\(\int x (a+b \text {arctanh}(c x^2))^3 \, dx\) [78]

Optimal result

Integrand size = 14, antiderivative size = 134 \[ \int x \left (a+b \text {arctanh}\left (c x^2\right )\right )^3 \, dx=\frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^3}{2 c}+\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )^3-\frac {3 b \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \log \left (\frac {2}{1-c x^2}\right )}{2 c}-\frac {3 b^2 \left (a+b \text {arctanh}\left (c x^2\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^2}\right )}{2 c}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^2}\right )}{4 c} \] Output:

1/2*(a+b*arctanh(c*x^2))^3/c+1/2*x^2*(a+b*arctanh(c*x^2))^3-3/2*b*(a+b*arc 
tanh(c*x^2))^2*ln(2/(-c*x^2+1))/c-3/2*b^2*(a+b*arctanh(c*x^2))*polylog(2,1 
-2/(-c*x^2+1))/c+3/4*b^3*polylog(3,1-2/(-c*x^2+1))/c
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.59 \[ \int x \left (a+b \text {arctanh}\left (c x^2\right )\right )^3 \, dx=\frac {a^3 x^2}{2}+\frac {3}{2} a^2 b x^2 \text {arctanh}\left (c x^2\right )+\frac {3 a^2 b \log \left (1-c^2 x^4\right )}{4 c}+\frac {3 a b^2 \left (\text {arctanh}\left (c x^2\right ) \left (-\text {arctanh}\left (c x^2\right )+c x^2 \text {arctanh}\left (c x^2\right )-2 \log \left (1+e^{-2 \text {arctanh}\left (c x^2\right )}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^2\right )}\right )\right )}{2 c}+\frac {b^3 \left (\text {arctanh}\left (c x^2\right )^2 \left (-\text {arctanh}\left (c x^2\right )+c x^2 \text {arctanh}\left (c x^2\right )-3 \log \left (1+e^{-2 \text {arctanh}\left (c x^2\right )}\right )\right )+3 \text {arctanh}\left (c x^2\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^2\right )}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (c x^2\right )}\right )\right )}{2 c} \] Input:

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

Output:

(a^3*x^2)/2 + (3*a^2*b*x^2*ArcTanh[c*x^2])/2 + (3*a^2*b*Log[1 - c^2*x^4])/ 
(4*c) + (3*a*b^2*(ArcTanh[c*x^2]*(-ArcTanh[c*x^2] + c*x^2*ArcTanh[c*x^2] - 
 2*Log[1 + E^(-2*ArcTanh[c*x^2])]) + PolyLog[2, -E^(-2*ArcTanh[c*x^2])]))/ 
(2*c) + (b^3*(ArcTanh[c*x^2]^2*(-ArcTanh[c*x^2] + c*x^2*ArcTanh[c*x^2] - 3 
*Log[1 + E^(-2*ArcTanh[c*x^2])]) + 3*ArcTanh[c*x^2]*PolyLog[2, -E^(-2*ArcT 
anh[c*x^2])] + (3*PolyLog[3, -E^(-2*ArcTanh[c*x^2])])/2))/(2*c)
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, 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.

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

\(\Big \downarrow \) 6454

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

\(\Big \downarrow \) 6436

\(\displaystyle \frac {1}{2} \left (x^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )^3-3 b c \int \frac {x^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{1-c^2 x^4}dx^2\right )\)

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

(x^2*(a + b*ArcTanh[c*x^2])^3 - 3*b*c*(-1/3*(a + b*ArcTanh[c*x^2])^3/(b*c^ 
2) + (((a + b*ArcTanh[c*x^2])^2*Log[2/(1 - c*x^2)])/c - 2*b*(-1/2*((a + b* 
ArcTanh[c*x^2])*PolyLog[2, 1 - 2/(1 - c*x^2)])/c + (b*PolyLog[3, 1 - 2/(1 
- c*x^2)])/(4*c)))/c))/2
 

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 = 1.67 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.98

