\(\int \frac {(a+b \coth ^{-1}(\frac {\sqrt {1-c x}}{\sqrt {1+c x}}))^2}{1-c^2 x^2} \, dx\) [124]

Optimal result

Integrand size = 40, antiderivative size = 302 \[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=-\frac {2 \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {b \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {b \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {1-c x}}{\sqrt {1+c x} \left (1+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 \sqrt {1-c x}}{\sqrt {1+c x} \left (1+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c} \]

[Out]

-2*arccoth(1-2/(1-(-c*x+1)^(1/2)/(c*x+1)^(1/2)))*(a+b*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/c-b*(a+b*arccot
h((-c*x+1)^(1/2)/(c*x+1)^(1/2)))*polylog(2,1-2/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))/c+b*(a+b*arccoth((-c*x+1)^(1/
2)/(c*x+1)^(1/2)))*polylog(2,1-2*(-c*x+1)^(1/2)/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)/(c*x+1)^(1/2))/c-1/2*b^2*poly
log(3,1-2/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))/c+1/2*b^2*polylog(3,1-2*(-c*x+1)^(1/2)/((-c*x+1)^(1/2)/(c*x+1)^(1/
2)+1)/(c*x+1)^(1/2))/c

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6813, 6034, 6200, 6096, 6204, 6745} \[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+1}\right ) \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {1-c x}}{\sqrt {c x+1} \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+1\right )}\right ) \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{c}-\frac {2 \coth ^{-1}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{c}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+1}\right )}{2 c}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 \sqrt {1-c x}}{\sqrt {c x+1} \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+1\right )}\right )}{2 c} \]

[In]

Int[(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2),x]

[Out]

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

Rule 6034

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcCoth[c*x])^p*ArcCoth[1 - 2/(1
- c*x)], x] - Dist[2*b*c*p, Int[(a + b*ArcCoth[c*x])^(p - 1)*(ArcCoth[1 - 2/(1 - c*x)]/(1 - c^2*x^2)), x], x]
/; FreeQ[{a, b, c}, x] && IGtQ[p, 1]

Rule 6096

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]

Rule 6200

Int[(ArcCoth[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[
Log[SimplifyIntegrand[1 + 1/u, x]]*((a + b*ArcCoth[c*x])^p/(d + e*x^2)), x], x] - Dist[1/2, Int[Log[SimplifyIn
tegrand[1 - 1/u, x]]*((a + b*ArcCoth[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 6204

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCot
h[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] - Dist[b*(p/2), Int[(a + b*ArcCoth[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 6745

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]

Rule 6813

Int[((a_.) + (b_.)*(F_)[((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.)*(x_)]])^(n_.)/((A_.) + (C_.)*(x_)^
2), x_Symbol] :> Dist[2*e*(g/(C*(e*f - d*g))), Subst[Int[(a + b*F[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x
]], x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (a+b \coth ^{-1}(x)\right )^2}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c} \\ & = -\frac {2 \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {(4 b) \text {Subst}\left (\int \frac {\coth ^{-1}\left (1-\frac {2}{1-x}\right ) \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c} \\ & = -\frac {2 \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {(2 b) \text {Subst}\left (\int \frac {\left (a+b \coth ^{-1}(x)\right ) \log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}+\frac {(2 b) \text {Subst}\left (\int \frac {\left (a+b \coth ^{-1}(x)\right ) \log \left (\frac {2 x}{1+x}\right )}{1-x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c} \\ & = -\frac {2 \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {b \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {b \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {1-c x}}{\sqrt {1+c x} \left (1+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+x}\right )}{1-x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}-\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,1-\frac {2 x}{1+x}\right )}{1-x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c} \\ & = -\frac {2 \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {b \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {b \left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {1-c x}}{\sqrt {1+c x} \left (1+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 \sqrt {1-c x}}{\sqrt {1+c x} \left (1+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx \]

[In]

Integrate[(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2),x]

[Out]

Integrate[(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(925\) vs. \(2(258)=516\).

Time = 0.90 (sec) , antiderivative size = 926, normalized size of antiderivative = 3.07

method	result	size
default	\(-\frac {a^{2} \ln \left (c x -1\right )}{2 c}+\frac {a^{2} \ln \left (c x +1\right )}{2 c}-b^{2} \left (-\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}-\frac {2 \,\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}+\frac {2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}+\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{c}+\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{c}-\frac {\operatorname {polylog}\left (3, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{2 c}-\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}-\frac {2 \,\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}+\frac {2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}\right )-2 a b \left (-\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}-\frac {\operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}+\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{c}+\frac {\operatorname {polylog}\left (2, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{2 c}-\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}-\frac {\operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}\right )\)	\(926\)
parts	\(-\frac {a^{2} \ln \left (c x -1\right )}{2 c}+\frac {a^{2} \ln \left (c x +1\right )}{2 c}-b^{2} \left (-\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}-\frac {2 \,\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}+\frac {2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}+\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{c}+\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{c}-\frac {\operatorname {polylog}\left (3, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{2 c}-\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}-\frac {2 \,\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}+\frac {2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}\right )-2 a b \left (-\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}-\frac {\operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}+\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{c}+\frac {\operatorname {polylog}\left (2, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\right )}{2 c}-\frac {\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}-\frac {\operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}{\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}}}\right )}{c}\right )\)	\(926\)

