3.146 \(\int (a+b \tanh ^{-1}(\frac{c}{x}))^2 \, dx\)

Optimal. Leaf size=74 \[ b^2 (-c) \text{PolyLog}\left (2,-\frac{c+x}{c-x}\right )+c \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2+x \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2-2 b c \log \left (\frac{2 c}{c-x}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right ) \]

[Out]

c*(a + b*ArcCoth[x/c])^2 + x*(a + b*ArcCoth[x/c])^2 - 2*b*c*(a + b*ArcCoth[x/c])*Log[(2*c)/(c - x)] - b^2*c*Po
lyLog[2, -((c + x)/(c - x))]

________________________________________________________________________________________

Rubi [B]  time = 0.402079, antiderivative size = 370, normalized size of antiderivative = 5., number of steps used = 31, number of rules used = 14, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.167, Rules used = {6093, 2448, 263, 31, 2449, 2391, 2556, 12, 2462, 260, 2416, 2394, 2315, 2393} \[ -\frac{1}{2} b^2 c \text{PolyLog}\left (2,\frac{c-x}{2 c}\right )+\frac{1}{2} b^2 c \text{PolyLog}\left (2,-\frac{c}{x}\right )-\frac{1}{2} b^2 c \text{PolyLog}\left (2,\frac{c}{x}\right )+\frac{1}{2} b^2 c \text{PolyLog}\left (2,\frac{c+x}{2 c}\right )+\frac{1}{2} b^2 c \text{PolyLog}\left (2,1-\frac{x}{c}\right )-\frac{1}{2} b^2 c \text{PolyLog}\left (2,\frac{x}{c}+1\right )+a^2 x-a b x \log \left (1-\frac{c}{x}\right )+a b x \log \left (\frac{c}{x}+1\right )+a b c \log (c-x)+a b c \log (c+x)-\frac{1}{4} b^2 (c-x) \log ^2\left (1-\frac{c}{x}\right )+\frac{1}{4} b^2 (c+x) \log ^2\left (\frac{c}{x}+1\right )-\frac{1}{2} b^2 x \log \left (1-\frac{c}{x}\right ) \log \left (\frac{c}{x}+1\right )-\frac{1}{2} b^2 c \log \left (1-\frac{c}{x}\right ) \log (-c-x)+\frac{1}{2} b^2 c \log (-c-x) \log \left (\frac{c-x}{2 c}\right )-\frac{1}{2} b^2 c \log (-c-x) \log \left (-\frac{x}{c}\right )+\frac{1}{2} b^2 c \log \left (\frac{c}{x}+1\right ) \log (x-c)+\frac{1}{2} b^2 c \log \left (\frac{x}{c}\right ) \log (x-c)-\frac{1}{2} b^2 c \log (x-c) \log \left (\frac{c+x}{2 c}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*ArcTanh[c/x])^2,x]

[Out]

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

Rule 6093

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + (b*Log[1 + c*x^n])/2
- (b*Log[1 - c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && IntegerQ[n]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2449

Int[((a_.) + Log[(c_.)*((d_) + (e_.)/(x_))^(p_.)]*(b_.))^(q_), x_Symbol] :> Simp[((e + d*x)*(a + b*Log[c*(d +
e/x)^p])^q)/d, x] + Dist[(b*e*p*q)/d, Int[(a + b*Log[c*(d + e/x)^p])^(q - 1)/x, x], x] /; FreeQ[{a, b, c, d, e
, p}, x] && IGtQ[q, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2556

Int[Log[v_]*Log[w_], x_Symbol] :> Simp[x*Log[v]*Log[w], x] + (-Int[SimplifyIntegrand[(x*Log[w]*D[v, x])/v, x],
x] - Int[SimplifyIntegrand[(x*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFreeQ[v, x] && InverseFunctionFree
Q[w, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2462

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[f +
g*x]*(a + b*Log[c*(d + e*x^n)^p]))/g, x] - Dist[(b*e*n*p)/g, Int[(x^(n - 1)*Log[f + g*x])/(d + e*x^n), x], x]
/; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rubi steps

