\(\int \frac {(a+b \text {arctanh}(c x^{3/2}))^2}{x} \, dx\) [221]

Optimal result

Integrand size = 18, antiderivative size = 156 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2}{x} \, dx=\frac {4}{3} \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2 \text {arctanh}\left (1-\frac {2}{1-c x^{3/2}}\right )-\frac {2}{3} b \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^{3/2}}\right )+\frac {2}{3} b \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x^{3/2}}\right )+\frac {1}{3} b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^{3/2}}\right )-\frac {1}{3} b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x^{3/2}}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2}{x} \, dx=a^2 \log (x)+\frac {2}{3} a b \left (-\operatorname {PolyLog}\left (2,-c x^{3/2}\right )+\operatorname {PolyLog}\left (2,c x^{3/2}\right )\right )+\frac {2}{3} b^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}\left (c x^{3/2}\right )^3-\text {arctanh}\left (c x^{3/2}\right )^2 \log \left (1+e^{-2 \text {arctanh}\left (c x^{3/2}\right )}\right )+\text {arctanh}\left (c x^{3/2}\right )^2 \log \left (1-e^{2 \text {arctanh}\left (c x^{3/2}\right )}\right )+\text {arctanh}\left (c x^{3/2}\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^{3/2}\right )}\right )+\text {arctanh}\left (c x^{3/2}\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (c x^{3/2}\right )}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (c x^{3/2}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (c x^{3/2}\right )}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, 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.

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

\(\Big \downarrow \) 6450

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

\(\Big \downarrow \) 6448

\(\displaystyle \frac {2}{3} \left (2 \text {arctanh}\left (1-\frac {2}{1-c x^{3/2}}\right ) \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2-4 b c \int \frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \text {arctanh}\left (1-\frac {2}{1-c x^{3/2}}\right )}{1-c^2 x^3}dx^{3/2}\right )\)

\(\Big \downarrow \) 6614

\(\displaystyle \frac {2}{3} \left (2 \text {arctanh}\left (1-\frac {2}{1-c x^{3/2}}\right ) \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2-4 b c \left (\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \log \left (2-\frac {2}{1-c x^{3/2}}\right )}{1-c^2 x^3}dx^{3/2}-\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \log \left (\frac {2}{1-c x^{3/2}}\right )}{1-c^2 x^3}dx^{3/2}\right )\right )\)

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

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 [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 18.05 (sec) , antiderivative size = 706, normalized size of antiderivative = 4.53

method	result	size
parts	\(a^{2} \ln \left (x \right )+b^{2} \left (\frac {2 \ln \left (c \,x^{\frac {3}{2}}\right ) \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2}}{3}-\frac {2 \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \operatorname {polylog}\left (2, -\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{-c^{2} x^{3}+1}\right )}{3}+\frac {\operatorname {polylog}\left (3, -\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{-c^{2} x^{3}+1}\right )}{3}-\frac {2 \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2} \ln \left (\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{-c^{2} x^{3}+1}-1\right )}{3}+\frac {2 \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2} \ln \left (1-\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )}{3}+\frac {4 \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \operatorname {polylog}\left (2, \frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )}{3}-\frac {4 \operatorname {polylog}\left (3, \frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )}{3}+\frac {2 \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2} \ln \left (1+\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )}{3}+\frac {4 \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \operatorname {polylog}\left (2, -\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )}{3}-\frac {4 \operatorname {polylog}\left (3, -\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )}{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right )}^{2}\right ) \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2}}{3}\right )+2 a b \left (\frac {2 \ln \left (c \,x^{\frac {3}{2}}\right ) \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{3}-\frac {\operatorname {dilog}\left (c \,x^{\frac {3}{2}}\right )}{3}-\frac {\operatorname {dilog}\left (c \,x^{\frac {3}{2}}+1\right )}{3}-\frac {\ln \left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{3}\right )\)	\(706\)
derivativedivides	\(\frac {2 a^{2} \ln \left (c \,x^{\frac {3}{2}}\right )}{3}+\frac {2 b^{2} \left (\ln \left (c \,x^{\frac {3}{2}}\right ) \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2}-\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \operatorname {polylog}\left (2, -\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{-c^{2} x^{3}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{-c^{2} x^{3}+1}\right )}{2}-\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2} \ln \left (\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{-c^{2} x^{3}+1}-1\right )+\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2} \ln \left (1+\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \operatorname {polylog}\left (2, -\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )-2 \operatorname {polylog}\left (3, -\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )+\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2} \ln \left (1-\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \operatorname {polylog}\left (2, \frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right )}^{2}\right ) \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2}}{2}\right )}{3}+\frac {4 a b \left (\ln \left (c \,x^{\frac {3}{2}}\right ) \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )-\frac {\operatorname {dilog}\left (c \,x^{\frac {3}{2}}+1\right )}{2}-\frac {\ln \left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{2}-\frac {\operatorname {dilog}\left (c \,x^{\frac {3}{2}}\right )}{2}\right )}{3}\)	\(708\)
default	\(\frac {2 a^{2} \ln \left (c \,x^{\frac {3}{2}}\right )}{3}+\frac {2 b^{2} \left (\ln \left (c \,x^{\frac {3}{2}}\right ) \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2}-\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \operatorname {polylog}\left (2, -\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{-c^{2} x^{3}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{-c^{2} x^{3}+1}\right )}{2}-\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2} \ln \left (\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{-c^{2} x^{3}+1}-1\right )+\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2} \ln \left (1+\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \operatorname {polylog}\left (2, -\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )-2 \operatorname {polylog}\left (3, -\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )+\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2} \ln \left (1-\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \operatorname {polylog}\left (2, \frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-c^{2} x^{3}+1}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{c^{2} x^{3}-1}}\right )}^{2}\right ) \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2}}{2}\right )}{3}+\frac {4 a b \left (\ln \left (c \,x^{\frac {3}{2}}\right ) \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )-\frac {\operatorname {dilog}\left (c \,x^{\frac {3}{2}}+1\right )}{2}-\frac {\ln \left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{2}-\frac {\operatorname {dilog}\left (c \,x^{\frac {3}{2}}\right )}{2}\right )}{3}\)	\(708\)

