3.123 \(\int \frac{(a+b \coth ^{-1}(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}))^3}{1-c^2 x^2} \, dx\)

Optimal. Leaf size=460 \[ -\frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1}\right ) \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}+\frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1\right )}\right ) \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}-\frac{3 b \text{PolyLog}\left (2,1-\frac{2}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1}\right ) \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}+\frac{3 b \text{PolyLog}\left (2,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1\right )}\right ) \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}-\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1}\right )}{4 c}+\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1\right )}\right )}{4 c}-\frac{2 \coth ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{c} \]

[Out]

(-2*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*ArcCoth[1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/c - (3*b*
(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 1 - 2/(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) + (3
*b*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(1 + Sqrt[1
- c*x]/Sqrt[1 + c*x]))])/(2*c) - (3*b^2*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, 1 - 2/(1 + Sqr
t[1 - c*x]/Sqrt[1 + c*x])])/(2*c) + (3*b^2*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, 1 - (2*Sqrt
[1 - c*x])/(Sqrt[1 + c*x]*(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/(2*c) - (3*b^3*PolyLog[4, 1 - 2/(1 + Sqrt[1 - c
*x]/Sqrt[1 + c*x])])/(4*c) + (3*b^3*PolyLog[4, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(1 + Sqrt[1 - c*x]/Sqrt[1
+ c*x]))])/(4*c)

________________________________________________________________________________________

Rubi [A]  time = 0.598234, antiderivative size = 460, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {6681, 5915, 6053, 5949, 6057, 6061, 6610} \[ -\frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1}\right ) \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}+\frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1\right )}\right ) \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}-\frac{3 b \text{PolyLog}\left (2,1-\frac{2}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1}\right ) \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}+\frac{3 b \text{PolyLog}\left (2,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1\right )}\right ) \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}-\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1}\right )}{4 c}+\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1\right )}\right )}{4 c}-\frac{2 \coth ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*ArcCoth[1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/c - (3*b*
(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 1 - 2/(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) + (3
*b*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(1 + Sqrt[1
- c*x]/Sqrt[1 + c*x]))])/(2*c) - (3*b^2*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, 1 - 2/(1 + Sqr
t[1 - c*x]/Sqrt[1 + c*x])])/(2*c) + (3*b^2*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, 1 - (2*Sqrt
[1 - c*x])/(Sqrt[1 + c*x]*(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/(2*c) - (3*b^3*PolyLog[4, 1 - 2/(1 + Sqrt[1 - c
*x]/Sqrt[1 + c*x])])/(4*c) + (3*b^3*PolyLog[4, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(1 + Sqrt[1 - c*x]/Sqrt[1
+ c*x]))])/(4*c)

Rule 6681

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]

Rule 5915

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 6053

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[SimplifyI
ntegrand[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 5949

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 6057

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a + b*ArcCo
th[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 6061

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

Rule 6610

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \coth ^{-1}(x)\right )^3}{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 )^3 \coth ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{(6 b) \operatorname{Subst}\left (\int \frac{\coth ^{-1}\left (1-\frac{2}{1-x}\right ) \left (a+b \coth ^{-1}(x)\right )^2}{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 )^3 \coth ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{\left (a+b \coth ^{-1}(x)\right )^2 \log \left (\frac{2}{1+x}\right )}{1-x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\left (a+b \coth ^{-1}(x)\right )^2 \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 )^3 \coth ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{3 b \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (1-\frac{2}{1+\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}+\frac{3 b \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (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}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \coth ^{-1}(x)\right ) \text{Li}_2\left (1-\frac{2}{1+x}\right )}{1-x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \coth ^{-1}(x)\right ) \text{Li}_2\left (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 )^3 \coth ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{3 b \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (1-\frac{2}{1+\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}+\frac{3 b \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (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}-\frac{3 b^2 \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (1-\frac{2}{1+\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}+\frac{3 b^2 \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (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}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (1-\frac{2}{1+x}\right )}{1-x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{2 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (1-\frac{2 x}{1+x}\right )}{1-x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{2 c}\\ &=-\frac{2 \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \coth ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{3 b \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (1-\frac{2}{1+\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}+\frac{3 b \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (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}-\frac{3 b^2 \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (1-\frac{2}{1+\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}+\frac{3 b^2 \left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (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}-\frac{3 b^3 \text{Li}_4\left (1-\frac{2}{1+\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{4 c}+\frac{3 b^3 \text{Li}_4\left (1-\frac{2 \sqrt{1-c x}}{\sqrt{1+c x} \left (1+\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{4 c}\\ \end{align*}

