\(\int \frac {(a+b \text {arctanh}(c x^2))^2}{x} \, dx\) [68]

Optimal result

Integrand size = 16, antiderivative size = 137 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x} \, dx=\left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \text {arctanh}\left (1-\frac {2}{1-c x^2}\right )-\frac {1}{2} b \left (a+b \text {arctanh}\left (c x^2\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^2}\right )+\frac {1}{2} b \left (a+b \text {arctanh}\left (c x^2\right )\right ) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x^2}\right )+\frac {1}{4} b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^2}\right )-\frac {1}{4} b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x^2}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x} \, dx=a^2 \log (x)+\frac {1}{2} a b \left (-\operatorname {PolyLog}\left (2,-c x^2\right )+\operatorname {PolyLog}\left (2,c x^2\right )\right )+\frac {1}{2} b^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}\left (c x^2\right )^3-\text {arctanh}\left (c x^2\right )^2 \log \left (1+e^{-2 \text {arctanh}\left (c x^2\right )}\right )+\text {arctanh}\left (c x^2\right )^2 \log \left (1-e^{2 \text {arctanh}\left (c x^2\right )}\right )+\text {arctanh}\left (c x^2\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^2\right )}\right )+\text {arctanh}\left (c x^2\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (c x^2\right )}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (c x^2\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (c x^2\right )}\right )\right ) \] 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*(( 
I/24)*Pi^3 - (2*ArcTanh[c*x^2]^3)/3 - ArcTanh[c*x^2]^2*Log[1 + E^(-2*ArcTa 
nh[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*Ar 
cTanh[c*x^2])] + PolyLog[3, -E^(-2*ArcTanh[c*x^2])]/2 - PolyLog[3, E^(2*Ar 
cTanh[c*x^2])]/2))/2
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.19, 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.

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

\(\Big \downarrow \) 6450

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

\(\Big \downarrow \) 6448

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

\(\Big \downarrow \) 6614

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

(2*(a + b*ArcTanh[c*x^2])^2*ArcTanh[1 - 2/(1 - c*x^2)] - 4*b*c*((((a + b*A 
rcTanh[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
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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