Input:

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

Output:

a^2*ln(x)+b^2*(2/3*ln(c*x^(3/2))*arctanh(c*x^(3/2))^2-2/3*arctanh(c*x^(3/2 
))*polylog(2,-(c*x^(3/2)+1)^2/(-c^2*x^3+1))+1/3*polylog(3,-(c*x^(3/2)+1)^2 
/(-c^2*x^3+1))-2/3*arctanh(c*x^(3/2))^2*ln((c*x^(3/2)+1)^2/(-c^2*x^3+1)-1) 
+2/3*arctanh(c*x^(3/2))^2*ln(1-(c*x^(3/2)+1)/(-c^2*x^3+1)^(1/2))+4/3*arcta 
nh(c*x^(3/2))*polylog(2,(c*x^(3/2)+1)/(-c^2*x^3+1)^(1/2))-4/3*polylog(3,(c 
*x^(3/2)+1)/(-c^2*x^3+1)^(1/2))+2/3*arctanh(c*x^(3/2))^2*ln(1+(c*x^(3/2)+1 
)/(-c^2*x^3+1)^(1/2))+4/3*arctanh(c*x^(3/2))*polylog(2,-(c*x^(3/2)+1)/(-c^ 
2*x^3+1)^(1/2))-4/3*polylog(3,-(c*x^(3/2)+1)/(-c^2*x^3+1)^(1/2))+1/3*I*Pi* 
csgn(I*(-(c*x^(3/2)+1)^2/(c^2*x^3-1)-1)/(1-(c*x^(3/2)+1)^2/(c^2*x^3-1)))*( 
csgn(I*(-(c*x^(3/2)+1)^2/(c^2*x^3-1)-1))*csgn(I/(1-(c*x^(3/2)+1)^2/(c^2*x^ 
3-1)))-csgn(I*(-(c*x^(3/2)+1)^2/(c^2*x^3-1)-1))*csgn(I*(-(c*x^(3/2)+1)^2/( 
c^2*x^3-1)-1)/(1-(c*x^(3/2)+1)^2/(c^2*x^3-1)))-csgn(I*(-(c*x^(3/2)+1)^2/(c 
^2*x^3-1)-1)/(1-(c*x^(3/2)+1)^2/(c^2*x^3-1)))*csgn(I/(1-(c*x^(3/2)+1)^2/(c 
^2*x^3-1)))+csgn(I*(-(c*x^(3/2)+1)^2/(c^2*x^3-1)-1)/(1-(c*x^(3/2)+1)^2/(c^ 
2*x^3-1)))^2)*arctanh(c*x^(3/2))^2)+2*a*b*(2/3*ln(c*x^(3/2))*arctanh(c*x^( 
3/2))-1/3*dilog(c*x^(3/2))-1/3*dilog(c*x^(3/2)+1)-1/3*ln(c*x^(3/2))*ln(c*x 
^(3/2)+1))
 

Fricas [F]

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

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

Output:

integral((b^2*arctanh(c*x^(3/2))^2 + 2*a*b*arctanh(c*x^(3/2)) + a^2)/x, x)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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