Integrand size = 16, antiderivative size = 96 \[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{3 c}+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-\frac {2 b \left (a+b \text {arctanh}\left (c x^3\right )\right ) \log \left (\frac {2}{1-c x^3}\right )}{3 c}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right )}{3 c} \]
1/3*(a+b*arctanh(c*x^3))^2/c+1/3*x^3*(a+b*arctanh(c*x^3))^2-2/3*b*(a+b*arc tanh(c*x^3))*ln(2/(-c*x^3+1))/c-1/3*b^2*polylog(2,1-2/(-c*x^3+1))/c
Time = 0.12 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03 \[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\frac {b^2 \left (-1+c x^3\right ) \text {arctanh}\left (c x^3\right )^2+2 b \text {arctanh}\left (c x^3\right ) \left (a c x^3-b \log \left (1+e^{-2 \text {arctanh}\left (c x^3\right )}\right )\right )+a \left (a c x^3+b \log \left (1-c^2 x^6\right )\right )+b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^3\right )}\right )}{3 c} \]
(b^2*(-1 + c*x^3)*ArcTanh[c*x^3]^2 + 2*b*ArcTanh[c*x^3]*(a*c*x^3 - b*Log[1 + E^(-2*ArcTanh[c*x^3])]) + a*(a*c*x^3 + b*Log[1 - c^2*x^6]) + b^2*PolyLo g[2, -E^(-2*ArcTanh[c*x^3])])/(3*c)
Time = 0.54 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6454, 6436, 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.
| \(\displaystyle \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 6454 |
| \(\displaystyle \frac {1}{3} \int \left (a+b \text {arctanh}\left (c x^3\right )\right )^2dx^3\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle \frac {1}{3} \left (x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-2 b c \int \frac {x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )}{1-c^2 x^6}dx^3\right )\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {1}{3} \left (x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-2 b c \left (\frac {\int \frac {a+b \text {arctanh}\left (c x^3\right )}{1-c x^3}dx^3}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^2}\right )\right )\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {1}{3} \left (x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{c}-b \int \frac {\log \left (\frac {2}{1-c x^3}\right )}{1-c^2 x^6}dx^3}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^2}\right )\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{3} \left (x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-2 b c \left (\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c x^3}\right )}{1-\frac {2}{1-c x^3}}d\frac {1}{1-c x^3}}{c}+\frac {\log \left (\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{c}}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^2}\right )\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{3} \left (x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right )}{2 c}}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^2}\right )\right )\) |
(x^3*(a + b*ArcTanh[c*x^3])^2 - 2*b*c*(-1/2*(a + b*ArcTanh[c*x^3])^2/(b*c^ 2) + (((a + b*ArcTanh[c*x^3])*Log[2/(1 - c*x^3)])/c + (b*PolyLog[2, 1 - 2/ (1 - c*x^3)])/(2*c))/c))/3
3.2.19.3.1 Defintions of rubi rules used
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_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTanh[c*x^n]) ^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 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_.)*(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 = 1.52 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.40
| method | result | size |
