Integrand size = 18, antiderivative size = 145 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x} \, dx=4 \text {arctanh}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )+2 b \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c \sqrt {x}}\right )+b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c \sqrt {x}}\right )-b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c \sqrt {x}}\right ) \] Output:
-4*arctanh(-1+2/(1-c*x^(1/2)))*(a+b*arctanh(c*x^(1/2)))^2-2*b*(a+b*arctanh (c*x^(1/2)))*polylog(2,1-2/(1-c*x^(1/2)))+2*b*(a+b*arctanh(c*x^(1/2)))*pol ylog(2,-1+2/(1-c*x^(1/2)))+b^2*polylog(3,1-2/(1-c*x^(1/2)))-b^2*polylog(3, -1+2/(1-c*x^(1/2)))
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x} \, dx=a^2 \log (x)+2 a b \left (-\operatorname {PolyLog}\left (2,-c \sqrt {x}\right )+\operatorname {PolyLog}\left (2,c \sqrt {x}\right )\right )+2 b^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}\left (c \sqrt {x}\right )^3-\text {arctanh}\left (c \sqrt {x}\right )^2 \log \left (1+e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+\text {arctanh}\left (c \sqrt {x}\right )^2 \log \left (1-e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+\text {arctanh}\left (c \sqrt {x}\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+\text {arctanh}\left (c \sqrt {x}\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right ) \] Input:
Integrate[(a + b*ArcTanh[c*Sqrt[x]])^2/x,x]
Output:
a^2*Log[x] + 2*a*b*(-PolyLog[2, -(c*Sqrt[x])] + PolyLog[2, c*Sqrt[x]]) + 2 *b^2*((I/24)*Pi^3 - (2*ArcTanh[c*Sqrt[x]]^3)/3 - ArcTanh[c*Sqrt[x]]^2*Log[ 1 + E^(-2*ArcTanh[c*Sqrt[x]])] + ArcTanh[c*Sqrt[x]]^2*Log[1 - E^(2*ArcTanh [c*Sqrt[x]])] + ArcTanh[c*Sqrt[x]]*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])] + ArcTanh[c*Sqrt[x]]*PolyLog[2, E^(2*ArcTanh[c*Sqrt[x]])] + PolyLog[3, -E^ (-2*ArcTanh[c*Sqrt[x]])]/2 - PolyLog[3, E^(2*ArcTanh[c*Sqrt[x]])]/2)
Time = 0.85 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.22, 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.
| \(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x} \, dx\) |
| \(\Big \downarrow \) 6450 |
| \(\displaystyle 2 \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{\sqrt {x}}d\sqrt {x}\) |
| \(\Big \downarrow \) 6448 |
| \(\displaystyle 2 \left (2 \text {arctanh}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-4 b c \int \frac {\text {arctanh}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}\right )\) |
| \(\Big \downarrow \) 6614 |
| \(\displaystyle 2 \left (2 \text {arctanh}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-4 b c \left (\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (2-\frac {2}{1-c \sqrt {x}}\right )}{1-c^2 x}d\sqrt {x}-\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-c^2 x}d\sqrt {x}\right )\right )\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle 2 \left (2 \text {arctanh}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-4 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 c}-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{1-c^2 x}d\sqrt {x}\right )+\frac {1}{2} \left (\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c \sqrt {x}}-1\right )}{1-c^2 x}d\sqrt {x}-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c \sqrt {x}}-1\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 c}\right )\right )\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle 2 \left (2 \text {arctanh}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-4 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 c}-\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{1-c \sqrt {x}}\right )}{4 c}\right )+\frac {1}{2} \left (\frac {b \operatorname {PolyLog}\left (3,\frac {2}{1-c \sqrt {x}}-1\right )}{4 c}-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c \sqrt {x}}-1\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 c}\right )\right )\right )\) |
Input:
Int[(a + b*ArcTanh[c*Sqrt[x]])^2/x,x]
Output:
2*(2*ArcTanh[1 - 2/(1 - c*Sqrt[x])]*(a + b*ArcTanh[c*Sqrt[x]])^2 - 4*b*c*( (((a + b*ArcTanh[c*Sqrt[x]])*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/(2*c) - (b *PolyLog[3, 1 - 2/(1 - c*Sqrt[x])])/(4*c))/2 + (-1/2*((a + b*ArcTanh[c*Sqr t[x]])*PolyLog[2, -1 + 2/(1 - c*Sqrt[x])])/c + (b*PolyLog[3, -1 + 2/(1 - c *Sqrt[x])])/(4*c))/2))
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 = 12.20 (sec) , antiderivative size = 662, normalized size of antiderivative = 4.57
| method | result | size |
| parts | \(a^{2} \ln \left (x \right )+b^{2} \left (2 \ln \left (c \sqrt {x}\right ) \operatorname {arctanh}\left (c \sqrt {x}\right )^{2}-2 \,\operatorname {arctanh}\left (c \sqrt {x}\right ) \operatorname {polylog}\left (2, -\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}\right )+\operatorname {polylog}\left (3, -\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}\right )-2 \operatorname {arctanh}\left (c \sqrt {x}\right )^{2} \ln \left (\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}-1\right )+2 \operatorname {arctanh}\left (c \sqrt {x}\right )^{2} \ln \left (1-\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+4 \,\operatorname {arctanh}\left (c \sqrt {x}\right ) \operatorname {polylog}\left (2, \frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )-4 \operatorname {polylog}\left (3, \frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+2 \operatorname {arctanh}\left (c \sqrt {x}\right )^{2} \ln \left (1+\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+4 \,\operatorname {arctanh}\left (c \sqrt {x}\right ) \operatorname {polylog}\left (2, -\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )-4 \operatorname {polylog}\left (3, -\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right )}^{2}\right ) \operatorname {arctanh}\left (c \sqrt {x}\right )^{2}\right )+2 a b \left (2 \ln \left (c \sqrt {x}\right ) \operatorname {arctanh}\left (c \sqrt {x}\right )-\operatorname {dilog}\left (c \sqrt {x}\right )-\operatorname {dilog}\left (1+c \sqrt {x}\right )-\ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )\right )\) | \(662\) |
| derivativedivides | \(2 a^{2} \ln \left (c \sqrt {x}\right )+2 b^{2} \left (\ln \left (c \sqrt {x}\right ) \operatorname {arctanh}\left (c \sqrt {x}\right )^{2}-\operatorname {arctanh}\left (c \sqrt {x}\right ) \operatorname {polylog}\left (2, -\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}\right )}{2}-\operatorname {arctanh}\left (c \sqrt {x}\right )^{2} \ln \left (\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}-1\right )+\operatorname {arctanh}\left (c \sqrt {x}\right )^{2} \ln \left (1+\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+2 \,\operatorname {arctanh}\left (c \sqrt {x}\right ) \operatorname {polylog}\left (2, -\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )-2 \operatorname {polylog}\left (3, -\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+\operatorname {arctanh}\left (c \sqrt {x}\right )^{2} \ln \left (1-\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+2 \,\operatorname {arctanh}\left (c \sqrt {x}\right ) \operatorname {polylog}\left (2, \frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )-2 \operatorname {polylog}\left (3, \frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right )}^{2}\right ) \operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{2}\right )+4 a b \left (\ln \left (c \sqrt {x}\right ) \operatorname {arctanh}\left (c \sqrt {x}\right )-\frac {\operatorname {dilog}\left (c \sqrt {x}\right )}{2}-\frac {\operatorname {dilog}\left (1+c \sqrt {x}\right )}{2}-\frac {\ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}\right )\) | \(666\) |
