\(\int (a+b \text {arctanh}(\frac {c}{x}))^2 \, dx\) [146]

Optimal result

Integrand size = 12, antiderivative size = 74 \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-2 b c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (\frac {2 c}{c-x}\right )-b^2 c \operatorname {PolyLog}\left (2,1-\frac {2 c}{c-x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.31 \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=b^2 (-c+x) \text {arctanh}\left (\frac {c}{x}\right )^2+2 b \text {arctanh}\left (\frac {c}{x}\right ) \left (a x-b c \log \left (1-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )+a \left (a x+b c \log \left (1-\frac {c^2}{x^2}\right )-2 b c \log \left (\frac {c}{x}\right )\right )+b^2 c \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6440, 6437, 27, 6547, 27, 6471, 27, 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.

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

\(\Big \downarrow \) 6440

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

\(\Big \downarrow \) 6437

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6547

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6471

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 6437

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
 + b*ArcCoth[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcCoth[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 6440

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_), x_Symbol] :> Int[(a + b* 
ArcCoth[1/(x^n*c)])^p, x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && ILtQ[n, 0 
]
 

rule 6471

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcCoth[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcCoth[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 6547

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ 
(c*d)   Int[(a + b*ArcCoth[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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(234\) vs. \(2(74)=148\).

Time = 0.81 (sec) , antiderivative size = 235, normalized size of antiderivative = 3.18

method	result	size
parts	\(x \,a^{2}-b^{2} c \left (-\frac {x \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{c}-\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )+2 \ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )-\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )+\operatorname {dilog}\left (\frac {c}{2 x}+\frac {1}{2}\right )+\frac {\ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{2}-\frac {\ln \left (\frac {c}{x}-1\right )^{2}}{4}+\frac {\ln \left (1+\frac {c}{x}\right )^{2}}{4}-\frac {\left (\ln \left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{2 x}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right )}{2}-\operatorname {dilog}\left (\frac {c}{x}\right )-\operatorname {dilog}\left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )\right )-2 a b c \left (-\frac {x \,\operatorname {arctanh}\left (\frac {c}{x}\right )}{c}-\frac {\ln \left (\frac {c}{x}-1\right )}{2}-\frac {\ln \left (1+\frac {c}{x}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )\)	\(235\)
derivativedivides	\(-c \left (-\frac {a^{2} x}{c}+b^{2} \left (-\frac {x \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{c}-\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )+2 \ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )-\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )+\operatorname {dilog}\left (\frac {c}{2 x}+\frac {1}{2}\right )+\frac {\ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{2}-\frac {\ln \left (\frac {c}{x}-1\right )^{2}}{4}+\frac {\ln \left (1+\frac {c}{x}\right )^{2}}{4}-\frac {\left (\ln \left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{2 x}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right )}{2}-\operatorname {dilog}\left (\frac {c}{x}\right )-\operatorname {dilog}\left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )\right )+2 a b \left (-\frac {x \,\operatorname {arctanh}\left (\frac {c}{x}\right )}{c}-\frac {\ln \left (\frac {c}{x}-1\right )}{2}-\frac {\ln \left (1+\frac {c}{x}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )\right )\)	\(239\)
default	\(-c \left (-\frac {a^{2} x}{c}+b^{2} \left (-\frac {x \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{c}-\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )+2 \ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )-\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )+\operatorname {dilog}\left (\frac {c}{2 x}+\frac {1}{2}\right )+\frac {\ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{2}-\frac {\ln \left (\frac {c}{x}-1\right )^{2}}{4}+\frac {\ln \left (1+\frac {c}{x}\right )^{2}}{4}-\frac {\left (\ln \left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{2 x}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right )}{2}-\operatorname {dilog}\left (\frac {c}{x}\right )-\operatorname {dilog}\left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )\right )+2 a b \left (-\frac {x \,\operatorname {arctanh}\left (\frac {c}{x}\right )}{c}-\frac {\ln \left (\frac {c}{x}-1\right )}{2}-\frac {\ln \left (1+\frac {c}{x}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )\right )\)	\(239\)
risch	\(\text {Expression too large to display}\)	\(4119\)

Input:

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

Output:

x*a^2-b^2*c*(-1/c*x*arctanh(c/x)^2-arctanh(c/x)*ln(c/x-1)+2*ln(c/x)*arctan 
h(c/x)-arctanh(c/x)*ln(1+c/x)+dilog(1/2*c/x+1/2)+1/2*ln(c/x-1)*ln(1/2*c/x+ 
1/2)-1/4*ln(c/x-1)^2+1/4*ln(1+c/x)^2-1/2*(ln(1+c/x)-ln(1/2*c/x+1/2))*ln(-1 
/2*c/x+1/2)-dilog(c/x)-dilog(1+c/x)-ln(c/x)*ln(1+c/x))-2*a*b*c*(-1/c*x*arc 
tanh(c/x)-1/2*ln(c/x-1)-1/2*ln(1+c/x)+ln(c/x))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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