\(\int x^2 (a+b \text {arctanh}(c x^3))^3 \, dx\) [126]

Optimal result

Integrand size = 16, antiderivative size = 130 \[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 c}+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-\frac {b \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \log \left (\frac {2}{1-c x^3}\right )}{c}-\frac {b^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right )}{c}+\frac {b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^3}\right )}{2 c} \] Output:

1/3*(a+b*arctanh(c*x^3))^3/c+1/3*x^3*(a+b*arctanh(c*x^3))^3-b*(a+b*arctanh 
(c*x^3))^2*ln(2/(-c*x^3+1))/c-b^2*(a+b*arctanh(c*x^3))*polylog(2,1-2/(-c*x 
^3+1))/c+1/2*b^3*polylog(3,1-2/(-c*x^3+1))/c
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.47 \[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\frac {2 a^3 c x^3+6 a^2 b c x^3 \text {arctanh}\left (c x^3\right )+3 a^2 b \log \left (1-c^2 x^6\right )+6 a b^2 \left (\text {arctanh}\left (c x^3\right ) \left (\left (-1+c x^3\right ) \text {arctanh}\left (c x^3\right )-2 \log \left (1+e^{-2 \text {arctanh}\left (c x^3\right )}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^3\right )}\right )\right )+b^3 \left (2 \text {arctanh}\left (c x^3\right )^2 \left (\left (-1+c x^3\right ) \text {arctanh}\left (c x^3\right )-3 \log \left (1+e^{-2 \text {arctanh}\left (c x^3\right )}\right )\right )+6 \text {arctanh}\left (c x^3\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^3\right )}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (c x^3\right )}\right )\right )}{6 c} \] Input:

Integrate[x^2*(a + b*ArcTanh[c*x^3])^3,x]
 

Output:

(2*a^3*c*x^3 + 6*a^2*b*c*x^3*ArcTanh[c*x^3] + 3*a^2*b*Log[1 - c^2*x^6] + 6 
*a*b^2*(ArcTanh[c*x^3]*((-1 + c*x^3)*ArcTanh[c*x^3] - 2*Log[1 + E^(-2*ArcT 
anh[c*x^3])]) + PolyLog[2, -E^(-2*ArcTanh[c*x^3])]) + b^3*(2*ArcTanh[c*x^3 
]^2*((-1 + c*x^3)*ArcTanh[c*x^3] - 3*Log[1 + E^(-2*ArcTanh[c*x^3])]) + 6*A 
rcTanh[c*x^3]*PolyLog[2, -E^(-2*ArcTanh[c*x^3])] + 3*PolyLog[3, -E^(-2*Arc 
Tanh[c*x^3])]))/(6*c)
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.10, 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, 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.

\(\displaystyle \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx\)

\(\Big \downarrow \) 6454

\(\displaystyle \frac {1}{3} \int \left (a+b \text {arctanh}\left (c x^3\right )\right )^3dx^3\)

\(\Big \downarrow \) 6436

\(\displaystyle \frac {1}{3} \left (x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-3 b c \int \frac {x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c^2 x^6}dx^3\right )\)

\(\Big \downarrow \) 6546

\(\displaystyle \frac {1}{3} \left (x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-3 b c \left (\frac {\int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c x^3}dx^3}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 b c^2}\right )\right )\)

\(\Big \downarrow \) 6470

\(\displaystyle \frac {1}{3} \left (x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-3 b c \left (\frac {\frac {\log \left (\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{c}-2 b \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right ) \log \left (\frac {2}{1-c x^3}\right )}{1-c^2 x^6}dx^3}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 b c^2}\right )\right )\)

\(\Big \downarrow \) 6620

\(\displaystyle \frac {1}{3} \left (x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-3 b c \left (\frac {\frac {\log \left (\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{c}-2 b \left (\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right )}{1-c^2 x^6}dx^3-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{2 c}\right )}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 b c^2}\right )\right )\)

\(\Big \downarrow \) 7164

\(\displaystyle \frac {1}{3} \left (x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-3 b c \left (\frac {\frac {\log \left (\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{c}-2 b \left (\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^3}\right )}{4 c}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{2 c}\right )}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 b c^2}\right )\right )\)

Input:

Int[x^2*(a + b*ArcTanh[c*x^3])^3,x]
 

Output:

(x^3*(a + b*ArcTanh[c*x^3])^3 - 3*b*c*(-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))/3
 

Defintions of rubi rules used

rule 6436

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])
 

rule 6454

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]]
 

rule 6470

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]
 

rule 6546

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]
 

rule 6620

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]
 

rule 7164

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(264\) vs. \(2(124)=248\).