[In]

int((a+b*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x,method=_RETURNVERBOSE)

[Out]

-1/2*a^2/c*ln(c*x-1)+1/2*a^2/c*ln(c*x+1)-b^2*(-1/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))^2*ln(1+1/(((-c*x+1)^(
1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))^(1/2))-2/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))*polyl
og(2,-1/(((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))^(1/2))+2/c*polylog(3,-1/(((-c*x+1)
^(1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))^(1/2))+1/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))^2*l
n(1+1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))+1/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1
/2))*polylog(2,-1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))-1/2/c*polylog(3,-1/((-c*x
+1)^(1/2)/(c*x+1)^(1/2)-1)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))-1/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))^2*ln(1-
1/(((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))^(1/2))-2/c*arccoth((-c*x+1)^(1/2)/(c*x+1
)^(1/2))*polylog(2,1/(((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))^(1/2))+2/c*polylog(3,
1/(((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))^(1/2)))-2*a*b*(-1/c*arccoth((-c*x+1)^(1/
2)/(c*x+1)^(1/2))*ln(1+1/(((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))^(1/2))-1/c*polylo
g(2,-1/(((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))^(1/2))+1/c*arccoth((-c*x+1)^(1/2)/(
c*x+1)^(1/2))*ln(1+1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))+1/2/c*polylog(2,-1/((-
c*x+1)^(1/2)/(c*x+1)^(1/2)-1)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))-1/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))*ln(1
-1/(((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))^(1/2))-1/c*polylog(2,1/(((-c*x+1)^(1/2)
/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1))^(1/2)))

Fricas [F]

\[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcoth}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]

[In]

integrate((a+b*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x, algorithm="fricas")

[Out]

integral(-(b^2*arccoth(sqrt(-c*x + 1)/sqrt(c*x + 1))^2 + 2*a*b*arccoth(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a^2)/(c
^2*x^2 - 1), x)

Sympy [F]

\[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=- \int \frac {a^{2}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{2} \operatorname {acoth}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {2 a b \operatorname {acoth}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]

[In]

integrate((a+b*acoth((-c*x+1)**(1/2)/(c*x+1)**(1/2)))**2/(-c**2*x**2+1),x)

[Out]

-Integral(a**2/(c**2*x**2 - 1), x) - Integral(b**2*acoth(sqrt(-c*x + 1)/sqrt(c*x + 1))**2/(c**2*x**2 - 1), x)
- Integral(2*a*b*acoth(sqrt(-c*x + 1)/sqrt(c*x + 1))/(c**2*x**2 - 1), x)

Maxima [F]

\[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcoth}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]

[In]

integrate((a+b*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x, algorithm="maxima")

[Out]

1/2*a^2*(log(c*x + 1)/c - log(c*x - 1)/c) + 1/8*(b^2*log(c*x + 1) - b^2*log(-c*x + 1))*log(-sqrt(c*x + 1) + sq
rt(-c*x + 1))^2/c + integrate(-1/8*(2*(sqrt(c*x + 1)*b^2 - sqrt(-c*x + 1)*b^2)*log(sqrt(c*x + 1) + sqrt(-c*x +
1))^2 + 8*(sqrt(c*x + 1)*a*b - sqrt(-c*x + 1)*a*b)*log(sqrt(c*x + 1) + sqrt(-c*x + 1)) - (4*(sqrt(c*x + 1)*b^
2 - sqrt(-c*x + 1)*b^2)*log(sqrt(c*x + 1) + sqrt(-c*x + 1)) + (8*a*b - (b^2*c*x - b^2)*log(c*x + 1) + (b^2*c*x
- b^2)*log(-c*x + 1))*sqrt(c*x + 1) - (8*a*b - (b^2*c*x + b^2)*log(c*x + 1) + (b^2*c*x + b^2)*log(-c*x + 1))*
sqrt(-c*x + 1))*log(-sqrt(c*x + 1) + sqrt(-c*x + 1)))/((c^2*x^2 - 1)*sqrt(c*x + 1) - (c^2*x^2 - 1)*sqrt(-c*x +
1)), x)

Giac [F]

\[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcoth}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]

[In]

integrate((a+b*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x, algorithm="giac")

[Out]

integrate(-(b*arccoth(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a)^2/(c^2*x^2 - 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\mathrm {acoth}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2}{c^2\,x^2-1} \,d x \]

[In]

int(-(a + b*acoth((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^2/(c^2*x^2 - 1),x)

[Out]

int(-(a + b*acoth((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^2/(c^2*x^2 - 1), x)