\(\int \frac {(a+b \arccos (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}))^3}{1-c^2 x^2} \, dx\) [102]

Optimal result

Integrand size = 40, antiderivative size = 279 \[ \int \frac {\left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\frac {i \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1+e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {3 i b \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}-\frac {3 b^2 \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{4 c} \]

[Out]

1/4*I*(a+b*arccos((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^4/b/c-(a+b*arccos((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3*ln(1+((-c*
x+1)^(1/2)/(c*x+1)^(1/2)+I*(1-(-c*x+1)/(c*x+1))^(1/2))^2)/c+3/2*I*b*(a+b*arccos((-c*x+1)^(1/2)/(c*x+1)^(1/2)))
^2*polylog(2,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+I*(1-(-c*x+1)/(c*x+1))^(1/2))^2)/c-3/2*b^2*(a+b*arccos((-c*x+1)^(1
/2)/(c*x+1)^(1/2)))*polylog(3,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+I*(1-(-c*x+1)/(c*x+1))^(1/2))^2)/c-3/4*I*b^3*poly
log(4,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+I*(1-(-c*x+1)/(c*x+1))^(1/2))^2)/c

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 279, 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, 4722, 3800, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {\left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=-\frac {3 b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{2 c}+\frac {3 i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{2 c}+\frac {i \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^4}{4 b c}-\frac {\log \left (1+e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{c}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{4 c} \]

[In]

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

[Out]

((I/4)*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^4)/(b*c) - ((a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*L
og[1 + E^((2*I)*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (((3*I)/2)*b*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c
*x]])^2*PolyLog[2, -E^((2*I)*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c - (3*b^2*(a + b*ArcCos[Sqrt[1 - c*x]/Sqr
t[1 + c*x]])*PolyLog[3, -E^((2*I)*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c) - (((3*I)/4)*b^3*PolyLog[4, -E^
((2*I)*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/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 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 4722

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcCos[c
*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 \arccos (x))^3}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c} \\ & = \frac {\text {Subst}\left (\int (a+b x)^3 \tan (x) \, dx,x,\arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c} \\ & = \frac {i \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {(2 i) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^3}{1+e^{2 i x}} \, dx,x,\arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c} \\ & = \frac {i \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1+e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {(3 b) \text {Subst}\left (\int (a+b x)^2 \log \left (1+e^{2 i x}\right ) \, dx,x,\arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c} \\ & = \frac {i \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1+e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {3 i b \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}-\frac {\left (3 i b^2\right ) \text {Subst}\left (\int (a+b x) \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \, dx,x,\arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c} \\ & = \frac {i \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1+e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {3 i b \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}-\frac {3 b^2 \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{2 i x}\right ) \, dx,x,\arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{2 c} \\ & = \frac {i \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1+e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {3 i b \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}-\frac {3 b^2 \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}-\frac {\left (3 i b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{4 c} \\ & = \frac {i \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1+e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {3 i b \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}-\frac {3 b^2 \left (a+b \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \arccos \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{4 c} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 680 vs. \(2 (308 ) = 616\).

Time = 6.02 (sec) , antiderivative size = 681, normalized size of antiderivative = 2.44

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 {i \arccos \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{4}}{4 c}+\frac {\arccos \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}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{c}-\frac {3 i \arccos \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}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{2 c}+\frac {3 \arccos \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}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{2 c}+\frac {3 i \operatorname {polylog}\left (4, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{4 c}\right )-3 a \,b^{2} \left (-\frac {i \arccos \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3}}{3 c}+\frac {\arccos \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}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{c}-\frac {i \arccos \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}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{c}+\frac {\operatorname {polylog}\left (3, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{2 c}\right )-3 a^{2} b \left (-\frac {i \arccos \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}+\frac {\arccos \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{c}-\frac {i \operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{2 c}\right )\)	\(681\)
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 {i \arccos \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{4}}{4 c}+\frac {\arccos \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}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{c}-\frac {3 i \arccos \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}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{2 c}+\frac {3 \arccos \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}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{2 c}+\frac {3 i \operatorname {polylog}\left (4, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{4 c}\right )-3 a \,b^{2} \left (-\frac {i \arccos \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3}}{3 c}+\frac {\arccos \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}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{c}-\frac {i \arccos \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}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{c}+\frac {\operatorname {polylog}\left (3, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{2 c}\right )-3 a^{2} b \left (-\frac {i \arccos \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}+\frac {\arccos \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{c}-\frac {i \operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+i \sqrt {1-\frac {-c x +1}{c x +1}}\right )^{2}\right )}{2 c}\right )\)	\(681\)

[In]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

integrate((a+b*arccos((-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) - integrate((b^3*arctan2(sqrt(2)*sqrt(c)*sqrt(x), sqrt(-c*x + 1))^3
+ 3*a*b^2*arctan2(sqrt(2)*sqrt(c)*sqrt(x), sqrt(-c*x + 1))^2 + 3*a^2*b*arctan2(sqrt(2)*sqrt(c)*sqrt(x), sqrt(-
c*x + 1)))/(c^2*x^2 - 1), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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