Mathematica [F]  time = 0.283333, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \coth ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 2.439, size = 1492, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x)

[Out]

-3*a*b^2/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))^2*ln(1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)*((-c*x+1)^(1/2)/(c*x+
1)^(1/2)+1)+1)-3*a*b^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))+3*a*b^2/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))+6*a*b^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))+3*a*b^2/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))+6*a*b^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))-1/2*a^3/c*ln(c*x-1)+1/2*a^3/c*ln(c*x+1)+6*b^3/c*polylog(4,-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))+6*b^3/c*polylog(4,1/(((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1
/2)/(c*x+1)^(1/2)+1))^(1/2))-3/4*b^3/c*polylog(4,-1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)*((-c*x+1)^(1/2)/(c*x+1)^(
1/2)+1))+b^3/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))^3*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))+3*b^3/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))^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))-6*b^3/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))*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))+b^3/c*arccoth((-c*x+1)^(1/2)/(c*x
+1)^(1/2))^3*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))+3*b^3/c*arccoth
((-c*x+1)^(1/2)/(c*x+1)^(1/2))^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))-6*b^3/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))*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))-b^3/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))^3*ln(1/((-c*x+1)^(1/2)/(c*x+1)^(1
/2)-1)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)+1)-3/2*b^3/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))^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))+3/2*b^3/c*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2))
*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))-6*a*b^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))+3/2*a*b^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))-6*a*b^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))+3*a^2*b/c*dilog(((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)/((-c*x+1)^(1/2)/(c*x+1)
^(1/2)+1))-3/4*a^2*b/c*dilog(((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)^2/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{3}{\left (\frac{\log \left (c x + 1\right )}{c} - \frac{\log \left (c x - 1\right )}{c}\right )} - \frac{{\left (b^{3} \log \left (c x + 1\right ) - b^{3} \log \left (-c x + 1\right )\right )} \log \left (-\sqrt{c x + 1} + \sqrt{-c x + 1}\right )^{3}}{16 \, c} - \int \frac{4 \,{\left (\sqrt{c x + 1} b^{3} - \sqrt{-c x + 1} b^{3}\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right )^{3} + 24 \,{\left (\sqrt{c x + 1} a b^{2} - \sqrt{-c x + 1} a b^{2}\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right )^{2} + 3 \,{\left (4 \,{\left (\sqrt{c x + 1} b^{3} - \sqrt{-c x + 1} b^{3}\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right ) +{\left (8 \, a b^{2} -{\left (b^{3} c x - b^{3}\right )} \log \left (c x + 1\right ) +{\left (b^{3} c x - b^{3}\right )} \log \left (-c x + 1\right )\right )} \sqrt{c x + 1} -{\left (8 \, a b^{2} -{\left (b^{3} c x + b^{3}\right )} \log \left (c x + 1\right ) +{\left (b^{3} c x + b^{3}\right )} \log \left (-c x + 1\right )\right )} \sqrt{-c x + 1}\right )} \log \left (-\sqrt{c x + 1} + \sqrt{-c x + 1}\right )^{2} + 48 \,{\left (\sqrt{c x + 1} a^{2} b - \sqrt{-c x + 1} a^{2} b\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right ) - 12 \,{\left (4 \, \sqrt{c x + 1} a^{2} b - 4 \, \sqrt{-c x + 1} a^{2} b +{\left (\sqrt{c x + 1} b^{3} - \sqrt{-c x + 1} b^{3}\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right )^{2} + 4 \,{\left (\sqrt{c x + 1} a b^{2} - \sqrt{-c x + 1} a b^{2}\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right )\right )} \log \left (-\sqrt{c x + 1} + \sqrt{-c x + 1}\right )}{32 \,{\left ({\left (c^{2} x^{2} - 1\right )} \sqrt{c x + 1} -{\left (c^{2} x^{2} - 1\right )} \sqrt{-c x + 1}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x, algorithm="maxima")