Time = 1.94 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.04

method	result	size
derivativedivides	\(\frac {a^{3} c \,x^{3}+b^{3} \left (\operatorname {arctanh}\left (c \,x^{3}\right )^{3} \left (c \,x^{3}-1\right )+2 \operatorname {arctanh}\left (c \,x^{3}\right )^{3}-3 \operatorname {arctanh}\left (c \,x^{3}\right )^{2} \ln \left (1+\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )-3 \,\operatorname {arctanh}\left (c \,x^{3}\right ) \operatorname {polylog}\left (2, -\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )}{2}\right )+3 a \,b^{2} \left (\operatorname {arctanh}\left (c \,x^{3}\right )^{2} \left (c \,x^{3}-1\right )+2 \operatorname {arctanh}\left (c \,x^{3}\right )^{2}-2 \,\operatorname {arctanh}\left (c \,x^{3}\right ) \ln \left (1+\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )\right )+3 a^{2} b \left (c \,x^{3} \operatorname {arctanh}\left (c \,x^{3}\right )+\frac {\ln \left (-c^{2} x^{6}+1\right )}{2}\right )}{3 c}\)	\(265\)
default	\(\frac {a^{3} c \,x^{3}+b^{3} \left (\operatorname {arctanh}\left (c \,x^{3}\right )^{3} \left (c \,x^{3}-1\right )+2 \operatorname {arctanh}\left (c \,x^{3}\right )^{3}-3 \operatorname {arctanh}\left (c \,x^{3}\right )^{2} \ln \left (1+\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )-3 \,\operatorname {arctanh}\left (c \,x^{3}\right ) \operatorname {polylog}\left (2, -\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )}{2}\right )+3 a \,b^{2} \left (\operatorname {arctanh}\left (c \,x^{3}\right )^{2} \left (c \,x^{3}-1\right )+2 \operatorname {arctanh}\left (c \,x^{3}\right )^{2}-2 \,\operatorname {arctanh}\left (c \,x^{3}\right ) \ln \left (1+\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )\right )+3 a^{2} b \left (c \,x^{3} \operatorname {arctanh}\left (c \,x^{3}\right )+\frac {\ln \left (-c^{2} x^{6}+1\right )}{2}\right )}{3 c}\)	\(265\)
parts	\(\frac {a^{3} x^{3}}{3}+\frac {b^{3} \left (\operatorname {arctanh}\left (c \,x^{3}\right )^{3} \left (c \,x^{3}-1\right )+2 \operatorname {arctanh}\left (c \,x^{3}\right )^{3}-3 \operatorname {arctanh}\left (c \,x^{3}\right )^{2} \ln \left (1+\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )-3 \,\operatorname {arctanh}\left (c \,x^{3}\right ) \operatorname {polylog}\left (2, -\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )}{2}\right )}{3 c}+\frac {a \,b^{2} \left (\operatorname {arctanh}\left (c \,x^{3}\right )^{2} \left (c \,x^{3}-1\right )+2 \operatorname {arctanh}\left (c \,x^{3}\right )^{2}-2 \,\operatorname {arctanh}\left (c \,x^{3}\right ) \ln \left (1+\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c \,x^{3}+1\right )^{2}}{-c^{2} x^{6}+1}\right )\right )}{c}+\frac {a^{2} b \left (c \,x^{3} \operatorname {arctanh}\left (c \,x^{3}\right )+\frac {\ln \left (-c^{2} x^{6}+1\right )}{2}\right )}{c}\)	\(268\)

Input:

int(x^2*(a+b*arctanh(c*x^3))^3,x,method=_RETURNVERBOSE)
 

Output:

1/3/c*(a^3*c*x^3+b^3*(arctanh(c*x^3)^3*(c*x^3-1)+2*arctanh(c*x^3)^3-3*arct 
anh(c*x^3)^2*ln(1+(c*x^3+1)^2/(-c^2*x^6+1))-3*arctanh(c*x^3)*polylog(2,-(c 
*x^3+1)^2/(-c^2*x^6+1))+3/2*polylog(3,-(c*x^3+1)^2/(-c^2*x^6+1)))+3*a*b^2* 
(arctanh(c*x^3)^2*(c*x^3-1)+2*arctanh(c*x^3)^2-2*arctanh(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)))+3*a^2*b*(c*x^3* 
arctanh(c*x^3)+1/2*ln(-c^2*x^6+1)))
 

Fricas [F]

\[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3} x^{2} \,d x } \] Input:

integrate(x^2*(a+b*arctanh(c*x^3))^3,x, algorithm="fricas")
 

Output:

integral(b^3*x^2*arctanh(c*x^3)^3 + 3*a*b^2*x^2*arctanh(c*x^3)^2 + 3*a^2*b 
*x^2*arctanh(c*x^3) + a^3*x^2, x)
 

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\text {Timed out} \] Input:

integrate(x**2*(a+b*atanh(c*x**3))**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3} x^{2} \,d x } \] Input:

integrate(x^2*(a+b*arctanh(c*x^3))^3,x, algorithm="maxima")
 

Output:

1/3*a^3*x^3 + 1/2*(2*c*x^3*arctanh(c*x^3) + log(-c^2*x^6 + 1))*a^2*b/c - 1 
/24*((b^3*c*x^3 - b^3)*log(-c*x^3 + 1)^3 - 3*(2*a*b^2*c*x^3 + (b^3*c*x^3 + 
 b^3)*log(c*x^3 + 1))*log(-c*x^3 + 1)^2)/c - integrate(-1/8*((b^3*c*x^5 - 
b^3*x^2)*log(c*x^3 + 1)^3 + 6*(a*b^2*c*x^5 - a*b^2*x^2)*log(c*x^3 + 1)^2 - 
 3*(4*a*b^2*c*x^5 + (b^3*c*x^5 - b^3*x^2)*log(c*x^3 + 1)^2 + 2*((2*a*b^2*c 
 + b^3*c)*x^5 - (2*a*b^2 - b^3)*x^2)*log(c*x^3 + 1))*log(-c*x^3 + 1))/(c*x 
^3 - 1), x)
 

Giac [F]

\[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3} x^{2} \,d x } \] Input:

integrate(x^2*(a+b*arctanh(c*x^3))^3,x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x^3) + a)^3*x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\int x^2\,{\left (a+b\,\mathrm {atanh}\left (c\,x^3\right )\right )}^3 \,d x \] Input:

int(x^2*(a + b*atanh(c*x^3))^3,x)
 

Output:

int(x^2*(a + b*atanh(c*x^3))^3, x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\frac {3 \mathit {atanh} \left (c \,x^{3}\right ) a^{2} b c \,x^{3}-3 \mathit {atanh} \left (c \,x^{3}\right ) a^{2} b +3 \left (\int \mathit {atanh} \left (c \,x^{3}\right )^{3} x^{2}d x \right ) b^{3} c +9 \left (\int \mathit {atanh} \left (c \,x^{3}\right )^{2} x^{2}d x \right ) a \,b^{2} c +3 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} x +1\right ) a^{2} b +3 \,\mathrm {log}\left (c^{\frac {2}{3}} x +c^{\frac {1}{3}}\right ) a^{2} b +a^{3} c \,x^{3}}{3 c} \] Input:

int(x^2*(a+b*atanh(c*x^3))^3,x)
 

Output:

(3*atanh(c*x**3)*a**2*b*c*x**3 - 3*atanh(c*x**3)*a**2*b + 3*int(atanh(c*x* 
*3)**3*x**2,x)*b**3*c + 9*int(atanh(c*x**3)**2*x**2,x)*a*b**2*c + 3*log(c* 
*(2/3)*x**2 - c**(1/3)*x + 1)*a**2*b + 3*log(c**(2/3)*x + c**(1/3))*a**2*b 
 + a**3*c*x**3)/(3*c)