\(\int \frac {(a+b \text {arctanh}(c x))^3}{x^2} \, dx\) [31]

Optimal result

Integrand size = 14, antiderivative size = 102 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{x^2} \, dx=c (a+b \text {arctanh}(c x))^3-\frac {(a+b \text {arctanh}(c x))^3}{x}+3 b c (a+b \text {arctanh}(c x))^2 \log \left (2-\frac {2}{1+c x}\right )-3 b^2 c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )-\frac {3}{2} b^3 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+c x}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.92 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{x^2} \, dx=-\frac {a^3}{x}-\frac {3 a^2 b \text {arctanh}(c x)}{x}+3 a^2 b c \log (x)-\frac {3}{2} a^2 b c \log \left (1-c^2 x^2\right )+3 a b^2 c \left (\text {arctanh}(c x) \left (\text {arctanh}(c x)-\frac {\text {arctanh}(c x)}{c x}+2 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )+b^3 c \left (\frac {i \pi ^3}{8}-\text {arctanh}(c x)^3-\frac {\text {arctanh}(c x)^3}{c x}+3 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+3 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right ) \] Input:

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

Output:

-(a^3/x) - (3*a^2*b*ArcTanh[c*x])/x + 3*a^2*b*c*Log[x] - (3*a^2*b*c*Log[1 
- c^2*x^2])/2 + 3*a*b^2*c*(ArcTanh[c*x]*(ArcTanh[c*x] - ArcTanh[c*x]/(c*x) 
 + 2*Log[1 - E^(-2*ArcTanh[c*x])]) - PolyLog[2, E^(-2*ArcTanh[c*x])]) + b^ 
3*c*((I/8)*Pi^3 - ArcTanh[c*x]^3 - ArcTanh[c*x]^3/(c*x) + 3*ArcTanh[c*x]^2 
*Log[1 - E^(2*ArcTanh[c*x])] + 3*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x] 
)] - (3*PolyLog[3, E^(2*ArcTanh[c*x])])/2)
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6452, 6550, 6494, 6618, 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.

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 6452

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]
 

rule 6494

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x 
_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - 
Simp[b*c*(p/d)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 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 6550

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ 
d   Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
 

rule 6618

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 [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 9.42 (sec) , antiderivative size = 1329, normalized size of antiderivative = 13.03

method	result	size
parts	\(\text {Expression too large to display}\)	\(1329\)
derivativedivides	\(\text {Expression too large to display}\)	\(1331\)
default	\(\text {Expression too large to display}\)	\(1331\)

Input:

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

Output:

-1/x*a^3+b^3*c*(-1/c/x*arctanh(c*x)^3-3/2*arctanh(c*x)^2*ln(c*x-1)+3*ln(c* 
x)*arctanh(c*x)^2-3/2*arctanh(c*x)^2*ln(c*x+1)+3*arctanh(c*x)^2*ln((c*x+1) 
/(-c^2*x^2+1)^(1/2))-arctanh(c*x)^3+3/4*(2*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x 
^2-1)))^3+I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I*(c*x+1)^2/(c^2*x^2 
-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2+2*I*Pi*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/ 
(1-(c*x+1)^2/(c^2*x^2-1)))^3+2*I*Pi-2*I*Pi*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)- 
1))*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2-I*Pi*cs 
gn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2 
*x^2-1)))^2-2*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))^2+I*Pi*csgn(I*(c*x+1) 
/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))+I*Pi*csgn(I*(c*x+1)^2 
/(c^2*x^2-1))^3+2*I*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2 
/(c^2*x^2-1))^2+I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1) 
))^3-2*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I*(-(c*x+1)^2/(c^2*x^2- 
1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2-I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))* 
csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c 
^2*x^2-1)))+2*I*Pi*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1))*csgn(I/(1-(c*x+1)^2/ 
(c^2*x^2-1)))*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1))) 
+4*ln(2))*arctanh(c*x)^2-3*arctanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1)+3*a 
rctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+6*arctanh(c*x)*polylog(2,-( 
c*x+1)/(-c^2*x^2+1)^(1/2))-6*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2))+3*a...
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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