\(\int \frac {(a+b \text {arctanh}(\frac {c}{x}))^2}{x^2} \, dx\) [148]

Optimal result

Integrand size = 16, antiderivative size = 87 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x^2} \, dx=-\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2}{c}-\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2}{x}+\frac {2 b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (\frac {2}{1-\frac {c}{x}}\right )}{c}+\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x}}\right )}{c} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x^2} \, dx=\frac {b^2 (-c+x) \text {arctanh}\left (\frac {c}{x}\right )^2+2 b \text {arctanh}\left (\frac {c}{x}\right ) \left (-a c+b x \log \left (1+e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )+a \left (-a c+2 b x \log \left (\frac {1}{\sqrt {1-\frac {c^2}{x^2}}}\right )\right )-b^2 x \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )}{c x} \] Input:

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

Output:

(b^2*(-c + x)*ArcTanh[c/x]^2 + 2*b*ArcTanh[c/x]*(-(a*c) + b*x*Log[1 + E^(- 
2*ArcTanh[c/x])]) + a*(-(a*c) + 2*b*x*Log[1/Sqrt[1 - c^2/x^2]]) - b^2*x*Po 
lyLog[2, -E^(-2*ArcTanh[c/x])])/(c*x)
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.16, 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, 2849, 2752}

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.

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

\(\Big \downarrow \) 6454

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

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

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]
 

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.54

method	result	size
derivativedivides	\(-\frac {\frac {c \,a^{2}}{x}+b^{2} \left (\operatorname {arctanh}\left (\frac {c}{x}\right )^{2} \left (\frac {c}{x}-1\right )+2 \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}-2 \,\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )-\operatorname {polylog}\left (2, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )\right )+\frac {2 a b c \,\operatorname {arctanh}\left (\frac {c}{x}\right )}{x}+a b \ln \left (1-\frac {c^{2}}{x^{2}}\right )}{c}\)	\(134\)
default	\(-\frac {\frac {c \,a^{2}}{x}+b^{2} \left (\operatorname {arctanh}\left (\frac {c}{x}\right )^{2} \left (\frac {c}{x}-1\right )+2 \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}-2 \,\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )-\operatorname {polylog}\left (2, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )\right )+\frac {2 a b c \,\operatorname {arctanh}\left (\frac {c}{x}\right )}{x}+a b \ln \left (1-\frac {c^{2}}{x^{2}}\right )}{c}\)	\(134\)
parts	\(-\frac {a^{2}}{x}-\frac {b^{2} \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{x}-\frac {b^{2} \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{c}+\frac {2 b^{2} \operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )}{c}+\frac {b^{2} \operatorname {polylog}\left (2, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )}{c}-\frac {2 a b \,\operatorname {arctanh}\left (\frac {c}{x}\right )}{x}-\frac {a b \ln \left (1-\frac {c^{2}}{x^{2}}\right )}{c}\)	\(144\)

Input:

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

Output:

-1/c*(c/x*a^2+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)))+2*a*b*c/x*a 
rctanh(c/x)+a*b*ln(1-c^2/x^2))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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