Integrand size = 35, antiderivative size = 496 \[ \int \frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=-\frac {i m (a+b \arccos (c x))^4}{12 b^2 c}+\frac {m (a+b \arccos (c x))^3 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{3 b c}+\frac {m (a+b \arccos (c x))^3 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{3 b c}-\frac {(a+b \arccos (c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {i m (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {i m (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {2 b m (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {2 b m (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {2 i b^2 m \operatorname {PolyLog}\left (4,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {2 i b^2 m \operatorname {PolyLog}\left (4,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c} \]
-1/12*I*m*(a+b*arccos(c*x))^4/b^2/c-1/3*(a+b*arccos(c*x))^3*ln(h*(g*x+f)^m )/b/c+1/3*m*(a+b*arccos(c*x))^3*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f-(c^ 2*f^2-g^2)^(1/2)))/b/c+1/3*m*(a+b*arccos(c*x))^3*ln(1+(c*x+I*(-c^2*x^2+1)^ (1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/b/c-I*m*(a+b*arccos(c*x))^2*polylog(2, -(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/c-I*m*(a+b*arccos (c*x))^2*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)) )/c+2*b*m*(a+b*arccos(c*x))*polylog(3,-(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f-( c^2*f^2-g^2)^(1/2)))/c+2*b*m*(a+b*arccos(c*x))*polylog(3,-(c*x+I*(-c^2*x^2 +1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/c+2*I*b^2*m*polylog(4,-(c*x+I*(-c^ 2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/c+2*I*b^2*m*polylog(4,-(c*x+I *(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/c
\[ \int \frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx \]
Time = 1.70 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {5279, 5241, 5031, 2620, 3011, 7163, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx\) |
\(\Big \downarrow \) 5279 |
\(\displaystyle \frac {g m \int \frac {(a+b \arccos (c x))^3}{f+g x}dx}{3 b c}-\frac {(a+b \arccos (c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}\) |
\(\Big \downarrow \) 5241 |
\(\displaystyle -\frac {g m \int \frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^3}{c f+c g x}d\arccos (c x)}{3 b c}-\frac {(a+b \arccos (c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}\) |
\(\Big \downarrow \) 5031 |
\(\displaystyle -\frac {(a+b \arccos (c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-i \int \frac {e^{i \arccos (c x)} (a+b \arccos (c x))^3}{c f+e^{i \arccos (c x)} g-\sqrt {c^2 f^2-g^2}}d\arccos (c x)-i \int \frac {e^{i \arccos (c x)} (a+b \arccos (c x))^3}{c f+e^{i \arccos (c x)} g+\sqrt {c^2 f^2-g^2}}d\arccos (c x)+\frac {i (a+b \arccos (c x))^4}{4 b g}\right )}{3 b c}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {(a+b \arccos (c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-i \left (\frac {3 i b \int (a+b \arccos (c x))^2 \log \left (\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )d\arccos (c x)}{g}-\frac {i (a+b \arccos (c x))^3 \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}\right )-i \left (\frac {3 i b \int (a+b \arccos (c x))^2 \log \left (\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )d\arccos (c x)}{g}-\frac {i (a+b \arccos (c x))^3 \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}\right )+\frac {i (a+b \arccos (c x))^4}{4 b g}\right )}{3 b c}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {(a+b \arccos (c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-i \left (\frac {3 i b \left (i (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-2 i b \int (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )d\arccos (c x)\right )}{g}-\frac {i (a+b \arccos (c x))^3 \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}\right )-i \left (\frac {3 i b \left (i (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-2 i b \int (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )d\arccos (c x)\right )}{g}-\frac {i (a+b \arccos (c x))^3 \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}\right )+\frac {i (a+b \arccos (c x))^4}{4 b g}\right )}{3 b c}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle -\frac {(a+b \arccos (c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-i \left (\frac {3 i b \left (i (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-2 i b \left (i b \int \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )d\arccos (c x)-i (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}-\frac {i (a+b \arccos (c x))^3 \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}\right )-i \left (\frac {3 i b \left (i (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-2 i b \left (i b \int \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )d\arccos (c x)-i (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}-\frac {i (a+b \arccos (c x))^3 \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}\right )+\frac {i (a+b \arccos (c x))^4}{4 b g}\right )}{3 b