\begin{align*} \int \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^2 \, dx &=\int \left (a^2-a b \log \left (1-\frac{c}{x}\right )+\frac{1}{4} b^2 \log ^2\left (1-\frac{c}{x}\right )+a b \log \left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 \log ^2\left (1+\frac{c}{x}\right )\right ) \, dx\\ &=a^2 x-(a b) \int \log \left (1-\frac{c}{x}\right ) \, dx+(a b) \int \log \left (1+\frac{c}{x}\right ) \, dx+\frac{1}{4} b^2 \int \log ^2\left (1-\frac{c}{x}\right ) \, dx+\frac{1}{4} b^2 \int \log ^2\left (1+\frac{c}{x}\right ) \, dx-\frac{1}{2} b^2 \int \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right ) \, dx\\ &=a^2 x-a b x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} b^2 (c-x) \log ^2\left (1-\frac{c}{x}\right )+a b x \log \left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 x \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 (c+x) \log ^2\left (1+\frac{c}{x}\right )+\frac{1}{2} b^2 \int \frac{c \log \left (1-\frac{c}{x}\right )}{-c-x} \, dx+\frac{1}{2} b^2 \int \frac{c \log \left (1+\frac{c}{x}\right )}{-c+x} \, dx+(a b c) \int \frac{1}{\left (1-\frac{c}{x}\right ) x} \, dx+(a b c) \int \frac{1}{\left (1+\frac{c}{x}\right ) x} \, dx-\frac{1}{2} \left (b^2 c\right ) \int \frac{\log \left (1-\frac{c}{x}\right )}{x} \, dx+\frac{1}{2} \left (b^2 c\right ) \int \frac{\log \left (1+\frac{c}{x}\right )}{x} \, dx\\ &=a^2 x-a b x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} b^2 (c-x) \log ^2\left (1-\frac{c}{x}\right )+a b x \log \left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 x \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 (c+x) \log ^2\left (1+\frac{c}{x}\right )+\frac{1}{2} b^2 c \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{2} b^2 c \text{Li}_2\left (\frac{c}{x}\right )+(a b c) \int \frac{1}{-c+x} \, dx+(a b c) \int \frac{1}{c+x} \, dx+\frac{1}{2} \left (b^2 c\right ) \int \frac{\log \left (1-\frac{c}{x}\right )}{-c-x} \, dx+\frac{1}{2} \left (b^2 c\right ) \int \frac{\log \left (1+\frac{c}{x}\right )}{-c+x} \, dx\\ &=a^2 x-a b x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} b^2 (c-x) \log ^2\left (1-\frac{c}{x}\right )+a b x \log \left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 x \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 (c+x) \log ^2\left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 c \log \left (1-\frac{c}{x}\right ) \log (-c-x)+a b c \log (c-x)+\frac{1}{2} b^2 c \log \left (1+\frac{c}{x}\right ) \log (-c+x)+a b c \log (c+x)+\frac{1}{2} b^2 c \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{2} b^2 c \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{2} \left (b^2 c^2\right ) \int \frac{\log (-c-x)}{\left (1-\frac{c}{x}\right ) x^2} \, dx+\frac{1}{2} \left (b^2 c^2\right ) \int \frac{\log (-c+x)}{\left (1+\frac{c}{x}\right ) x^2} \, dx\\ &=a^2 x-a b x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} b^2 (c-x) \log ^2\left (1-\frac{c}{x}\right )+a b x \log \left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 x \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 (c+x) \log ^2\left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 c \log \left (1-\frac{c}{x}\right ) \log (-c-x)+a b c \log (c-x)+\frac{1}{2} b^2 c \log \left (1+\frac{c}{x}\right ) \log (-c+x)+a b c \log (c+x)+\frac{1}{2} b^2 c \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{2} b^2 c \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{2} \left (b^2 c^2\right ) \int \left (-\frac{\log (-c-x)}{c (c-x)}-\frac{\log (-c-x)}{c x}\right ) \, dx+\frac{1}{2} \left (b^2 c^2\right ) \int \left (\frac{\log (-c+x)}{c x}-\frac{\log (-c+x)}{c (c+x)}\right ) \, dx\\ &=a^2 x-a b x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} b^2 (c-x) \log ^2\left (1-\frac{c}{x}\right )+a b x \log \left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 x \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 (c+x) \log ^2\left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 c \log \left (1-\frac{c}{x}\right ) \log (-c-x)+a b c \log (c-x)+\frac{1}{2} b^2 c \log \left (1+\frac{c}{x}\right ) \log (-c+x)+a b c \log (c+x)+\frac{1}{2} b^2 c \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{2} b^2 c \text{Li}_2\left (\frac{c}{x}\right )-\frac{1}{2} \left (b^2 c\right ) \int \frac{\log (-c-x)}{c-x} \, dx-\frac{1}{2} \left (b^2 c\right ) \int \frac{\log (-c-x)}{x} \, dx+\frac{1}{2} \left (b^2 c\right ) \int \frac{\log (-c+x)}{x} \, dx-\frac{1}{2} \left (b^2 c\right ) \int \frac{\log (-c+x)}{c+x} \, dx\\ &=a^2 x-a b x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} b^2 (c-x) \log ^2\left (1-\frac{c}{x}\right )+a b x \log \left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 x \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 (c+x) \log ^2\left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 c \log \left (1-\frac{c}{x}\right ) \log (-c-x)+a b c \log (c-x)+\frac{1}{2} b^2 c \log (-c-x) \log \left (\frac{c-x}{2 c}\right )-\frac{1}{2} b^2 c \log (-c-x) \log \left (-\frac{x}{c}\right )+\frac{1}{2} b^2 c \log \left (1+\frac{c}{x}\right ) \log (-c+x)+\frac{1}{2} b^2 c \log \left (\frac{x}{c}\right ) \log (-c+x)+a b c \log (c+x)-\frac{1}{2} b^2 c \log (-c+x) \log \left (\frac{c+x}{2 c}\right )+\frac{1}{2} b^2 c \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{2} b^2 c \text{Li}_2\left (\frac{c}{x}\right )-\frac{1}{2} \left (b^2 c\right ) \int \frac{\log \left (-\frac{x}{c}\right )}{-c-x} \, dx-\frac{1}{2} \left (b^2 c\right ) \int \frac{\log \left (\frac{x}{c}\right )}{-c+x} \, dx+\frac{1}{2} \left (b^2 c\right ) \int \frac{\log \left (-\frac{-c+x}{2 c}\right )}{-c-x} \, dx+\frac{1}{2} \left (b^2 c\right ) \int \frac{\log \left (\frac{c+x}{2 c}\right )}{-c+x} \, dx\\ &=a^2 x-a b x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} b^2 (c-x) \log ^2\left (1-\frac{c}{x}\right )+a b x \log \left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 x \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 (c+x) \log ^2\left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 c \log \left (1-\frac{c}{x}\right ) \log (-c-x)+a b c \log (c-x)+\frac{1}{2} b^2 c \log (-c-x) \log \left (\frac{c-x}{2 c}\right )-\frac{1}{2} b^2 c \log (-c-x) \log \left (-\frac{x}{c}\right )+\frac{1}{2} b^2 c \log \left (1+\frac{c}{x}\right ) \log (-c+x)+\frac{1}{2} b^2 c \log \left (\frac{x}{c}\right ) \log (-c+x)+a b c \log (c+x)-\frac{1}{2} b^2 c \log (-c+x) \log \left (\frac{c+x}{2 c}\right )+\frac{1}{2} b^2 c \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{2} b^2 c \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{2} b^2 c \text{Li}_2\left (1-\frac{x}{c}\right )-\frac{1}{2} b^2 c \text{Li}_2\left (1+\frac{x}{c}\right )-\frac{1}{2} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{x}{2 c}\right )}{x} \, dx,x,-c-x\right )+\frac{1}{2} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{x}{2 c}\right )}{x} \, dx,x,-c+x\right )\\ &=a^2 x-a b x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} b^2 (c-x) \log ^2\left (1-\frac{c}{x}\right )+a b x \log \left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 x \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 (c+x) \log ^2\left (1+\frac{c}{x}\right )-\frac{1}{2} b^2 c \log \left (1-\frac{c}{x}\right ) \log (-c-x)+a b c \log (c-x)+\frac{1}{2} b^2 c \log (-c-x) \log \left (\frac{c-x}{2 c}\right )-\frac{1}{2} b^2 c \log (-c-x) \log \left (-\frac{x}{c}\right )+\frac{1}{2} b^2 c \log \left (1+\frac{c}{x}\right ) \log (-c+x)+\frac{1}{2} b^2 c \log \left (\frac{x}{c}\right ) \log (-c+x)+a b c \log (c+x)-\frac{1}{2} b^2 c \log (-c+x) \log \left (\frac{c+x}{2 c}\right )-\frac{1}{2} b^2 c \text{Li}_2\left (\frac{c-x}{2 c}\right )+\frac{1}{2} b^2 c \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{2} b^2 c \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{2} b^2 c \text{Li}_2\left (\frac{c+x}{2 c}\right )+\frac{1}{2} b^2 c \text{Li}_2\left (1-\frac{x}{c}\right )-\frac{1}{2} b^2 c \text{Li}_2\left (1+\frac{x}{c}\right )\\ \end{align*}

