3.2.73 \(\int \frac {(a+b \text {arctanh}(\frac {c}{x^2}))^2}{x} \, dx\) [173]

3.2.73.1 Optimal result

Integrand size = 16, antiderivative size = 144 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2}{x} \, dx=-\left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right )^2 \text {arctanh}\left (1-\frac {2}{1-\frac {c}{x^2}}\right )+\frac {1}{2} b \left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x^2}}\right )-\frac {1}{2} b \left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right ) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-\frac {c}{x^2}}\right )-\frac {1}{4} b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-\frac {c}{x^2}}\right )+\frac {1}{4} b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-\frac {c}{x^2}}\right ) \]

output

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

3.2.73.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2}{x} \, dx=a^2 \log (x)+\frac {1}{2} a b \left (\operatorname {PolyLog}\left (2,-\frac {c}{x^2}\right )-\operatorname {PolyLog}\left (2,\frac {c}{x^2}\right )\right )+\frac {1}{2} b^2 \left (-\frac {i \pi ^3}{24}+\frac {2}{3} \text {arctanh}\left (\frac {c}{x^2}\right )^3+\text {arctanh}\left (\frac {c}{x^2}\right )^2 \log \left (1+e^{-2 \text {arctanh}\left (\frac {c}{x^2}\right )}\right )-\text {arctanh}\left (\frac {c}{x^2}\right )^2 \log \left (1-e^{2 \text {arctanh}\left (\frac {c}{x^2}\right )}\right )-\text {arctanh}\left (\frac {c}{x^2}\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (\frac {c}{x^2}\right )}\right )-\text {arctanh}\left (\frac {c}{x^2}\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (\frac {c}{x^2}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (\frac {c}{x^2}\right )}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (\frac {c}{x^2}\right )}\right )\right ) \]

input

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

output

a^2*Log[x] + (a*b*(PolyLog[2, -(c/x^2)] - PolyLog[2, c/x^2]))/2 + (b^2*((- 
1/24*I)*Pi^3 + (2*ArcTanh[c/x^2]^3)/3 + ArcTanh[c/x^2]^2*Log[1 + E^(-2*Arc 
Tanh[c/x^2])] - ArcTanh[c/x^2]^2*Log[1 - E^(2*ArcTanh[c/x^2])] - ArcTanh[c 
/x^2]*PolyLog[2, -E^(-2*ArcTanh[c/x^2])] - ArcTanh[c/x^2]*PolyLog[2, E^(2* 
ArcTanh[c/x^2])] - PolyLog[3, -E^(-2*ArcTanh[c/x^2])]/2 + PolyLog[3, E^(2* 
ArcTanh[c/x^2])]/2))/2
 

3.2.73.3 Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6450, 6448, 6614, 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.

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

\(\Big \downarrow \) 6450

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

\(\Big \downarrow \) 6448

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

\(\Big \downarrow \) 6614

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 7164

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

input

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

output

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

3.2.73.3.1 Defintions of rubi rules used

rule 6448

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + 
 b*ArcTanh[c*x])^p*ArcTanh[1 - 2/(1 - c*x)], x] - Simp[2*b*c*p   Int[(a + b 
*ArcTanh[c*x])^(p - 1)*(ArcTanh[1 - 2/(1 - c*x)]/(1 - c^2*x^2)), x], x] /; 
FreeQ[{a, b, c}, x] && IGtQ[p, 1]
 

rule 6450

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

rule 6614

Int[(ArcTanh[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*( 
x_)^2), x_Symbol] :> Simp[1/2   Int[Log[1 + u]*((a + b*ArcTanh[c*x])^p/(d + 
 e*x^2)), x], x] - Simp[1/2   Int[Log[1 - u]*((a + b*ArcTanh[c*x])^p/(d + e 
*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 
 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 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]
 

3.2.73.4 Maple [F]

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

input

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

output

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

3.2.73.5 Fricas [F]

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

input

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

output

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

3.2.73.6 Sympy [F]

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

input

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

output

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

3.2.73.7 Maxima [F]

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

input

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

output

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

3.2.73.8 Giac [F]

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

input

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

output

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

3.2.73.9 Mupad [F(-1)]

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

input

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

output

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