\(\int \frac {(a+b \text {arctanh}(c x^2))^3}{x^3} \, dx\) [80]

Optimal result

Integrand size = 16, antiderivative size = 125 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^3}{x^3} \, dx=\frac {1}{2} c \left (a+b \text {arctanh}\left (c x^2\right )\right )^3-\frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^3}{2 x^2}+\frac {3}{2} b c \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \log \left (2-\frac {2}{1+c x^2}\right )-\frac {3}{2} b^2 c \left (a+b \text {arctanh}\left (c x^2\right )\right ) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x^2}\right )-\frac {3}{4} b^3 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+c x^2}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^3}{x^3} \, dx=\frac {1}{4} \left (-\frac {2 a^3}{x^2}-\frac {6 a^2 b \text {arctanh}\left (c x^2\right )}{x^2}+12 a^2 b c \log (x)-3 a^2 b c \log \left (1-c^2 x^4\right )+6 a b^2 c \left (\text {arctanh}\left (c x^2\right ) \left (\left (1-\frac {1}{c x^2}\right ) \text {arctanh}\left (c x^2\right )+2 \log \left (1-e^{-2 \text {arctanh}\left (c x^2\right )}\right )\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (c x^2\right )}\right )\right )+2 b^3 c \left (\frac {i \pi ^3}{8}-\text {arctanh}\left (c x^2\right )^3-\frac {\text {arctanh}\left (c x^2\right )^3}{c x^2}+3 \text {arctanh}\left (c x^2\right )^2 \log \left (1-e^{2 \text {arctanh}\left (c x^2\right )}\right )+3 \text {arctanh}\left (c x^2\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (c x^2\right )}\right )-\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (c x^2\right )}\right )\right )\right ) \] Input:

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

Output:

((-2*a^3)/x^2 - (6*a^2*b*ArcTanh[c*x^2])/x^2 + 12*a^2*b*c*Log[x] - 3*a^2*b 
*c*Log[1 - c^2*x^4] + 6*a*b^2*c*(ArcTanh[c*x^2]*((1 - 1/(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*b^3*c*((I/8)*Pi^3 - ArcTanh[c*x^2]^3 - ArcTanh[c*x^2]^3/(c*x^2) + 
 3*ArcTanh[c*x^2]^2*Log[1 - E^(2*ArcTanh[c*x^2])] + 3*ArcTanh[c*x^2]*PolyL 
og[2, E^(2*ArcTanh[c*x^2])] - (3*PolyLog[3, E^(2*ArcTanh[c*x^2])])/2))/4
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6454, 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.

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

\(\Big \downarrow \) 6454

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

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 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 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 [F]

\[\int \frac {{\left (a +b \,\operatorname {arctanh}\left (c \,x^{2}\right )\right )}^{3}}{x^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-3/4*(c*(log(c^2*x^4 - 1) - log(x^4)) + 2*arctanh(c*x^2)/x^2)*a^2*b - 1/2* 
a^3/x^2 - 1/16*((b^3*c*x^2 - b^3)*log(-c*x^2 + 1)^3 + 3*(2*a*b^2 + (b^3*c* 
x^2 + b^3)*log(c*x^2 + 1))*log(-c*x^2 + 1)^2)/x^2 - integrate(-1/8*((b^3*c 
*x^2 - b^3)*log(c*x^2 + 1)^3 + 6*(a*b^2*c*x^2 - a*b^2)*log(c*x^2 + 1)^2 + 
3*(4*a*b^2*c*x^2 - (b^3*c*x^2 - b^3)*log(c*x^2 + 1)^2 + 2*(b^3*c^2*x^4 + 2 
*a*b^2 - (2*a*b^2*c - b^3*c)*x^2)*log(c*x^2 + 1))*log(-c*x^2 + 1))/(c*x^5 
- x^3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^3}{x^3} \, dx=\frac {-\mathit {atanh} \left (c \,x^{2}\right )^{3} b^{3}-3 \mathit {atanh} \left (c \,x^{2}\right )^{2} a \,b^{2}+3 \mathit {atanh} \left (c \,x^{2}\right ) a^{2} b c \,x^{2}-3 \mathit {atanh} \left (c \,x^{2}\right ) a^{2} b -12 \left (\int \frac {\mathit {atanh} \left (c \,x^{2}\right )}{c^{2} x^{5}-x}d x \right ) a \,b^{2} c \,x^{2}-6 \left (\int \frac {\mathit {atanh} \left (c \,x^{2}\right )^{2}}{c^{2} x^{5}-x}d x \right ) b^{3} c \,x^{2}-3 \,\mathrm {log}\left (c \,x^{2}+1\right ) a^{2} b c \,x^{2}+6 \,\mathrm {log}\left (x \right ) a^{2} b c \,x^{2}-a^{3}}{2 x^{2}} \] Input:

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

Output:

( - atanh(c*x**2)**3*b**3 - 3*atanh(c*x**2)**2*a*b**2 + 3*atanh(c*x**2)*a* 
*2*b*c*x**2 - 3*atanh(c*x**2)*a**2*b - 12*int(atanh(c*x**2)/(c**2*x**5 - x 
),x)*a*b**2*c*x**2 - 6*int(atanh(c*x**2)**2/(c**2*x**5 - x),x)*b**3*c*x**2 
 - 3*log(c*x**2 + 1)*a**2*b*c*x**2 + 6*log(x)*a**2*b*c*x**2 - a**3)/(2*x** 
2)