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

Optimal. Leaf size=265 \[ \frac {3 b^2 \text {Li}_3\left (-e^{-2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{2 c}+\frac {3 b \text {Li}_2\left (-e^{-2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{2 c}-\frac {\left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^4}{4 b c}-\frac {\log \left (e^{-2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}+1\right ) \left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{c}+\frac {3 b^3 \text {Li}_4\left (-e^{-2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{4 c} \]

[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

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Rubi [A]  time = 0.22, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6681, 5660, 3718, 2190, 2531, 6609, 2282, 6589} \[ \frac {3 b^2 \text {PolyLog}\left (3,-e^{2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{2 c}-\frac {3 b \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{2 c}-\frac {3 b^3 \text {PolyLog}\left (4,-e^{2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{4 c}+\frac {\left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^4}{4 b c}-\frac {\log \left (e^{2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}+1\right ) \left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^4/(4*b*c) - ((a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*Log[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*PolyLo
g[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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2282

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 2531

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 3718

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 5660

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

Rule 6589

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 6609

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 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]

Rubi steps

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

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Mathematica [F]  time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*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]

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fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b^{3} \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{3} + 3 \, a b^{2} \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{2} + 3 \, a^{2} b \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a^{3}}{c^{2} x^{2} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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,x):;OUTPUT:Warn
ing, choosing root of [1,0,-4,0,%%%{4,[2,2]%%%}] at parameters values [86,-97]Warning, choosing root of [1,0,-
4,0,%%%{4,[2,2]%%%}] at parameters values [-82,7]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur
& l) Error: Bad Argument Value

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maple [B]  time = 0.90, size = 870, normalized size = 3.28 \[ \frac {a^{3} \ln \left (c x +1\right )}{2 c}-\frac {a^{3} \ln \left (c x -1\right )}{2 c}+\frac {b^{3} \mathrm {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{4}}{4 c}-\frac {b^{3} \mathrm {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3} \ln \left (1+\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 {3 b^{3} \mathrm {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \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 b^{3} \mathrm {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \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 b^{3} \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}+\frac {a \,b^{2} \mathrm {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3}}{c}-\frac {3 a \,b^{2} \mathrm {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\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 {3 a \,b^{2} \mathrm {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \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 {3 a \,b^{2} \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 a^{2} b \mathrm {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}-\frac {3 a^{2} b \,\mathrm {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\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 {3 a^{2} b \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} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int -\frac {{\left (a+b\,\mathrm {acosh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^3}{c^2\,x^2-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {a^{3}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{3} \operatorname {acosh}^{3}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {3 a b^{2} \operatorname {acosh}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {3 a^{2} b \operatorname {acosh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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