c}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {(a+b \arccos (c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-i \left (\frac {3 i b \left (i (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-2 i b \left (b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )de^{i \arccos (c x)}-i (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}-\frac {i (a+b \arccos (c x))^3 \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}\right )-i \left (\frac {3 i b \left (i (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-2 i b \left (b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )de^{i \arccos (c x)}-i (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}-\frac {i (a+b \arccos (c x))^3 \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}\right )+\frac {i (a+b \arccos (c x))^4}{4 b g}\right )}{3 b c}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {(a+b \arccos (c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-i \left (\frac {3 i b \left (i (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-2 i b \left (b \operatorname {PolyLog}\left (4,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-i (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}-\frac {i (a+b \arccos (c x))^3 \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}\right )-i \left (\frac {3 i b \left (i (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-2 i b \left (b \operatorname {PolyLog}\left (4,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-i (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}-\frac {i (a+b \arccos (c x))^3 \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}\right )+\frac {i (a+b \arccos (c x))^4}{4 b g}\right )}{3 b c}\) |
-1/3*((a + b*ArcCos[c*x])^3*Log[h*(f + g*x)^m])/(b*c) - (g*m*(((I/4)*(a + b*ArcCos[c*x])^4)/(b*g) - I*(((-I)*(a + b*ArcCos[c*x])^3*Log[1 + (E^(I*Arc Cos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g + ((3*I)*b*(I*(a + b*ArcCos[c *x])^2*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))] - (2*I)*b*((-I)*(a + b*ArcCos[c*x])*PolyLog[3, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))] + b*PolyLog[4, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqr t[c^2*f^2 - g^2]))])))/g) - I*(((-I)*(a + b*ArcCos[c*x])^3*Log[1 + (E^(I*A rcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g + ((3*I)*b*(I*(a + b*ArcCos [c*x])^2*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))] - (2*I)*b*((-I)*(a + b*ArcCos[c*x])*PolyLog[3, -((E^(I*ArcCos[c*x])*g)/(c* f + Sqrt[c^2*f^2 - g^2]))] + b*PolyLog[4, -((E^(I*ArcCos[c*x])*g)/(c*f + S qrt[c^2*f^2 - g^2]))])))/g)))/(3*b*c)
3.1.20.3.1 Defintions of rubi rules used
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] - Si mp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)])/(Cos[(c_.) + (d_.) *(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[I*((e + f*x)^(m + 1)/(b*f*(m + 1))) , x] + (-Simp[I Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(I*(c + d*x)))), x], x] - Simp[I Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(I*(c + d*x)))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Subst[Int[(a + b*x)^n*(Sin[x]/(c*d + e*Cos[x])), x], x, ArcCos[c*x]] / ; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
Int[(Log[(h_.)*((f_.) + (g_.)*(x_))^(m_.)]*((a_.) + ArcCos[(c_.)*(x_)]*(b_. ))^(n_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-Log[h*(f + g*x)^m] )*((a + b*ArcCos[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))), x] + Simp[g*(m/(b*c* Sqrt[d]*(n + 1))) Int[(a + b*ArcCos[c*x])^(n + 1)/(f + g*x), x], x] /; Fr eeQ[{a, b, c, d, e, f, g, h, m}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IG tQ[n, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
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] - Simp[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]
\[\int \frac {\left (a +b \arccos \left (c x \right )\right )^{2} \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {-c^{2} x^{2}+1}}d x\]
\[ \int \frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]
integral(-sqrt(-c^2*x^2 + 1)*(b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^2) *log((g*x + f)^m*h)/(c^2*x^2 - 1), x)
\[ \int \frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2} \log {\left (h \left (f + g x\right )^{m} \right )}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
\[ \int \frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]
(b^2*c*integrate(arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x)*log(h) + 2*a*b*c*integrate(arctan2(sqrt(c*x + 1)*sq rt(-c*x + 1), c*x)/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x)*log(h) + b^2*c*integ rate(arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2*log((g*x + f)^m)/(sqrt(c *x + 1)*sqrt(-c*x + 1)), x) + 2*a*b*c*integrate(arctan2(sqrt(c*x + 1)*sqrt (-c*x + 1), c*x)*log((g*x + f)^m)/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + a^2 *c*integrate(log((g*x + f)^m)/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + a^2*arc tan2(c*x, sqrt(-c^2*x^2 + 1))*log(h))/c
\[ \int \frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{\sqrt {1-c^2\,x^2}} \,d x \]