[Out]

1/2*a^3*(log(c*x + 1)/c - log(c*x - 1)/c) - 1/16*(b^3*log(c*x + 1) - b^3*log(-c*x + 1))*log(-sqrt(c*x + 1) + s
qrt(-c*x + 1))^3/c - integrate(1/32*(4*(sqrt(c*x + 1)*b^3 - sqrt(-c*x + 1)*b^3)*log(sqrt(c*x + 1) + sqrt(-c*x
+ 1))^3 + 24*(sqrt(c*x + 1)*a*b^2 - sqrt(-c*x + 1)*a*b^2)*log(sqrt(c*x + 1) + sqrt(-c*x + 1))^2 + 3*(4*(sqrt(c
*x + 1)*b^3 - sqrt(-c*x + 1)*b^3)*log(sqrt(c*x + 1) + sqrt(-c*x + 1)) + (8*a*b^2 - (b^3*c*x - b^3)*log(c*x + 1
) + (b^3*c*x - b^3)*log(-c*x + 1))*sqrt(c*x + 1) - (8*a*b^2 - (b^3*c*x + b^3)*log(c*x + 1) + (b^3*c*x + b^3)*l
og(-c*x + 1))*sqrt(-c*x + 1))*log(-sqrt(c*x + 1) + sqrt(-c*x + 1))^2 + 48*(sqrt(c*x + 1)*a^2*b - sqrt(-c*x + 1
)*a^2*b)*log(sqrt(c*x + 1) + sqrt(-c*x + 1)) - 12*(4*sqrt(c*x + 1)*a^2*b - 4*sqrt(-c*x + 1)*a^2*b + (sqrt(c*x
+ 1)*b^3 - sqrt(-c*x + 1)*b^3)*log(sqrt(c*x + 1) + sqrt(-c*x + 1))^2 + 4*(sqrt(c*x + 1)*a*b^2 - sqrt(-c*x + 1)
*a*b^2)*log(sqrt(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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b^{3} \operatorname{arcoth}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{3} + 3 \, a b^{2} \operatorname{arcoth}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} + 3 \, a^{2} b \operatorname{arcoth}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a^{3}}{c^{2} x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x, algorithm="fricas")

[Out]

integral(-(b^3*arccoth(sqrt(-c*x + 1)/sqrt(c*x + 1))^3 + 3*a*b^2*arccoth(sqrt(-c*x + 1)/sqrt(c*x + 1))^2 + 3*a
^2*b*arccoth(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a^3)/(c^2*x^2 - 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{a^{3}}{c^{2} x^{2} - 1}\, dx - \int \frac{b^{3} \operatorname{acoth}^{3}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac{3 a b^{2} \operatorname{acoth}^{2}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac{3 a^{2} b \operatorname{acoth}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acoth((-c*x+1)**(1/2)/(c*x+1)**(1/2)))**3/(-c**2*x**2+1),x)

[Out]

-Integral(a**3/(c**2*x**2 - 1), x) - Integral(b**3*acoth(sqrt(-c*x + 1)/sqrt(c*x + 1))**3/(c**2*x**2 - 1), x)
- Integral(3*a*b**2*acoth(sqrt(-c*x + 1)/sqrt(c*x + 1))**2/(c**2*x**2 - 1), x) - Integral(3*a**2*b*acoth(sqrt(
-c*x + 1)/sqrt(c*x + 1))/(c**2*x**2 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccoth((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x, algorithm="giac")

[Out]

integrate(-(b*arccoth(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a)^3/(c^2*x^2 - 1), x)