}}+\frac {b^{2} \ln \left (c x +1\right )^{2} x^{5}}{20}-\frac {137 a b}{150 c^{5}}-\frac {413 b^{2}}{2250 c^{5}}+\frac {a^{2} x^{5}}{5}\) | \(406\) |
1/5*a^2*x^5+b^2/c^5*(1/5*c^5*x^5*arctanh(c*x)^2+1/10*c^4*x^4*arctanh(c*x)+ 1/5*c^2*x^2*arctanh(c*x)+1/5*arctanh(c*x)*ln(c*x-1)+1/5*arctanh(c*x)*ln(c* x+1)+1/30*c^3*x^3+3/10*c*x+3/20*ln(c*x-1)-3/20*ln(c*x+1)-1/5*dilog(1/2*c*x +1/2)-1/10*ln(c*x-1)*ln(1/2*c*x+1/2)+1/20*ln(c*x-1)^2+1/10*(ln(c*x+1)-ln(1 /2*c*x+1/2))*ln(-1/2*c*x+1/2)-1/20*ln(c*x+1)^2)+2*a*b/c^5*(1/5*c^5*x^5*arc tanh(c*x)+1/20*c^4*x^4+1/10*c^2*x^2+1/10*ln(c*x-1)+1/10*ln(c*x+1))
\[ \int x^4 (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{4} \,d x } \]
\[ \int x^4 (a+b \text {arctanh}(c x))^2 \, dx=\int x^{4} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}\, dx \]
\[ \int x^4 (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{4} \,d x } \]
1/5*a^2*x^5 + 1/10*(4*x^5*arctanh(c*x) + c*((c^2*x^4 + 2*x^2)/c^4 + 2*log( c^2*x^2 - 1)/c^6))*a*b - 1/36000*(24*c^6*(2*(3*c^4*x^5 + 5*c^2*x^3 + 15*x) /c^10 - 15*log(c*x + 1)/c^11 + 15*log(c*x - 1)/c^11) - 45*c^5*((c^2*x^4 + 2*x^2)/c^8 + 2*log(c^2*x^2 - 1)/c^10) - 1080000*c^5*integrate(1/150*x^5*lo g(c*x + 1)/(c^6*x^2 - c^4), x) + 50*c^4*(2*(c^2*x^3 + 3*x)/c^8 - 3*log(c*x + 1)/c^9 + 3*log(c*x - 1)/c^9) - 300*c^3*(x^2/c^6 + log(c^2*x^2 - 1)/c^8) + 900*c^2*(2*x/c^6 - log(c*x + 1)/c^7 + log(c*x - 1)/c^7) - 540000*c*inte grate(1/150*x*log(c*x + 1)/(c^6*x^2 - c^4), x) - 60*(30*c^5*x^5*log(c*x + 1)^2 + (12*c^5*x^5 - 15*c^4*x^4 + 20*c^3*x^3 - 30*c^2*x^2 + 60*c*x - 60*(c ^5*x^5 + 1)*log(c*x + 1))*log(-c*x + 1))/c^5 - (72*(c*x - 1)^5*(25*log(-c* x + 1)^2 - 10*log(-c*x + 1) + 2) + 1125*(c*x - 1)^4*(8*log(-c*x + 1)^2 - 4 *log(-c*x + 1) + 1) + 2000*(c*x - 1)^3*(9*log(-c*x + 1)^2 - 6*log(-c*x + 1 ) + 2) + 9000*(c*x - 1)^2*(2*log(-c*x + 1)^2 - 2*log(-c*x + 1) + 1) + 9000 *(c*x - 1)*(log(-c*x + 1)^2 - 2*log(-c*x + 1) + 2))/c^5 + 1800*log(150*c^6 *x^2 - 150*c^4)/c^5 - 540000*integrate(1/150*log(c*x + 1)/(c^6*x^2 - c^4), x))*b^2
\[ \int x^4 (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{4} \,d x } \]
Timed out. \[ \int x^4 (a+b \text {arctanh}(c x))^2 \, dx=\int x^4\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2 \,d x \]