Integrand size = 16, antiderivative size = 231 \[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\frac {a b^2 x^3}{3 c^2}+\frac {b^3 x^3 \text {arctanh}\left (c x^3\right )}{3 c^2}-\frac {b \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{6 c^3}+\frac {b x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{6 c}+\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{9 c^3}+\frac {1}{9} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-\frac {b \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \log \left (\frac {2}{1-c x^3}\right )}{3 c^3}+\frac {b^3 \log \left (1-c^2 x^6\right )}{6 c^3}-\frac {b^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right )}{3 c^3}+\frac {b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^3}\right )}{6 c^3} \]
1/3*a*b^2*x^3/c^2+1/3*b^3*x^3*arctanh(c*x^3)/c^2-1/6*b*(a+b*arctanh(c*x^3) )^2/c^3+1/6*b*x^6*(a+b*arctanh(c*x^3))^2/c+1/9*(a+b*arctanh(c*x^3))^3/c^3+ 1/9*x^9*(a+b*arctanh(c*x^3))^3-1/3*b*(a+b*arctanh(c*x^3))^2*ln(2/(-c*x^3+1 ))/c^3+1/6*b^3*ln(-c^2*x^6+1)/c^3-1/3*b^2*(a+b*arctanh(c*x^3))*polylog(2,1 -2/(-c*x^3+1))/c^3+1/6*b^3*polylog(3,1-2/(-c*x^3+1))/c^3
Time = 0.36 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.45 \[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\frac {6 a b^2 c x^3+3 a^2 b c^2 x^6+2 a^3 c^3 x^9-6 a b^2 \text {arctanh}\left (c x^3\right )+6 b^3 c x^3 \text {arctanh}\left (c x^3\right )+6 a b^2 c^2 x^6 \text {arctanh}\left (c x^3\right )+6 a^2 b c^3 x^9 \text {arctanh}\left (c x^3\right )-6 a b^2 \text {arctanh}\left (c x^3\right )^2-3 b^3 \text {arctanh}\left (c x^3\right )^2+3 b^3 c^2 x^6 \text {arctanh}\left (c x^3\right )^2+6 a b^2 c^3 x^9 \text {arctanh}\left (c x^3\right )^2-2 b^3 \text {arctanh}\left (c x^3\right )^3+2 b^3 c^3 x^9 \text {arctanh}\left (c x^3\right )^3-12 a b^2 \text {arctanh}\left (c x^3\right ) \log \left (1+e^{-2 \text {arctanh}\left (c x^3\right )}\right )-6 b^3 \text {arctanh}\left (c x^3\right )^2 \log \left (1+e^{-2 \text {arctanh}\left (c x^3\right )}\right )+3 a^2 b \log \left (1-c^2 x^6\right )+3 b^3 \log \left (1-c^2 x^6\right )+6 b^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^3\right )}\right )+3 b^3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (c x^3\right )}\right )}{18 c^3} \]
(6*a*b^2*c*x^3 + 3*a^2*b*c^2*x^6 + 2*a^3*c^3*x^9 - 6*a*b^2*ArcTanh[c*x^3] + 6*b^3*c*x^3*ArcTanh[c*x^3] + 6*a*b^2*c^2*x^6*ArcTanh[c*x^3] + 6*a^2*b*c^ 3*x^9*ArcTanh[c*x^3] - 6*a*b^2*ArcTanh[c*x^3]^2 - 3*b^3*ArcTanh[c*x^3]^2 + 3*b^3*c^2*x^6*ArcTanh[c*x^3]^2 + 6*a*b^2*c^3*x^9*ArcTanh[c*x^3]^2 - 2*b^3 *ArcTanh[c*x^3]^3 + 2*b^3*c^3*x^9*ArcTanh[c*x^3]^3 - 12*a*b^2*ArcTanh[c*x^ 3]*Log[1 + E^(-2*ArcTanh[c*x^3])] - 6*b^3*ArcTanh[c*x^3]^2*Log[1 + E^(-2*A rcTanh[c*x^3])] + 3*a^2*b*Log[1 - c^2*x^6] + 3*b^3*Log[1 - c^2*x^6] + 6*b^ 2*(a + b*ArcTanh[c*x^3])*PolyLog[2, -E^(-2*ArcTanh[c*x^3])] + 3*b^3*PolyLo g[3, -E^(-2*ArcTanh[c*x^3])])/(18*c^3)
Time = 1.74 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {6454, 6452, 6542, 6452, 6542, 2009, 6510, 6546, 6470, 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.
