Integrand size = 18, antiderivative size = 156 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2}{x} \, dx=\frac {4}{3} \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2 \text {arctanh}\left (1-\frac {2}{1-c x^{3/2}}\right )-\frac {2}{3} b \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^{3/2}}\right )+\frac {2}{3} b \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x^{3/2}}\right )+\frac {1}{3} b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^{3/2}}\right )-\frac {1}{3} b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x^{3/2}}\right ) \]
-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)))
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2}{x} \, dx=a^2 \log (x)+\frac {2}{3} a b \left (-\operatorname {PolyLog}\left (2,-c x^{3/2}\right )+\operatorname {PolyLog}\left (2,c x^{3/2}\right )\right )+\frac {2}{3} b^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}\left (c x^{3/2}\right )^3-\text {arctanh}\left (c x^{3/2}\right )^2 \log \left (1+e^{-2 \text {arctanh}\left (c x^{3/2}\right )}\right )+\text {arctanh}\left (c x^{3/2}\right )^2 \log \left (1-e^{2 \text {arctanh}\left (c x^{3/2}\right )}\right )+\text {arctanh}\left (c x^{3/2}\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^{3/2}\right )}\right )+\text {arctanh}\left (c x^{3/2}\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (c x^{3/2}\right )}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (c x^{3/2}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (c x^{3/2}\right )}\right )\right ) \]
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
Time = 0.81 (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 |
| \(\displaystyle \frac {2}{3} \left (2 \text {arctanh}\left (1-\frac {2}{1-c x^{3/2}}\right ) \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2-4 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^{3/2}}\right ) \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )}{2 c}-\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^{3/2}}\right )}{4 c}\right )+\frac {1}{2} \left (\frac {b \operatorname {PolyLog}\left (3,\frac {2}{1-c x^{3/2}}-1\right )}{4 c}-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c x^{3/2}}-1\right ) \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )}{2 c}\right )\right )\right )\) |
(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
3.3.22.3.1 Defintions of rubi rules used
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]
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]
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]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 31.97 (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}}{-x^{3} c^{2}+1}\right )}{3}+\frac {\operatorname {polylog}\left (3, -\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{-x^{3} c^{2}+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}}{-x^{3} c^{2}+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 {-x^{3} c^{2}+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 {-x^{3} c^{2}+1}}\right )}{3}-\frac {4 \operatorname {polylog}\left (3, -\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+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 {-x^{3} c^{2}+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 {-x^{3} c^{2}+1}}\right )}{3}-\frac {4 \operatorname {polylog}\left (3, \frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+1}}\right )}{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-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}}+1\right )}{3}-\frac {\ln \left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{3}-\frac {\operatorname {dilog}\left (c \,x^{\frac {3}{2}}\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}}{-x^{3} c^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{-x^{3} c^{2}+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}}{-x^{3} c^{2}+1}-1\right )+\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2} \ln \left (1-\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \operatorname {polylog}\left (2, \frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+1}}\right )+\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2} \ln \left (1+\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \operatorname {polylog}\left (2, -\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+1}}\right )-2 \operatorname {polylog}\left (3, -\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+1}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-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}}{-x^{3} c^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{-x^{3} c^{2}+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}}{-x^{3} c^{2}+1}-1\right )+\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2} \ln \left (1-\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \operatorname {polylog}\left (2, \frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+1}}\right )+\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2} \ln \left (1+\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+1}}\right )+2 \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \operatorname {polylog}\left (2, -\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+1}}\right )-2 \operatorname {polylog}\left (3, -\frac {c \,x^{\frac {3}{2}}+1}{\sqrt {-x^{3} c^{2}+1}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-1}-1\right )}{1-\frac {\left (c \,x^{\frac {3}{2}}+1\right )^{2}}{x^{3} c^{2}-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}\) | \(708\) |
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)-1/3*ln(c*x^(3/2))*ln(c*x^(3/2)+1)-1/3*dilog(c *x^(3/2)))
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + a\right )}^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + a\right )}^{2}}{x} \,d x } \]
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)
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + a\right )}^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^{3/2}\right )\right )}^2}{x} \,d x \]