Integrand size = 14, antiderivative size = 145 \[ \int x^5 (a+b \text {arctanh}(c x))^2 \, dx=\frac {a b x}{3 c^5}+\frac {4 b^2 x^2}{45 c^4}+\frac {b^2 x^4}{60 c^2}+\frac {b^2 x \text {arctanh}(c x)}{3 c^5}+\frac {b x^3 (a+b \text {arctanh}(c x))}{9 c^3}+\frac {b x^5 (a+b \text {arctanh}(c x))}{15 c}-\frac {(a+b \text {arctanh}(c x))^2}{6 c^6}+\frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2+\frac {23 b^2 \log \left (1-c^2 x^2\right )}{90 c^6} \] Output:
1/3*a*b*x/c^5+4/45*b^2*x^2/c^4+1/60*b^2*x^4/c^2+1/3*b^2*x*arctanh(c*x)/c^5 +1/9*b*x^3*(a+b*arctanh(c*x))/c^3+1/15*b*x^5*(a+b*arctanh(c*x))/c-1/6*(a+b *arctanh(c*x))^2/c^6+1/6*x^6*(a+b*arctanh(c*x))^2+23/90*b^2*ln(-c^2*x^2+1) /c^6
Time = 0.04 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.13 \[ \int x^5 (a+b \text {arctanh}(c x))^2 \, dx=\frac {60 a b c x+16 b^2 c^2 x^2+20 a b c^3 x^3+3 b^2 c^4 x^4+12 a b c^5 x^5+30 a^2 c^6 x^6+4 b c x \left (15 a c^5 x^5+b \left (15+5 c^2 x^2+3 c^4 x^4\right )\right ) \text {arctanh}(c x)+30 b^2 \left (-1+c^6 x^6\right ) \text {arctanh}(c x)^2+2 b (15 a+23 b) \log (1-c x)-30 a b \log (1+c x)+46 b^2 \log (1+c x)}{180 c^6} \] Input:
Integrate[x^5*(a + b*ArcTanh[c*x])^2,x]
Output:
(60*a*b*c*x + 16*b^2*c^2*x^2 + 20*a*b*c^3*x^3 + 3*b^2*c^4*x^4 + 12*a*b*c^5 *x^5 + 30*a^2*c^6*x^6 + 4*b*c*x*(15*a*c^5*x^5 + b*(15 + 5*c^2*x^2 + 3*c^4* x^4))*ArcTanh[c*x] + 30*b^2*(-1 + c^6*x^6)*ArcTanh[c*x]^2 + 2*b*(15*a + 23 *b)*Log[1 - c*x] - 30*a*b*Log[1 + c*x] + 46*b^2*Log[1 + c*x])/(180*c^6)
Time = 1.32 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.41, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6452, 6542, 6452, 243, 49, 2009, 6542, 6452, 243, 49, 2009, 6542, 2009, 6510}
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.
| \(\displaystyle \int x^5 (a+b \text {arctanh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \int \frac {x^6 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \left (\frac {\int \frac {x^4 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\int x^4 (a+b \text {arctanh}(c x))dx}{c^2}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \left (\frac {\int \frac {x^4 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))-\frac {1}{5} b c \int \frac {x^5}{1-c^2 x^2}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \left (\frac {\int \frac {x^4 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))-\frac {1}{10} b c \int \frac {x^4}{1-c^2 x^2}dx^2}{c^2}\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \left (\frac {\int \frac {x^4 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))-\frac {1}{10} b c \int \left (-\frac {x^2}{c^2}-\frac {1}{c^4 \left (c^2 x^2-1\right )}-\frac {1}{c^4}\right )dx^2}{c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \left (\frac {\int \frac {x^4 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))-\frac {1}{10} b c \left (-\frac {x^2}{c^4}-\frac {x^4}{2 c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^6}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \left (\frac {\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\int x^2 (a+b \text {arctanh}(c x))dx}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))-\frac {1}{10} b c \left (-\frac {x^2}{c^4}-\frac {x^4}{2 c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^6}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \left (\frac {\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{3} b c \int \frac {x^3}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))-\frac {1}{10} b c \left (-\frac {x^2}{c^4}-\frac {x^4}{2 c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^6}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \left (\frac {\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \int \frac {x^2}{1-c^2 x^2}dx^2}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))-\frac {1}{10} b c \left (-\frac {x^2}{c^4}-\frac {x^4}{2 c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^6}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \left (\frac {\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (c^2 x^2-1\right )}\right )dx^2}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))-\frac {1}{10} b c \left (-\frac {x^2}{c^4}-\frac {x^4}{2 c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^6}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \left (\frac {\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))-\frac {1}{10} b c \left (-\frac {x^2}{c^4}-\frac {x^4}{2 c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^6}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \left (\frac {\frac {\frac {\int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx}{c^2}-\frac {\int (a+b \text {arctanh}(c x))dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))-\frac {1}{10} b c \left (-\frac {x^2}{c^4}-\frac {x^4}{2 c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^6}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \left (\frac {\frac {\frac {\int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx}{c^2}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))-\frac {1}{10} b c \left (-\frac {x^2}{c^4}-\frac {x^4}{2 c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^6}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^2-\frac {1}{3} b c \left (\frac {\frac {\frac {(a+b \text {arctanh}(c x))^2}{2 b c^3}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))-\frac {1}{10} b c \left (-\frac {x^2}{c^4}-\frac {x^4}{2 c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^6}\right )}{c^2}\right )\) |
Input:
Int[x^5*(a + b*ArcTanh[c*x])^2,x]
Output:
(x^6*(a + b*ArcTanh[c*x])^2)/6 - (b*c*(-(((x^5*(a + b*ArcTanh[c*x]))/5 - ( b*c*(-(x^2/c^4) - x^4/(2*c^2) - Log[1 - c^2*x^2]/c^6))/10)/c^2) + (-(((x^3 *(a + b*ArcTanh[c*x]))/3 - (b*c*(-(x^2/c^2) - Log[1 - c^2*x^2]/c^4))/6)/c^ 2) + ((a + b*ArcTanh[c*x])^2/(2*b*c^3) - (a*x + b*x*ArcTanh[c*x] + (b*Log[ 1 - c^2*x^2])/(2*c))/c^2)/c^2)/c^2))/3
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_)^2), x_Symb ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
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]
Time = 0.59 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.26
| method | result | size |
| parallelrisch | \(\frac {30 b^{2} \operatorname {arctanh}\left (c x \right )^{2} x^{6} c^{6}+60 a b \,\operatorname {arctanh}\left (c x \right ) x^{6} c^{6}+30 a^{2} c^{6} x^{6}+12 b^{2} \operatorname {arctanh}\left (c x \right ) x^{5} c^{5}+12 a b \,c^{5} x^{5}+3 b^{2} c^{4} x^{4}+20 b^{2} \operatorname {arctanh}\left (c x \right ) x^{3} c^{3}+20 a b \,c^{3} x^{3}+16 b^{2} c^{2} x^{2}+60 b^{2} \operatorname {arctanh}\left (c x \right ) x c +60 a b c x -30 b^{2} \operatorname {arctanh}\left (c x \right )^{2}+92 \ln \left (c x -1\right ) b^{2}-60 \,\operatorname {arctanh}\left (c x \right ) a b +92 \,\operatorname {arctanh}\left (c x \right ) b^{2}+16 b^{2}}{180 c^{6}}\) | \(182\) |
