\(\int \frac {(a+b \text {arccosh}(\frac {\sqrt {1-c x}}{\sqrt {1+c x}}))^3}{1-c^2 x^2} \, dx\) [269]

Optimal result

Integrand size = 40, antiderivative size = 265 \[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=-\frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1+e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {3 b \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}+\frac {3 b^2 \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}+\frac {3 b^3 \operatorname {PolyLog}\left (4,-e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{4 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6813, 5882, 3799, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\frac {3 b^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{2 c}+\frac {3 b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{2 c}-\frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^4}{4 b c}-\frac {\log \left (e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}+1\right ) \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{c}+\frac {3 b^3 \operatorname {PolyLog}\left (4,-e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{4 c} \]

[In]

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

[Out]

-1/4*(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^4/(b*c) - ((a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*Lo
g[1 + E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (3*b*(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*Po
lyLog[2, -E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c) + (3*b^2*(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c
*x]])*PolyLog[3, -E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c) + (3*b^3*PolyLog[4, -E^(-2*ArcCosh[Sqrt[
1 - c*x]/Sqrt[1 + c*x]])])/(4*c)

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5882

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Tanh[-a/b + x/b], x],
x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 6724

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

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

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

Mathematica [F]

\[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(843\) vs. \(2(381)=762\).

Time = 2.54 (sec) , antiderivative size = 844, normalized size of antiderivative = 3.18

method	result	size
default	\(\frac {a^{3} \ln \left (c x +1\right )}{2 c}-\frac {a^{3} \ln \left (c x -1\right )}{2 c}-b^{3} \left (-\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{4}}{4 c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3} \ln \left (\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}+1\right )}{c}+\frac {3 \operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}-\frac {3 \,\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (3, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}+\frac {3 \operatorname {polylog}\left (4, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{4 c}\right )-3 a \,b^{2} \left (-\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3}}{3 c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}+1\right )}{c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{c}-\frac {\operatorname {polylog}\left (3, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}\right )-3 a^{2} b \left (-\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}+1\right )}{c}+\frac {\operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}\right )\)	\(844\)
parts	\(\frac {a^{3} \ln \left (c x +1\right )}{2 c}-\frac {a^{3} \ln \left (c x -1\right )}{2 c}-b^{3} \left (-\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{4}}{4 c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3} \ln \left (\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}+1\right )}{c}+\frac {3 \operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}-\frac {3 \,\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (3, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}+\frac {3 \operatorname {polylog}\left (4, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{4 c}\right )-3 a \,b^{2} \left (-\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3}}{3 c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}+1\right )}{c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{c}-\frac {\operatorname {polylog}\left (3, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}\right )-3 a^{2} b \left (-\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}+1\right )}{c}+\frac {\operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}\right )\)	\(844\)

[In]

int((a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=- \int \frac {a^{3}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{3} \operatorname {acosh}^{3}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {3 a b^{2} \operatorname {acosh}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {3 a^{2} b \operatorname {acosh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]

[In]

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

Maxima [F]

\[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \]

[In]

integrate((a+b*arccosh((-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) + 1/2*(b^3*log(c*x + 1) - b^3*log(-c*x + 1))*log(sqrt(sqrt(c*x + 1)
+ sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + sqrt(-c*x + 1))^3/c + integrate(1/8*(((c*x + 1)*sqrt
(-c*x + 1)*b^3 - (-c*x + 1)^(3/2)*b^3)*log(c*x + 1)^3 - 6*((c*x + 1)*sqrt(-c*x + 1)*a*b^2 - (-c*x + 1)^(3/2)*a
*b^2)*log(c*x + 1)^2 - 6*((4*a*b^2 + (b^3*c*x + b^3)*log(c*x + 1) - (b^3*c*x + b^3)*log(-c*x + 1))*(c*x + 1)*s
qrt(-c*x + 1) - (4*a*b^2 + (b^3*c*x + b^3)*log(c*x + 1) - (b^3*c*x + b^3)*log(-c*x + 1))*(-c*x + 1)^(3/2) + ((
4*a*b^2 + (b^3*c*x - b^3)*log(c*x + 1) - (b^3*c*x - b^3)*log(-c*x + 1))*(c*x + 1) + (4*a*b^2 + (b^3*c*x + b^3)
*log(c*x + 1) - (b^3*c*x + b^3)*log(-c*x + 1))*(c*x - 1) - 2*((c*x + 1)*b^3 + (c*x - 1)*b^3)*log(c*x + 1))*sqr
t(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) - 2*((c*x + 1)*sqrt(-c*x + 1)*b^3 - (-
c*x + 1)^(3/2)*b^3)*log(c*x + 1))*log(sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1
)) + sqrt(-c*x + 1))^2 + (((c*x + 1)*b^3 + (c*x - 1)*b^3)*log(c*x + 1)^3 - 6*((c*x + 1)*a*b^2 + (c*x - 1)*a*b^
2)*log(c*x + 1)^2 + 12*((c*x + 1)*a^2*b + (c*x - 1)*a^2*b)*log(c*x + 1))*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*
sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + 12*((c*x + 1)*sqrt(-c*x + 1)*a^2*b - (-c*x + 1)^(3/2)*a^2*b)*log(c*x +
1) - 6*(4*(c*x + 1)*sqrt(-c*x + 1)*a^2*b - 4*(-c*x + 1)^(3/2)*a^2*b + ((c*x + 1)*sqrt(-c*x + 1)*b^3 - (-c*x +
1)^(3/2)*b^3)*log(c*x + 1)^2 + (4*(c*x + 1)*a^2*b + 4*(c*x - 1)*a^2*b + ((c*x + 1)*b^3 + (c*x - 1)*b^3)*log(c
*x + 1)^2 - 4*((c*x + 1)*a*b^2 + (c*x - 1)*a*b^2)*log(c*x + 1))*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqr
t(c*x + 1) + sqrt(-c*x + 1)) - 4*((c*x + 1)*sqrt(-c*x + 1)*a*b^2 - (-c*x + 1)^(3/2)*a*b^2)*log(c*x + 1))*log(s
qrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + sqrt(-c*x + 1)))/((c^2*x^2 - 1)*(c
*x + 1)*sqrt(-c*x + 1) - (c^2*x^2 - 1)*(-c*x + 1)^(3/2) + ((c^2*x^2 - 1)*(c*x + 1) + (c^2*x^2 - 1)*(c*x - 1))*
sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1))), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\mathrm {acosh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^3}{c^2\,x^2-1} \,d x \]

[In]

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

[Out]

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