| derivativedivides | \(\frac {a^{2} c \,x^{3}+b^{2} \left (\operatorname {arctanh}\left (c \,x^{3}\right )^{2} \left (c \,x^{3}-1\right )+2 \operatorname {arctanh}\left (c \,x^{3}\right )^{2}-2 \,\operatorname {arctanh}\left (c \,x^{3}\right ) \ln \left (1+\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )\right )+2 a b c \,x^{3} \operatorname {arctanh}\left (c \,x^{3}\right )+a b \ln \left (-c^{2} x^{6}+1\right )}{3 c}\) | \(134\) |
| default | \(\frac {a^{2} c \,x^{3}+b^{2} \left (\operatorname {arctanh}\left (c \,x^{3}\right )^{2} \left (c \,x^{3}-1\right )+2 \operatorname {arctanh}\left (c \,x^{3}\right )^{2}-2 \,\operatorname {arctanh}\left (c \,x^{3}\right ) \ln \left (1+\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )\right )+2 a b c \,x^{3} \operatorname {arctanh}\left (c \,x^{3}\right )+a b \ln \left (-c^{2} x^{6}+1\right )}{3 c}\) | \(134\) |
| parts | \(\frac {a^{2} x^{3}}{3}+\frac {b^{2} \left (\operatorname {arctanh}\left (c \,x^{3}\right )^{2} \left (c \,x^{3}-1\right )+2 \operatorname {arctanh}\left (c \,x^{3}\right )^{2}-2 \,\operatorname {arctanh}\left (c \,x^{3}\right ) \ln \left (1+\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )\right )}{3 c}+\frac {2 a b \,x^{3} \operatorname {arctanh}\left (c \,x^{3}\right )}{3}+\frac {a b \ln \left (-c^{2} x^{6}+1\right )}{3 c}\) | \(136\) |
| risch | \(-\frac {b^{2} \operatorname {dilog}\left (\frac {c \,x^{3}}{2}+\frac {1}{2}\right )}{3 c}-\frac {b^{2} \ln \left (c \,x^{3}-1\right )}{3 c}+\frac {a^{2} x^{3}}{3}+\frac {\ln \left (-c \,x^{3}+1\right )^{2} b^{2} x^{3}}{12}-\frac {\ln \left (-c \,x^{3}+1\right )^{2} b^{2}}{12 c}+\frac {\ln \left (-c \,x^{3}+1\right ) b^{2}}{3 c}+\frac {b^{2} \ln \left (c \,x^{3}+1\right )^{2} x^{3}}{12}+\frac {b^{2} \ln \left (c \,x^{3}+1\right )^{2}}{12 c}-\frac {2 a b}{3 c}-\frac {\ln \left (-c \,x^{3}+1\right ) a b \,x^{3}}{3}+\frac {\ln \left (-c \,x^{3}+1\right ) a b}{3 c}-\frac {b^{2}}{3 c}-\frac {a^{2}}{3 c}+\frac {b a \ln \left (c \,x^{3}+1\right ) x^{3}}{3}+\frac {b a \ln \left (c \,x^{3}+1\right )}{3 c}-\frac {b^{2} \ln \left (-c \,x^{3}+1\right ) \ln \left (c \,x^{3}+1\right ) x^{3}}{6}-\frac {b^{2} \ln \left (-c \,x^{3}+1\right ) \ln \left (c \,x^{3}+1\right )}{6 c}+\frac {b^{2} \ln \left (\frac {1}{2}-\frac {c \,x^{3}}{2}\right ) \ln \left (c \,x^{3}+1\right )}{3 c}-\frac {b^{2} \ln \left (\frac {1}{2}-\frac {c \,x^{3}}{2}\right ) \ln \left (\frac {c \,x^{3}}{2}+\frac {1}{2}\right )}{3 c}\) | \(320\) |
1/3/c*(a^2*c*x^3+b^2*(arctanh(c*x^3)^2*(c*x^3-1)+2*arctanh(c*x^3)^2-2*arct anh(c*x^3)*ln(1+(c*x^3+1)^2/(-c^2*x^6+1))-polylog(2,-(c*x^3+1)^2/(-c^2*x^6 +1)))+2*a*b*c*x^3*arctanh(c*x^3)+a*b*ln(-c^2*x^6+1))
\[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{2} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\text {Timed out} \]
\[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{2} x^{2} \,d x } \]
1/3*a^2*x^3 + 1/12*(x^3*log(-c*x^3 + 1)^2 - c^2*(2*x^3/c^2 - log(c*x^3 + 1 )/c^3 + log(c*x^3 - 1)/c^3) - 2*(x^3/c + log(c*x^3 - 1)/c^2)*c*log(-c*x^3 + 1) + 18*c*integrate(x^5*log(c*x^3 + 1)/(c^2*x^6 - 1), x) + (c*x^3*log(c* x^3 + 1)^2 + 2*(c*x^3 - (c*x^3 + 1)*log(c*x^3 + 1))*log(-c*x^3 + 1))/c + ( 2*c*x^3 + log(c*x^3 - 1)^2 + 2*log(c*x^3 - 1))/c - log(c^2*x^6 - 1)/c + 6* integrate(x^2*log(c*x^3 + 1)/(c^2*x^6 - 1), x))*b^2 + 1/3*(2*c*x^3*arctanh (c*x^3) + log(-c^2*x^6 + 1))*a*b/c
\[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{2} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {atanh}\left (c\,x^3\right )\right )}^2 \,d x \]