method	result	size
derivativedivides	\(\frac {a^{3} c \,x^{2}+b^{3} \left (\operatorname {arctanh}\left (c \,x^{2}\right )^{3} \left (c \,x^{2}-1\right )+2 \operatorname {arctanh}\left (c \,x^{2}\right )^{3}-3 \operatorname {arctanh}\left (c \,x^{2}\right )^{2} \ln \left (1+\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )-3 \,\operatorname {arctanh}\left (c \,x^{2}\right ) \operatorname {polylog}\left (2, -\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )}{2}\right )+3 a \,b^{2} \left (\operatorname {arctanh}\left (c \,x^{2}\right )^{2} \left (c \,x^{2}-1\right )+2 \operatorname {arctanh}\left (c \,x^{2}\right )^{2}-2 \,\operatorname {arctanh}\left (c \,x^{2}\right ) \ln \left (1+\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )\right )+3 a^{2} b \left (c \,x^{2} \operatorname {arctanh}\left (c \,x^{2}\right )+\frac {\ln \left (-c^{2} x^{4}+1\right )}{2}\right )}{2 c}\)	\(265\)
default	\(\frac {a^{3} c \,x^{2}+b^{3} \left (\operatorname {arctanh}\left (c \,x^{2}\right )^{3} \left (c \,x^{2}-1\right )+2 \operatorname {arctanh}\left (c \,x^{2}\right )^{3}-3 \operatorname {arctanh}\left (c \,x^{2}\right )^{2} \ln \left (1+\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )-3 \,\operatorname {arctanh}\left (c \,x^{2}\right ) \operatorname {polylog}\left (2, -\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )}{2}\right )+3 a \,b^{2} \left (\operatorname {arctanh}\left (c \,x^{2}\right )^{2} \left (c \,x^{2}-1\right )+2 \operatorname {arctanh}\left (c \,x^{2}\right )^{2}-2 \,\operatorname {arctanh}\left (c \,x^{2}\right ) \ln \left (1+\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )\right )+3 a^{2} b \left (c \,x^{2} \operatorname {arctanh}\left (c \,x^{2}\right )+\frac {\ln \left (-c^{2} x^{4}+1\right )}{2}\right )}{2 c}\)	\(265\)
parts	\(\frac {a^{3} x^{2}}{2}+\frac {b^{3} \left (\operatorname {arctanh}\left (c \,x^{2}\right )^{3} \left (c \,x^{2}-1\right )+2 \operatorname {arctanh}\left (c \,x^{2}\right )^{3}-3 \operatorname {arctanh}\left (c \,x^{2}\right )^{2} \ln \left (1+\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )-3 \,\operatorname {arctanh}\left (c \,x^{2}\right ) \operatorname {polylog}\left (2, -\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )}{2}\right )}{2 c}+\frac {3 a \,b^{2} \left (\operatorname {arctanh}\left (c \,x^{2}\right )^{2} \left (c \,x^{2}-1\right )+2 \operatorname {arctanh}\left (c \,x^{2}\right )^{2}-2 \,\operatorname {arctanh}\left (c \,x^{2}\right ) \ln \left (1+\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )\right )}{2 c}+\frac {3 a^{2} b \,x^{2} \operatorname {arctanh}\left (c \,x^{2}\right )}{2}+\frac {3 a^{2} b \ln \left (-c^{2} x^{4}+1\right )}{4 c}\)	\(271\)

Input:

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

Output:

1/2/c*(a^3*c*x^2+b^3*(arctanh(c*x^2)^3*(c*x^2-1)+2*arctanh(c*x^2)^3-3*arct 
anh(c*x^2)^2*ln(1+(c*x^2+1)^2/(-c^2*x^4+1))-3*arctanh(c*x^2)*polylog(2,-(c 
*x^2+1)^2/(-c^2*x^4+1))+3/2*polylog(3,-(c*x^2+1)^2/(-c^2*x^4+1)))+3*a*b^2* 
(arctanh(c*x^2)^2*(c*x^2-1)+2*arctanh(c*x^2)^2-2*arctanh(c*x^2)*ln(1+(c*x^ 
2+1)^2/(-c^2*x^4+1))-polylog(2,-(c*x^2+1)^2/(-c^2*x^4+1)))+3*a^2*b*(c*x^2* 
arctanh(c*x^2)+1/2*ln(-c^2*x^4+1)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*a^3*x^2 + 3/4*(2*c*x^2*arctanh(c*x^2) + log(-c^2*x^4 + 1))*a^2*b/c - 1 
/16*((b^3*c*x^2 - b^3)*log(-c*x^2 + 1)^3 - 3*(2*a*b^2*c*x^2 + (b^3*c*x^2 + 
 b^3)*log(c*x^2 + 1))*log(-c*x^2 + 1)^2)/c - integrate(-1/8*((b^3*c*x^3 - 
b^3*x)*log(c*x^2 + 1)^3 + 6*(a*b^2*c*x^3 - a*b^2*x)*log(c*x^2 + 1)^2 - 3*( 
4*a*b^2*c*x^3 + (b^3*c*x^3 - b^3*x)*log(c*x^2 + 1)^2 + 2*((2*a*b^2*c + b^3 
*c)*x^3 - (2*a*b^2 - b^3)*x)*log(c*x^2 + 1))*log(-c*x^2 + 1))/(c*x^2 - 1), 
 x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x \left (a+b \text {arctanh}\left (c x^2\right )\right )^3 \, dx=\frac {3 \mathit {atanh} \left (c \,x^{2}\right ) a^{2} b c \,x^{2}-3 \mathit {atanh} \left (c \,x^{2}\right ) a^{2} b +2 \left (\int \mathit {atanh} \left (c \,x^{2}\right )^{3} x d x \right ) b^{3} c +6 \left (\int \mathit {atanh} \left (c \,x^{2}\right )^{2} x d x \right ) a \,b^{2} c +3 \,\mathrm {log}\left (c \,x^{2}+1\right ) a^{2} b +a^{3} c \,x^{2}}{2 c} \] Input:

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

Output:

(3*atanh(c*x**2)*a**2*b*c*x**2 - 3*atanh(c*x**2)*a**2*b + 2*int(atanh(c*x* 
*2)**3*x,x)*b**3*c + 6*int(atanh(c*x**2)**2*x,x)*a*b**2*c + 3*log(c*x**2 + 
 1)*a**2*b + a**3*c*x**2)/(2*c)