| parts | \(\frac {a^{2} x^{6}}{6}+\frac {b^{2} \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )^{2}}{6}+\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{15}+\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{9}+\frac {c x \,\operatorname {arctanh}\left (c x \right )}{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{6}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (c x -1\right )^{2}}{24}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (c x +1\right )^{2}}{24}+\frac {c^{4} x^{4}}{60}+\frac {4 c^{2} x^{2}}{45}+\frac {23 \ln \left (c x -1\right )}{90}+\frac {23 \ln \left (c x +1\right )}{90}\right )}{c^{6}}+\frac {2 a b \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{6}+\frac {c^{5} x^{5}}{30}+\frac {c^{3} x^{3}}{18}+\frac {c x}{6}+\frac {\ln \left (c x -1\right )}{12}-\frac {\ln \left (c x +1\right )}{12}\right )}{c^{6}}\) | \(236\) |
| derivativedivides | \(\frac {\frac {a^{2} c^{6} x^{6}}{6}+b^{2} \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )^{2}}{6}+\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{15}+\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{9}+\frac {c x \,\operatorname {arctanh}\left (c x \right )}{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{6}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (c x -1\right )^{2}}{24}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (c x +1\right )^{2}}{24}+\frac {c^{4} x^{4}}{60}+\frac {4 c^{2} x^{2}}{45}+\frac {23 \ln \left (c x -1\right )}{90}+\frac {23 \ln \left (c x +1\right )}{90}\right )+2 a b \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{6}+\frac {c^{5} x^{5}}{30}+\frac {c^{3} x^{3}}{18}+\frac {c x}{6}+\frac {\ln \left (c x -1\right )}{12}-\frac {\ln \left (c x +1\right )}{12}\right )}{c^{6}}\) | \(237\) |
| default | \(\frac {\frac {a^{2} c^{6} x^{6}}{6}+b^{2} \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )^{2}}{6}+\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{15}+\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{9}+\frac {c x \,\operatorname {arctanh}\left (c x \right )}{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{6}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (c x -1\right )^{2}}{24}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (c x +1\right )^{2}}{24}+\frac {c^{4} x^{4}}{60}+\frac {4 c^{2} x^{2}}{45}+\frac {23 \ln \left (c x -1\right )}{90}+\frac {23 \ln \left (c x +1\right )}{90}\right )+2 a b \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{6}+\frac {c^{5} x^{5}}{30}+\frac {c^{3} x^{3}}{18}+\frac {c x}{6}+\frac {\ln \left (c x -1\right )}{12}-\frac {\ln \left (c x +1\right )}{12}\right )}{c^{6}}\) | \(237\) |
