3.2.28 \(\int \frac {(a+b \text {arctanh}(c x^3))^3}{x^4} \, dx\) [128]

3.2.28.1 Optimal result

Integrand size = 16, antiderivative size = 120 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x^4} \, dx=\frac {1}{3} c \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 x^3}+b c \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \log \left (2-\frac {2}{1+c x^3}\right )-b^2 c \left (a+b \text {arctanh}\left (c x^3\right )\right ) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x^3}\right )-\frac {1}{2} b^3 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+c x^3}\right ) \]

output

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

3.2.28.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.86 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x^4} \, dx=-\frac {a^3}{3 x^3}-\frac {a^2 b \text {arctanh}\left (c x^3\right )}{x^3}+3 a^2 b c \log (x)-\frac {1}{2} a^2 b c \log \left (1-c^2 x^6\right )+a b^2 c \left (\text {arctanh}\left (c x^3\right ) \left (\left (1-\frac {1}{c x^3}\right ) \text {arctanh}\left (c x^3\right )+2 \log \left (1-e^{-2 \text {arctanh}\left (c x^3\right )}\right )\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (c x^3\right )}\right )\right )+\frac {1}{3} b^3 c \left (\frac {i \pi ^3}{8}-\text {arctanh}\left (c x^3\right )^3-\frac {\text {arctanh}\left (c x^3\right )^3}{c x^3}+3 \text {arctanh}\left (c x^3\right )^2 \log \left (1-e^{2 \text {arctanh}\left (c x^3\right )}\right )+3 \text {arctanh}\left (c x^3\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (c x^3\right )}\right )-\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (c x^3\right )}\right )\right ) \]

input

Integrate[(a + b*ArcTanh[c*x^3])^3/x^4,x]
 

output

-1/3*a^3/x^3 - (a^2*b*ArcTanh[c*x^3])/x^3 + 3*a^2*b*c*Log[x] - (a^2*b*c*Lo 
g[1 - c^2*x^6])/2 + a*b^2*c*(ArcTanh[c*x^3]*((1 - 1/(c*x^3))*ArcTanh[c*x^3 
] + 2*Log[1 - E^(-2*ArcTanh[c*x^3])]) - PolyLog[2, E^(-2*ArcTanh[c*x^3])]) 
 + (b^3*c*((I/8)*Pi^3 - ArcTanh[c*x^3]^3 - ArcTanh[c*x^3]^3/(c*x^3) + 3*Ar 
cTanh[c*x^3]^2*Log[1 - E^(2*ArcTanh[c*x^3])] + 3*ArcTanh[c*x^3]*PolyLog[2, 
 E^(2*ArcTanh[c*x^3])] - (3*PolyLog[3, E^(2*ArcTanh[c*x^3])])/2))/3
 

3.2.28.3 Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.11, 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.

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

\(\Big \downarrow \) 6454

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 7164

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

input

Int[(a + b*ArcTanh[c*x^3])^3/x^4,x]
 

output

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

3.2.28.3.1 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]
 

3.2.28.4 Maple [F]

\[\int \frac {{\left (a +b \,\operatorname {arctanh}\left (c \,x^{3}\right )\right )}^{3}}{x^{4}}d x\]

input

int((a+b*arctanh(c*x^3))^3/x^4,x)
 

output

int((a+b*arctanh(c*x^3))^3/x^4,x)
 

3.2.28.5 Fricas [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x^4} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3}}{x^{4}} \,d x } \]

input

integrate((a+b*arctanh(c*x^3))^3/x^4,x, algorithm="fricas")
 

output

integral((b^3*arctanh(c*x^3)^3 + 3*a*b^2*arctanh(c*x^3)^2 + 3*a^2*b*arctan 
h(c*x^3) + a^3)/x^4, x)
 

3.2.28.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x^4} \, dx=\text {Timed out} \]

input

integrate((a+b*atanh(c*x**3))**3/x**4,x)
 

output

Timed out
 

3.2.28.7 Maxima [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x^4} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3}}{x^{4}} \,d x } \]

input

integrate((a+b*arctanh(c*x^3))^3/x^4,x, algorithm="maxima")
 

output

-1/2*(c*(log(c^2*x^6 - 1) - log(x^6)) + 2*arctanh(c*x^3)/x^3)*a^2*b - 1/3* 
a^3/x^3 - 1/24*((b^3*c*x^3 - b^3)*log(-c*x^3 + 1)^3 + 3*(2*a*b^2 + (b^3*c* 
x^3 + b^3)*log(c*x^3 + 1))*log(-c*x^3 + 1)^2)/x^3 - integrate(-1/8*((b^3*c 
*x^3 - b^3)*log(c*x^3 + 1)^3 + 6*(a*b^2*c*x^3 - a*b^2)*log(c*x^3 + 1)^2 + 
3*(4*a*b^2*c*x^3 - (b^3*c*x^3 - b^3)*log(c*x^3 + 1)^2 + 2*(b^3*c^2*x^6 - ( 
2*a*b^2*c - b^3*c)*x^3 + 2*a*b^2)*log(c*x^3 + 1))*log(-c*x^3 + 1))/(c*x^7 
- x^4), x)
 

3.2.28.8 Giac [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x^4} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3}}{x^{4}} \,d x } \]

input

integrate((a+b*arctanh(c*x^3))^3/x^4,x, algorithm="giac")
 

output

integrate((b*arctanh(c*x^3) + a)^3/x^4, x)
 

3.2.28.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^3\right )\right )}^3}{x^4} \,d x \]

input

int((a + b*atanh(c*x^3))^3/x^4,x)
 

output

int((a + b*atanh(c*x^3))^3/x^4, x)