\(\int (a+b \text {arctanh}(c x))^3 \, dx\) [29]

Optimal result

Integrand size = 10, antiderivative size = 108 \[ \int (a+b \text {arctanh}(c x))^3 \, dx=\frac {(a+b \text {arctanh}(c x))^3}{c}+x (a+b \text {arctanh}(c x))^3-\frac {3 b (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.49 \[ \int (a+b \text {arctanh}(c x))^3 \, dx=\frac {2 a^3 c x+6 a^2 b c x \text {arctanh}(c x)+3 a^2 b \log \left (1-c^2 x^2\right )+6 a b^2 \left (\text {arctanh}(c x) \left ((-1+c x) \text {arctanh}(c x)-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+b^3 \left (2 \text {arctanh}(c x)^2 \left ((-1+c x) \text {arctanh}(c x)-3 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+6 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )\right )}{2 c} \] Input:

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

Output:

(2*a^3*c*x + 6*a^2*b*c*x*ArcTanh[c*x] + 3*a^2*b*Log[1 - c^2*x^2] + 6*a*b^2 
*(ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log[1 + E^(-2*ArcTanh[c*x])]) 
+ PolyLog[2, -E^(-2*ArcTanh[c*x])]) + b^3*(2*ArcTanh[c*x]^2*((-1 + c*x)*Ar 
cTanh[c*x] - 3*Log[1 + E^(-2*ArcTanh[c*x])]) + 6*ArcTanh[c*x]*PolyLog[2, - 
E^(-2*ArcTanh[c*x])] + 3*PolyLog[3, -E^(-2*ArcTanh[c*x])]))/(2*c)
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {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.

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

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

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 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. \(229\) vs. \(2(106)=212\).

Time = 1.19 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.13

method	result	size
derivativedivides	\(\frac {a^{3} c x +b^{3} \left (\operatorname {arctanh}\left (c x \right )^{3} \left (c x -1\right )+2 \operatorname {arctanh}\left (c x \right )^{3}-3 \operatorname {arctanh}\left (c x \right )^{2} \ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-3 \,\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}{2}\right )+3 a \,b^{2} \left (\operatorname {arctanh}\left (c x \right )^{2} \left (c x -1\right )+2 \operatorname {arctanh}\left (c x \right )^{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )\right )+3 a^{2} b \left (c x \,\operatorname {arctanh}\left (c x \right )+\frac {\ln \left (-c^{2} x^{2}+1\right )}{2}\right )}{c}\)	\(230\)
default	\(\frac {a^{3} c x +b^{3} \left (\operatorname {arctanh}\left (c x \right )^{3} \left (c x -1\right )+2 \operatorname {arctanh}\left (c x \right )^{3}-3 \operatorname {arctanh}\left (c x \right )^{2} \ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-3 \,\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}{2}\right )+3 a \,b^{2} \left (\operatorname {arctanh}\left (c x \right )^{2} \left (c x -1\right )+2 \operatorname {arctanh}\left (c x \right )^{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )\right )+3 a^{2} b \left (c x \,\operatorname {arctanh}\left (c x \right )+\frac {\ln \left (-c^{2} x^{2}+1\right )}{2}\right )}{c}\)	\(230\)
parts	\(x \,a^{3}+\frac {b^{3} \left (\operatorname {arctanh}\left (c x \right )^{3} \left (c x -1\right )+2 \operatorname {arctanh}\left (c x \right )^{3}-3 \operatorname {arctanh}\left (c x \right )^{2} \ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-3 \,\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}{2}\right )}{c}+\frac {3 a \,b^{2} \left (\operatorname {arctanh}\left (c x \right )^{2} \left (c x -1\right )+2 \operatorname {arctanh}\left (c x \right )^{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )\right )}{c}+3 a^{2} b x \,\operatorname {arctanh}\left (c x \right )+\frac {3 a^{2} b \ln \left (-c^{2} x^{2}+1\right )}{2 c}\)	\(235\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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