| risch | \(\frac {b^{2} \left (c^{6} x^{6}-1\right ) \ln \left (c x +1\right )^{2}}{24 c^{6}}+\frac {b \left (-15 b \,x^{6} \ln \left (-c x +1\right ) c^{6}+30 a \,c^{6} x^{6}+6 b \,c^{5} x^{5}+10 b \,c^{3} x^{3}+30 b c x +15 b \ln \left (-c x +1\right )\right ) \ln \left (c x +1\right )}{180 c^{6}}+\frac {b^{2} x^{6} \ln \left (-c x +1\right )^{2}}{24}-\frac {a b \,x^{6} \ln \left (-c x +1\right )}{6}+\frac {a^{2} x^{6}}{6}-\frac {b^{2} x^{5} \ln \left (-c x +1\right )}{30 c}+\frac {a b \,x^{5}}{15 c}+\frac {b^{2} x^{4}}{60 c^{2}}-\frac {b^{2} x^{3} \ln \left (-c x +1\right )}{18 c^{3}}+\frac {a b \,x^{3}}{9 c^{3}}+\frac {4 b^{2} x^{2}}{45 c^{4}}-\frac {b^{2} x \ln \left (-c x +1\right )}{6 c^{5}}-\frac {b^{2} \ln \left (-c x +1\right )^{2}}{24 c^{6}}+\frac {a b x}{3 c^{5}}-\frac {b \ln \left (c x +1\right ) a}{6 c^{6}}+\frac {23 b^{2} \ln \left (c x +1\right )}{90 c^{6}}+\frac {b \ln \left (-c x +1\right ) a}{6 c^{6}}+\frac {23 b^{2} \ln \left (-c x +1\right )}{90 c^{6}}\) | \(312\) |
Input:
int(x^5*(a+b*arctanh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/180*(30*b^2*arctanh(c*x)^2*x^6*c^6+60*a*b*arctanh(c*x)*x^6*c^6+30*a^2*c^ 6*x^6+12*b^2*arctanh(c*x)*x^5*c^5+12*a*b*c^5*x^5+3*b^2*c^4*x^4+20*b^2*arct anh(c*x)*x^3*c^3+20*a*b*c^3*x^3+16*b^2*c^2*x^2+60*b^2*arctanh(c*x)*x*c+60* a*b*c*x-30*b^2*arctanh(c*x)^2+92*ln(c*x-1)*b^2-60*arctanh(c*x)*a*b+92*arct anh(c*x)*b^2+16*b^2)/c^6
Time = 0.09 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.33 \[ \int x^5 (a+b \text {arctanh}(c x))^2 \, dx=\frac {60 \, a^{2} c^{6} x^{6} + 24 \, a b c^{5} x^{5} + 6 \, b^{2} c^{4} x^{4} + 40 \, a b c^{3} x^{3} + 32 \, b^{2} c^{2} x^{2} + 120 \, a b c x + 15 \, {\left (b^{2} c^{6} x^{6} - b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} - 4 \, {\left (15 \, a b - 23 \, b^{2}\right )} \log \left (c x + 1\right ) + 4 \, {\left (15 \, a b + 23 \, b^{2}\right )} \log \left (c x - 1\right ) + 4 \, {\left (15 \, a b c^{6} x^{6} + 3 \, b^{2} c^{5} x^{5} + 5 \, b^{2} c^{3} x^{3} + 15 \, b^{2} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{360 \, c^{6}} \] Input:
integrate(x^5*(a+b*arctanh(c*x))^2,x, algorithm="fricas")
Output:
1/360*(60*a^2*c^6*x^6 + 24*a*b*c^5*x^5 + 6*b^2*c^4*x^4 + 40*a*b*c^3*x^3 + 32*b^2*c^2*x^2 + 120*a*b*c*x + 15*(b^2*c^6*x^6 - b^2)*log(-(c*x + 1)/(c*x - 1))^2 - 4*(15*a*b - 23*b^2)*log(c*x + 1) + 4*(15*a*b + 23*b^2)*log(c*x - 1) + 4*(15*a*b*c^6*x^6 + 3*b^2*c^5*x^5 + 5*b^2*c^3*x^3 + 15*b^2*c*x)*log( -(c*x + 1)/(c*x - 1)))/c^6
Time = 0.52 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.46 \[ \int x^5 (a+b \text {arctanh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} x^{6}}{6} + \frac {a b x^{6} \operatorname {atanh}{\left (c x \right )}}{3} + \frac {a b x^{5}}{15 c} + \frac {a b x^{3}}{9 c^{3}} + \frac {a b x}{3 c^{5}} - \frac {a b \operatorname {atanh}{\left (c x \right )}}{3 c^{6}} + \frac {b^{2} x^{6} \operatorname {atanh}^{2}{\left (c x \right )}}{6} + \frac {b^{2} x^{5} \operatorname {atanh}{\left (c x \right )}}{15 c} + \frac {b^{2} x^{4}}{60 c^{2}} + \frac {b^{2} x^{3} \operatorname {atanh}{\left (c x \right )}}{9 c^{3}} + \frac {4 b^{2} x^{2}}{45 c^{4}} + \frac {b^{2} x \operatorname {atanh}{\left (c x \right )}}{3 c^{5}} + \frac {23 b^{2} \log {\left (x - \frac {1}{c} \right )}}{45 c^{6}} - \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{6 c^{6}} + \frac {23 b^{2} \operatorname {atanh}{\left (c x \right )}}{45 c^{6}} & \text {for}\: c \neq 0 \\\frac {a^{2} x^{6}}{6} & \text {otherwise} \end {cases} \] Input:
integrate(x**5*(a+b*atanh(c*x))**2,x)
Output:
