\(\int \frac {(a+b \text {arctanh}(\frac {c}{x}))^3}{x^2} \, dx\) [155]

Optimal result

Integrand size = 16, antiderivative size = 126 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{x^2} \, dx=-\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3}{c}-\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3}{x}+\frac {3 b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \log \left (\frac {2}{1-\frac {c}{x}}\right )}{c}+\frac {3 b^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x}}\right )}{c}-\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-\frac {c}{x}}\right )}{2 c} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{x^2} \, dx=-\frac {a^3}{x}-\frac {3 a^2 b \text {arctanh}\left (\frac {c}{x}\right )}{x}-\frac {3 a^2 b \log \left (1-\frac {c^2}{x^2}\right )}{2 c}-\frac {3 a b^2 \left (\text {arctanh}\left (\frac {c}{x}\right ) \left (-\text {arctanh}\left (\frac {c}{x}\right )+\frac {c \text {arctanh}\left (\frac {c}{x}\right )}{x}-2 \log \left (1+e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )}{c}-\frac {b^3 \left (\text {arctanh}\left (\frac {c}{x}\right )^2 \left (-\text {arctanh}\left (\frac {c}{x}\right )+\frac {c \text {arctanh}\left (\frac {c}{x}\right )}{x}-3 \log \left (1+e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )+3 \text {arctanh}\left (\frac {c}{x}\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )}{c} \] 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*Log[1 - c^2/x^2])/(2*c) - ( 
3*a*b^2*(ArcTanh[c/x]*(-ArcTanh[c/x] + (c*ArcTanh[c/x])/x - 2*Log[1 + E^(- 
2*ArcTanh[c/x])]) + PolyLog[2, -E^(-2*ArcTanh[c/x])]))/c - (b^3*(ArcTanh[c 
/x]^2*(-ArcTanh[c/x] + (c*ArcTanh[c/x])/x - 3*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))/c
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 140, 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, 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.

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

\(\Big \downarrow \) 6454

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

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

-((a + b*ArcTanh[c/x])^3/x) + 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 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 = 2.18 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.10

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

Input:

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

Output:

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

Fricas [F]

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

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Reduce [F]

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

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

Output:

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