| default | \(2 a^{2} \ln \left (c \sqrt {x}\right )+2 b^{2} \left (\ln \left (c \sqrt {x}\right ) \operatorname {arctanh}\left (c \sqrt {x}\right )^{2}-\operatorname {arctanh}\left (c \sqrt {x}\right ) \operatorname {polylog}\left (2, -\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}\right )}{2}-\operatorname {arctanh}\left (c \sqrt {x}\right )^{2} \ln \left (\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}-1\right )+\operatorname {arctanh}\left (c \sqrt {x}\right )^{2} \ln \left (1+\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+2 \,\operatorname {arctanh}\left (c \sqrt {x}\right ) \operatorname {polylog}\left (2, -\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )-2 \operatorname {polylog}\left (3, -\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+\operatorname {arctanh}\left (c \sqrt {x}\right )^{2} \ln \left (1-\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+2 \,\operatorname {arctanh}\left (c \sqrt {x}\right ) \operatorname {polylog}\left (2, \frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )-2 \operatorname {polylog}\left (3, \frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}-1\right )}{1-\frac {\left (1+c \sqrt {x}\right )^{2}}{c^{2} x -1}}\right )}^{2}\right ) \operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{2}\right )+4 a b \left (\ln \left (c \sqrt {x}\right ) \operatorname {arctanh}\left (c \sqrt {x}\right )-\frac {\operatorname {dilog}\left (c \sqrt {x}\right )}{2}-\frac {\operatorname {dilog}\left (1+c \sqrt {x}\right )}{2}-\frac {\ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}\right )\) | \(666\) |
Input:
int((a+b*arctanh(c*x^(1/2)))^2/x,x,method=_RETURNVERBOSE)
Output:
a^2*ln(x)+b^2*(2*ln(c*x^(1/2))*arctanh(c*x^(1/2))^2-2*arctanh(c*x^(1/2))*p olylog(2,-(1+c*x^(1/2))^2/(-c^2*x+1))+polylog(3,-(1+c*x^(1/2))^2/(-c^2*x+1 ))-2*arctanh(c*x^(1/2))^2*ln((1+c*x^(1/2))^2/(-c^2*x+1)-1)+2*arctanh(c*x^( 1/2))^2*ln(1-(1+c*x^(1/2))/(-c^2*x+1)^(1/2))+4*arctanh(c*x^(1/2))*polylog( 2,(1+c*x^(1/2))/(-c^2*x+1)^(1/2))-4*polylog(3,(1+c*x^(1/2))/(-c^2*x+1)^(1/ 2))+2*arctanh(c*x^(1/2))^2*ln(1+(1+c*x^(1/2))/(-c^2*x+1)^(1/2))+4*arctanh( c*x^(1/2))*polylog(2,-(1+c*x^(1/2))/(-c^2*x+1)^(1/2))-4*polylog(3,-(1+c*x^ (1/2))/(-c^2*x+1)^(1/2))+I*Pi*csgn(I*(-(1+c*x^(1/2))^2/(c^2*x-1)-1)/(1-(1+ c*x^(1/2))^2/(c^2*x-1)))*(csgn(I*(-(1+c*x^(1/2))^2/(c^2*x-1)-1))*csgn(I/(1 -(1+c*x^(1/2))^2/(c^2*x-1)))-csgn(I*(-(1+c*x^(1/2))^2/(c^2*x-1)-1))*csgn(I *(-(1+c*x^(1/2))^2/(c^2*x-1)-1)/(1-(1+c*x^(1/2))^2/(c^2*x-1)))-csgn(I*(-(1 +c*x^(1/2))^2/(c^2*x-1)-1)/(1-(1+c*x^(1/2))^2/(c^2*x-1)))*csgn(I/(1-(1+c*x ^(1/2))^2/(c^2*x-1)))+csgn(I*(-(1+c*x^(1/2))^2/(c^2*x-1)-1)/(1-(1+c*x^(1/2 ))^2/(c^2*x-1)))^2)*arctanh(c*x^(1/2))^2)+2*a*b*(2*ln(c*x^(1/2))*arctanh(c *x^(1/2))-dilog(c*x^(1/2))-dilog(1+c*x^(1/2))-ln(c*x^(1/2))*ln(1+c*x^(1/2) ))
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{2}}{x} \,d x } \] Input:
integrate((a+b*arctanh(c*x^(1/2)))^2/x,x, algorithm="fricas")
Output:
integral((b^2*arctanh(c*sqrt(x))^2 + 2*a*b*arctanh(c*sqrt(x)) + a^2)/x, x)
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{2}}{x}\, dx \] Input:
integrate((a+b*atanh(c*x**(1/2)))**2/x,x)
Output:
Integral((a + b*atanh(c*sqrt(x)))**2/x, x)
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{2}}{x} \,d x } \] Input:
integrate((a+b*arctanh(c*x^(1/2)))^2/x,x, algorithm="maxima")
Output:
1/4*b^2*integrate(log(c*sqrt(x) + 1)^2/x, x) - 1/2*b^2*integrate(log(c*sqr t(x) + 1)*log(-c*sqrt(x) + 1)/x, x) + 1/4*b^2*integrate(log(-c*sqrt(x) + 1 )^2/x, x) + a*b*integrate(log(c*sqrt(x) + 1)/x, x) - a*b*integrate(log(-c* sqrt(x) + 1)/x, x) + a^2*log(x)
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{2}}{x} \,d x } \] Input:
integrate((a+b*arctanh(c*x^(1/2)))^2/x,x, algorithm="giac")
Output:
integrate((b*arctanh(c*sqrt(x)) + a)^2/x, x)
Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^2}{x} \,d x \] Input:
int((a + b*atanh(c*x^(1/2)))^2/x,x)
Output:
int((a + b*atanh(c*x^(1/2)))^2/x, x)
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x} \, dx=2 \left (\int \frac {\mathit {atanh} \left (\sqrt {x}\, c \right )}{x}d x \right ) a b +\left (\int \frac {\mathit {atanh} \left (\sqrt {x}\, c \right )^{2}}{x}d x \right ) b^{2}+\mathrm {log}\left (x \right ) a^{2} \] Input:
int((a+b*atanh(c*x^(1/2)))^2/x,x)
Output:
2*int(atanh(sqrt(x)*c)/x,x)*a*b + int(atanh(sqrt(x)*c)**2/x,x)*b**2 + log( x)*a**2