Piecewise((a**2*x**6/6 + a*b*x**6*atanh(c*x)/3 + a*b*x**5/(15*c) + a*b*x** 3/(9*c**3) + a*b*x/(3*c**5) - a*b*atanh(c*x)/(3*c**6) + b**2*x**6*atanh(c* x)**2/6 + b**2*x**5*atanh(c*x)/(15*c) + b**2*x**4/(60*c**2) + b**2*x**3*at anh(c*x)/(9*c**3) + 4*b**2*x**2/(45*c**4) + b**2*x*atanh(c*x)/(3*c**5) + 2 3*b**2*log(x - 1/c)/(45*c**6) - b**2*atanh(c*x)**2/(6*c**6) + 23*b**2*atan h(c*x)/(45*c**6), Ne(c, 0)), (a**2*x**6/6, True))
Time = 0.03 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.48 \[ \int x^5 (a+b \text {arctanh}(c x))^2 \, dx=\frac {1}{6} \, b^{2} x^{6} \operatorname {artanh}\left (c x\right )^{2} + \frac {1}{6} \, a^{2} x^{6} + \frac {1}{90} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} a b + \frac {1}{360} \, {\left (4 \, c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )} \operatorname {artanh}\left (c x\right ) + \frac {6 \, c^{4} x^{4} + 32 \, c^{2} x^{2} - 2 \, {\left (15 \, \log \left (c x - 1\right ) - 46\right )} \log \left (c x + 1\right ) + 15 \, \log \left (c x + 1\right )^{2} + 15 \, \log \left (c x - 1\right )^{2} + 92 \, \log \left (c x - 1\right )}{c^{6}}\right )} b^{2} \] Input:
integrate(x^5*(a+b*arctanh(c*x))^2,x, algorithm="maxima")
Output:
1/6*b^2*x^6*arctanh(c*x)^2 + 1/6*a^2*x^6 + 1/90*(30*x^6*arctanh(c*x) + c*( 2*(3*c^4*x^5 + 5*c^2*x^3 + 15*x)/c^6 - 15*log(c*x + 1)/c^7 + 15*log(c*x - 1)/c^7))*a*b + 1/360*(4*c*(2*(3*c^4*x^5 + 5*c^2*x^3 + 15*x)/c^6 - 15*log(c *x + 1)/c^7 + 15*log(c*x - 1)/c^7)*arctanh(c*x) + (6*c^4*x^4 + 32*c^2*x^2 - 2*(15*log(c*x - 1) - 46)*log(c*x + 1) + 15*log(c*x + 1)^2 + 15*log(c*x - 1)^2 + 92*log(c*x - 1))/c^6)*b^2
Leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (127) = 254\).
Time = 0.14 (sec) , antiderivative size = 889, normalized size of antiderivative = 6.13 \[ \int x^5 (a+b \text {arctanh}(c x))^2 \, dx=\text {Too large to display} \] Input:
integrate(x^5*(a+b*arctanh(c*x))^2,x, algorithm="giac")
Output:
1/90*(15*(3*(c*x + 1)^5*b^2/(c*x - 1)^5 + 10*(c*x + 1)^3*b^2/(c*x - 1)^3 + 3*(c*x + 1)*b^2/(c*x - 1))*log(-(c*x + 1)/(c*x - 1))^2/((c*x + 1)^6*c^7/( c*x - 1)^6 - 6*(c*x + 1)^5*c^7/(c*x - 1)^5 + 15*(c*x + 1)^4*c^7/(c*x - 1)^ 4 - 20*(c*x + 1)^3*c^7/(c*x - 1)^3 + 15*(c*x + 1)^2*c^7/(c*x - 1)^2 - 6*(c *x + 1)*c^7/(c*x - 1) + c^7) + 2*(90*(c*x + 1)^5*a*b/(c*x - 1)^5 + 300*(c* x + 1)^3*a*b/(c*x - 1)^3 + 90*(c*x + 1)*a*b/(c*x - 1) + 45*(c*x + 1)^5*b^2 /(c*x - 1)^5 - 135*(c*x + 1)^4*b^2/(c*x - 1)^4 + 230*(c*x + 1)^3*b^2/(c*x - 1)^3 - 210*(c*x + 1)^2*b^2/(c*x - 1)^2 + 93*(c*x + 1)*b^2/(c*x - 1) - 23 *b^2)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^6*c^7/(c*x - 1)^6 - 6*(c*x + 1) ^5*c^7/(c*x - 1)^5 + 15*(c*x + 1)^4*c^7/(c*x - 1)^4 - 20*(c*x + 1)^3*c^7/( c*x - 1)^3 + 15*(c*x + 1)^2*c^7/(c*x - 1)^2 - 6*(c*x + 1)*c^7/(c*x - 1) + c^7) + 4*(45*(c*x + 1)^5*a^2/(c*x - 1)^5 + 150*(c*x + 1)^3*a^2/(c*x - 1)^3 + 45*(c*x + 1)*a^2/(c*x - 1) + 45*(c*x + 1)^5*a*b/(c*x - 1)^5 - 135*(c*x + 1)^4*a*b/(c*x - 1)^4 + 230*(c*x + 1)^3*a*b/(c*x - 1)^3 - 210*(c*x + 1)^2 *a*b/(c*x - 1)^2 + 93*(c*x + 1)*a*b/(c*x - 1) - 23*a*b + 11*(c*x + 1)^5*b^ 2/(c*x - 1)^5 - 38*(c*x + 1)^4*b^2/(c*x - 1)^4 + 54*(c*x + 1)^3*b^2/(c*x - 1)^3 - 38*(c*x + 1)^2*b^2/(c*x - 1)^2 + 11*(c*x + 1)*b^2/(c*x - 1))/((c*x + 1)^6*c^7/(c*x - 1)^6 - 6*(c*x + 1)^5*c^7/(c*x - 1)^5 + 15*(c*x + 1)^4*c ^7/(c*x - 1)^4 - 20*(c*x + 1)^3*c^7/(c*x - 1)^3 + 15*(c*x + 1)^2*c^7/(c*x - 1)^2 - 6*(c*x + 1)*c^7/(c*x - 1) + c^7) - 46*b^2*log(-(c*x + 1)/(c*x ...