\(\displaystyle \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx\) |
\(\Big \downarrow \) 6454 |
\(\displaystyle \frac {1}{3} \int x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3dx^3\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \int \frac {x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c^2 x^6}dx^3\right )\) |
\(\Big \downarrow \) 6542 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\int \frac {x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c^2 x^6}dx^3}{c^2}-\frac {\int x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2dx^3}{c^2}\right )\right )\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\int \frac {x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c^2 x^6}dx^3}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \int \frac {x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )}{1-c^2 x^6}dx^3}{c^2}\right )\right )\) |
\(\Big \downarrow \) 6542 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\int \frac {x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c^2 x^6}dx^3}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\int \frac {a+b \text {arctanh}\left (c x^3\right )}{1-c^2 x^6}dx^3}{c^2}-\frac {\int \left (a+b \text {arctanh}\left (c x^3\right )\right )dx^3}{c^2}\right )}{c^2}\right )\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\int \frac {x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c^2 x^6}dx^3}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\int \frac {a+b \text {arctanh}\left (c x^3\right )}{1-c^2 x^6}dx^3}{c^2}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )}{c^2}\right )\right )\) |
\(\Big \downarrow \) 6510 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\int \frac {x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c^2 x^6}dx^3}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^3}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )}{c^2}\right )\right )\) |
\(\Big \downarrow \) 6546 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\frac {\int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c x^3}dx^3}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 b c^2}}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^3}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )}{c^2}\right )\right )\) |
\(\Big \downarrow \) 6470 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{c}-2 b \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right ) \log \left (\frac {2}{1-c x^3}\right )}{1-c^2 x^6}dx^3}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 b c^2}}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^3}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )}{c^2}\right )\right )\) |
\(\Big \downarrow \) 6620 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{c}-2 b \left (\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right )}{1-c^2 x^6}dx^3-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{2 c}\right )}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 b c^2}}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^3}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )}{c^2}\right )\right )\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{c}-2 b \left (\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^3}\right )}{4 c}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{2 c}\right )}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 b c^2}}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^3}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )}{c^2}\right )\right )\) |
((x^9*(a + b*ArcTanh[c*x^3])^3)/3 - b*c*(-(((x^6*(a + b*ArcTanh[c*x^3])^2) /2 - b*c*((a + b*ArcTanh[c*x^3])^2/(2*b*c^3) - (a*x^3 + b*x^3*ArcTanh[c*x^ 3] + (b*Log[1 - c^2*x^6])/(2*c))/c^2))/c^2) + (-1/3*(a + b*ArcTanh[c*x^3]) ^3/(b*c^2) + (((a + b*ArcTanh[c*x^3])^2*Log[2/(1 - c*x^3)])/c - 2*b*(-1/2* ((a + b*ArcTanh[c*x^3])*PolyLog[2, 1 - 2/(1 - c*x^3)])/c + (b*PolyLog[3, 1 - 2/(1 - c*x^3)])/(4*c)))/c)/c^2))/3
3.2.24.3.1 Defintions of rubi rules used
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_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTanh[c*x])^p, x ], x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simpl ify[(m + 1)/n]]
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_.)/((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]
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]
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]
\[\int x^{8} {\left (a +b \,\operatorname {arctanh}\left (c \,x^{3}\right )\right )}^{3}d x\]
\[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3} x^{8} \,d x } \]
integral(b^3*x^8*arctanh(c*x^3)^3 + 3*a*b^2*x^8*arctanh(c*x^3)^2 + 3*a^2*b *x^8*arctanh(c*x^3) + a^3*x^8, x)
Timed out. \[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\text {Timed out} \]
\[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3} x^{8} \,d x } \]
1/9*a^3*x^9 + 1/6*(2*x^9*arctanh(c*x^3) + (x^6/c^2 + log(c^2*x^6 - 1)/c^4) *c)*a^2*b - 1/72*((b^3*c^3*x^9 - b^3)*log(-c*x^3 + 1)^3 - 3*(2*a*b^2*c^3*x ^9 + b^3*c^2*x^6 + (b^3*c^3*x^9 + b^3)*log(c*x^3 + 1))*log(-c*x^3 + 1)^2)/ c^3 - integrate(-1/8*((b^3*c^3*x^11 - b^3*c^2*x^8)*log(c*x^3 + 1)^3 + 6*(a *b^2*c^3*x^11 - a*b^2*c^2*x^8)*log(c*x^3 + 1)^2 - (4*a*b^2*c^3*x^11 + 2*b^ 3*c^2*x^8 + 3*(b^3*c^3*x^11 - b^3*c^2*x^8)*log(c*x^3 + 1)^2 - 2*(6*a*b^2*c ^2*x^8 - (6*a*b^2*c^3 + b^3*c^3)*x^11 - b^3*x^2)*log(c*x^3 + 1))*log(-c*x^ 3 + 1))/(c^3*x^3 - c^2), x)
\[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3} x^{8} \,d x } \]
Timed out. \[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\int x^8\,{\left (a+b\,\mathrm {atanh}\left (c\,x^3\right )\right )}^3 \,d x \]