Integrand size = 16, antiderivative size = 133 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x} \, dx=-2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \text {arctanh}\left (1-\frac {2}{1-\frac {c}{x}}\right )+b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x}}\right )-b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-\frac {c}{x}}\right )-\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-\frac {c}{x}}\right )+\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-\frac {c}{x}}\right ) \]
2*(a+b*arccoth(x/c))^2*arctanh(-1+2/(1-c/x))+b*(a+b*arccoth(x/c))*polylog( 2,1-2/(1-c/x))-b*(a+b*arccoth(x/c))*polylog(2,-1+2/(1-c/x))-1/2*b^2*polylo g(3,1-2/(1-c/x))+1/2*b^2*polylog(3,-1+2/(1-c/x))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x} \, dx=a^2 \log (x)+a b \left (\operatorname {PolyLog}\left (2,-\frac {c}{x}\right )-\operatorname {PolyLog}\left (2,\frac {c}{x}\right )\right )+b^2 \left (-\frac {i \pi ^3}{24}+\frac {2}{3} \text {arctanh}\left (\frac {c}{x}\right )^3+\text {arctanh}\left (\frac {c}{x}\right )^2 \log \left (1+e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-\text {arctanh}\left (\frac {c}{x}\right )^2 \log \left (1-e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-\text {arctanh}\left (\frac {c}{x}\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-\text {arctanh}\left (\frac {c}{x}\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right ) \]
a^2*Log[x] + a*b*(PolyLog[2, -(c/x)] - PolyLog[2, c/x]) + b^2*((-1/24*I)*P i^3 + (2*ArcTanh[c/x]^3)/3 + ArcTanh[c/x]^2*Log[1 + E^(-2*ArcTanh[c/x])] - ArcTanh[c/x]^2*Log[1 - E^(2*ArcTanh[c/x])] - ArcTanh[c/x]*PolyLog[2, -E^( -2*ArcTanh[c/x])] - ArcTanh[c/x]*PolyLog[2, E^(2*ArcTanh[c/x])] - PolyLog[ 3, -E^(-2*ArcTanh[c/x])]/2 + PolyLog[3, E^(2*ArcTanh[c/x])]/2)
Time = 0.85 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.20, 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.
| \(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x} \, dx\) |
| \(\Big \downarrow \) 6450 |
| \(\displaystyle -\int x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 6448 |
| \(\displaystyle 4 b c \int \frac {\text {arctanh}\left (1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}-2 \text {arctanh}\left (1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\) |
| \(\Big \downarrow \) 6614 |
| \(\displaystyle 4 b c \left (\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right ) \log \left (2-\frac {2}{1-\frac {c}{x}}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}-\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right ) \log \left (\frac {2}{1-\frac {c}{x}}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}\right )-2 \text {arctanh}\left (1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle 4 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{2 c}-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x}}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}\right )+\frac {1}{2} \left (\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-\frac {c}{x}}-1\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-\frac {c}{x}}-1\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{2 c}\right )\right )-2 \text {arctanh}\left (1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle 4 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{2 c}-\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{1-\frac {c}{x}}\right )}{4 c}\right )+\frac {1}{2} \left (\frac {b \operatorname {PolyLog}\left (3,\frac {2}{1-\frac {c}{x}}-1\right )}{4 c}-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-\frac {c}{x}}-1\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{2 c}\right )\right )-2 \text {arctanh}\left (1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\) |
-2*ArcTanh[1 - 2/(1 - c/x)]*(a + b*ArcTanh[c/x])^2 + 4*b*c*((((a + b*ArcTa nh[c/x])*PolyLog[2, 1 - 2/(1 - c/x)])/(2*c) - (b*PolyLog[3, 1 - 2/(1 - c/x )])/(4*c))/2 + (-1/2*((a + b*ArcTanh[c/x])*PolyLog[2, -1 + 2/(1 - c/x)])/c + (b*PolyLog[3, -1 + 2/(1 - c/x)])/(4*c))/2)
3.2.47.3.1 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 = 56.96 (sec) , antiderivative size = 704, normalized size of antiderivative = 5.29
| method | result | size |
| parts | \(a^{2} \ln \left (x \right )+b^{2} \left (-\ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}+\operatorname {arctanh}\left (\frac {c}{x}\right ) \operatorname {polylog}\left (2, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )}{2}+\operatorname {arctanh}\left (\frac {c}{x}\right )^{2} \ln \left (\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}-1\right )-\operatorname {arctanh}\left (\frac {c}{x}\right )^{2} \ln \left (1+\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )-2 \,\operatorname {arctanh}\left (\frac {c}{x}\right ) \operatorname {polylog}\left (2, -\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )+2 \operatorname {polylog}\left (3, -\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )-\operatorname {arctanh}\left (\frac {c}{x}\right )^{2} \ln \left (1-\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )-2 \,\operatorname {arctanh}\left (\frac {c}{x}\right ) \operatorname {polylog}\left (2, \frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )+2 \operatorname {polylog}\left (3, \frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right )-\operatorname {csgn}\left (\frac {i}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right )}^{2}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{2}\right )+2 a b \left (-\ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )+\frac {\operatorname {dilog}\left (1+\frac {c}{x}\right )}{2}+\frac {\ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {c}{x}\right )}{2}\right )\) | \(704\) |