Time = 3.97 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.18 \[ \int x^5 (a+b \text {arctanh}(c x))^2 \, dx=\frac {46\,b^2\,\ln \left (c^2\,x^2-1\right )-30\,b^2\,{\mathrm {atanh}\left (c\,x\right )}^2+30\,a^2\,c^6\,x^6+16\,b^2\,c^2\,x^2+3\,b^2\,c^4\,x^4-60\,a\,b\,\mathrm {atanh}\left (c\,x\right )+20\,b^2\,c^3\,x^3\,\mathrm {atanh}\left (c\,x\right )+12\,b^2\,c^5\,x^5\,\mathrm {atanh}\left (c\,x\right )+60\,b^2\,c\,x\,\mathrm {atanh}\left (c\,x\right )+30\,b^2\,c^6\,x^6\,{\mathrm {atanh}\left (c\,x\right )}^2+20\,a\,b\,c^3\,x^3+12\,a\,b\,c^5\,x^5+60\,a\,b\,c\,x+60\,a\,b\,c^6\,x^6\,\mathrm {atanh}\left (c\,x\right )}{180\,c^6} \] Input:
int(x^5*(a + b*atanh(c*x))^2,x)
Output:
(46*b^2*log(c^2*x^2 - 1) - 30*b^2*atanh(c*x)^2 + 30*a^2*c^6*x^6 + 16*b^2*c ^2*x^2 + 3*b^2*c^4*x^4 - 60*a*b*atanh(c*x) + 20*b^2*c^3*x^3*atanh(c*x) + 1 2*b^2*c^5*x^5*atanh(c*x) + 60*b^2*c*x*atanh(c*x) + 30*b^2*c^6*x^6*atanh(c* x)^2 + 20*a*b*c^3*x^3 + 12*a*b*c^5*x^5 + 60*a*b*c*x + 60*a*b*c^6*x^6*atanh (c*x))/(180*c^6)
Time = 0.18 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.24 \[ \int x^5 (a+b \text {arctanh}(c x))^2 \, dx=\frac {30 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{6} x^{6}-30 \mathit {atanh} \left (c x \right )^{2} b^{2}+60 \mathit {atanh} \left (c x \right ) a b \,c^{6} x^{6}-60 \mathit {atanh} \left (c x \right ) a b +12 \mathit {atanh} \left (c x \right ) b^{2} c^{5} x^{5}+20 \mathit {atanh} \left (c x \right ) b^{2} c^{3} x^{3}+60 \mathit {atanh} \left (c x \right ) b^{2} c x +92 \mathit {atanh} \left (c x \right ) b^{2}+92 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2}+30 a^{2} c^{6} x^{6}+12 a b \,c^{5} x^{5}+20 a b \,c^{3} x^{3}+60 a b c x +3 b^{2} c^{4} x^{4}+16 b^{2} c^{2} x^{2}}{180 c^{6}} \] Input:
int(x^5*(a+b*atanh(c*x))^2,x)
Output:
(30*atanh(c*x)**2*b**2*c**6*x**6 - 30*atanh(c*x)**2*b**2 + 60*atanh(c*x)*a *b*c**6*x**6 - 60*atanh(c*x)*a*b + 12*atanh(c*x)*b**2*c**5*x**5 + 20*atanh (c*x)*b**2*c**3*x**3 + 60*atanh(c*x)*b**2*c*x + 92*atanh(c*x)*b**2 + 92*lo g(c**2*x - c)*b**2 + 30*a**2*c**6*x**6 + 12*a*b*c**5*x**5 + 20*a*b*c**3*x* *3 + 60*a*b*c*x + 3*b**2*c**4*x**4 + 16*b**2*c**2*x**2)/(180*c**6)