Mathematica [A]  time = 0.121809, size = 97, normalized size = 1.31 \[ b^2 c \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )+a \left (a x+b c \log \left (1-\frac{c^2}{x^2}\right )-2 b c \log \left (\frac{c}{x}\right )\right )+2 b \tanh ^{-1}\left (\frac{c}{x}\right ) \left (a x-b c \log \left (1-e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )\right )+b^2 (x-c) \tanh ^{-1}\left (\frac{c}{x}\right )^2 \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTanh[c/x])^2,x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.012, size = 282, normalized size = 3.8 \begin{align*}{a}^{2}x+{b}^{2}x \left ({\it Artanh} \left ({\frac{c}{x}} \right ) \right ) ^{2}+c{b}^{2}{\it Artanh} \left ({\frac{c}{x}} \right ) \ln \left ({\frac{c}{x}}-1 \right ) -2\,c{b}^{2}\ln \left ({\frac{c}{x}} \right ){\it Artanh} \left ({\frac{c}{x}} \right ) +c{b}^{2}{\it Artanh} \left ({\frac{c}{x}} \right ) \ln \left ( 1+{\frac{c}{x}} \right ) -c{b}^{2}{\it dilog} \left ({\frac{1}{2}}+{\frac{c}{2\,x}} \right ) -{\frac{{b}^{2}c}{2}\ln \left ({\frac{c}{x}}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2\,x}} \right ) }+{\frac{{b}^{2}c}{4} \left ( \ln \left ({\frac{c}{x}}-1 \right ) \right ) ^{2}}-{\frac{{b}^{2}c}{2}\ln \left ( -{\frac{c}{2\,x}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2\,x}} \right ) }+{\frac{{b}^{2}c}{2}\ln \left ( -{\frac{c}{2\,x}}+{\frac{1}{2}} \right ) \ln \left ( 1+{\frac{c}{x}} \right ) }-{\frac{{b}^{2}c}{4} \left ( \ln \left ( 1+{\frac{c}{x}} \right ) \right ) ^{2}}+c{b}^{2}{\it dilog} \left ({\frac{c}{x}} \right ) +c{b}^{2}{\it dilog} \left ( 1+{\frac{c}{x}} \right ) +c{b}^{2}\ln \left ({\frac{c}{x}} \right ) \ln \left ( 1+{\frac{c}{x}} \right ) +2\,abx{\it Artanh} \left ({\frac{c}{x}} \right ) +cab\ln \left ({\frac{c}{x}}-1 \right ) -2\,cab\ln \left ({\frac{c}{x}} \right ) +cab\ln \left ( 1+{\frac{c}{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*x+b^2*x*arctanh(c/x)^2+c*b^2*arctanh(c/x)*ln(c/x-1)-2*c*b^2*ln(c/x)*arctanh(c/x)+c*b^2*arctanh(c/x)*ln(1+c
/x)-c*b^2*dilog(1/2+1/2*c/x)-1/2*c*b^2*ln(c/x-1)*ln(1/2+1/2*c/x)+1/4*c*b^2*ln(c/x-1)^2-1/2*c*b^2*ln(-1/2*c/x+1
/2)*ln(1/2+1/2*c/x)+1/2*c*b^2*ln(-1/2*c/x+1/2)*ln(1+c/x)-1/4*c*b^2*ln(1+c/x)^2+c*b^2*dilog(c/x)+c*b^2*dilog(1+
c/x)+c*b^2*ln(c/x)*ln(1+c/x)+2*a*b*x*arctanh(c/x)+c*a*b*ln(c/x-1)-2*c*a*b*ln(c/x)+c*a*b*ln(1+c/x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (2 \, x \operatorname{artanh}\left (\frac{c}{x}\right ) + c \log \left (-c^{2} + x^{2}\right )\right )} a b + \frac{1}{4} \,{\left (x \log \left (c + x\right )^{2} - 2 \,{\left (c + x\right )} \log \left (c + x\right ) \log \left (-c + x\right ) -{\left (c - x\right )} \log \left (-c + x\right )^{2} + \int -\frac{2 \,{\left (c^{2} + 3 \, c x\right )} \log \left (c + x\right )}{c^{2} - x^{2}}\,{d x}\right )} b^{2} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} \operatorname{artanh}\left (\frac{c}{x}\right )^{2} + 2 \, a b \operatorname{artanh}\left (\frac{c}{x}\right ) + a^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{atanh}{\left (\frac{c}{x} \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c/x))**2, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (\frac{c}{x}\right ) + a\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctanh(c/x) + a)^2, x)