| derivativedivides | \(-a^{2} \ln \left (\frac {c}{x}\right )-b^{2} \left (\ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}-\operatorname {arctanh}\left (\frac {c}{x}\right ) \operatorname {polylog}\left (2, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )}{2}-\operatorname {arctanh}\left (\frac {c}{x}\right )^{2} \ln \left (\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}-1\right )+\operatorname {arctanh}\left (\frac {c}{x}\right )^{2} \ln \left (1-\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )+2 \,\operatorname {arctanh}\left (\frac {c}{x}\right ) \operatorname {polylog}\left (2, \frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )-2 \operatorname {polylog}\left (3, \frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )+\operatorname {arctanh}\left (\frac {c}{x}\right )^{2} \ln \left (1+\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )+2 \,\operatorname {arctanh}\left (\frac {c}{x}\right ) \operatorname {polylog}\left (2, -\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )-2 \operatorname {polylog}\left (3, -\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right )-\operatorname {csgn}\left (\frac {i}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right )}^{2}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{2}\right )-2 a b \left (\ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )-\frac {\operatorname {dilog}\left (1+\frac {c}{x}\right )}{2}-\frac {\ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c}{x}\right )}{2}\right )\) | \(708\) |
| default | \(-a^{2} \ln \left (\frac {c}{x}\right )-b^{2} \left (\ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}-\operatorname {arctanh}\left (\frac {c}{x}\right ) \operatorname {polylog}\left (2, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )}{2}-\operatorname {arctanh}\left (\frac {c}{x}\right )^{2} \ln \left (\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}-1\right )+\operatorname {arctanh}\left (\frac {c}{x}\right )^{2} \ln \left (1-\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )+2 \,\operatorname {arctanh}\left (\frac {c}{x}\right ) \operatorname {polylog}\left (2, \frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )-2 \operatorname {polylog}\left (3, \frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )+\operatorname {arctanh}\left (\frac {c}{x}\right )^{2} \ln \left (1+\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )+2 \,\operatorname {arctanh}\left (\frac {c}{x}\right ) \operatorname {polylog}\left (2, -\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )-2 \operatorname {polylog}\left (3, -\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right )-\operatorname {csgn}\left (\frac {i}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}-1\right )}{1-\frac {\left (1+\frac {c}{x}\right )^{2}}{\frac {c^{2}}{x^{2}}-1}}\right )}^{2}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{2}\right )-2 a b \left (\ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )-\frac {\operatorname {dilog}\left (1+\frac {c}{x}\right )}{2}-\frac {\ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c}{x}\right )}{2}\right )\) | \(708\) |
a^2*ln(x)+b^2*(-ln(c/x)*arctanh(c/x)^2+arctanh(c/x)*polylog(2,-(1+c/x)^2/( 1-c^2/x^2))-1/2*polylog(3,-(1+c/x)^2/(1-c^2/x^2))+arctanh(c/x)^2*ln((1+c/x )^2/(1-c^2/x^2)-1)-arctanh(c/x)^2*ln(1+(1+c/x)/(1-c^2/x^2)^(1/2))-2*arctan h(c/x)*polylog(2,-(1+c/x)/(1-c^2/x^2)^(1/2))+2*polylog(3,-(1+c/x)/(1-c^2/x ^2)^(1/2))-arctanh(c/x)^2*ln(1-(1+c/x)/(1-c^2/x^2)^(1/2))-2*arctanh(c/x)*p olylog(2,(1+c/x)/(1-c^2/x^2)^(1/2))+2*polylog(3,(1+c/x)/(1-c^2/x^2)^(1/2)) -1/2*I*Pi*csgn(I*(-(1+c/x)^2/(c^2/x^2-1)-1)/(1-(1+c/x)^2/(c^2/x^2-1)))*(cs gn(I*(-(1+c/x)^2/(c^2/x^2-1)-1))*csgn(I/(1-(1+c/x)^2/(c^2/x^2-1)))-csgn(I* (-(1+c/x)^2/(c^2/x^2-1)-1))*csgn(I*(-(1+c/x)^2/(c^2/x^2-1)-1)/(1-(1+c/x)^2 /(c^2/x^2-1)))-csgn(I/(1-(1+c/x)^2/(c^2/x^2-1)))*csgn(I*(-(1+c/x)^2/(c^2/x ^2-1)-1)/(1-(1+c/x)^2/(c^2/x^2-1)))+csgn(I*(-(1+c/x)^2/(c^2/x^2-1)-1)/(1-( 1+c/x)^2/(c^2/x^2-1)))^2)*arctanh(c/x)^2)+2*a*b*(-ln(c/x)*arctanh(c/x)+1/2 *dilog(1+c/x)+1/2*ln(c/x)*ln(1+c/x)+1/2*dilog(c/x))
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{2}}{x} \,d x } \]
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (\frac {c}{x} \right )}\right )^{2}}{x}\, dx \]
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{2}}{x} \,d x } \]
a^2*log(x) + integrate(1/4*b^2*(log(c/x + 1) - log(-c/x + 1))^2/x + a*b*(l og(c/x + 1) - log(-c/x + 1))/x, x)
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )\right )}^2}{x} \,d x \]