Integrand size = 22, antiderivative size = 236 \[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {x \arccos (a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {i \sqrt {1-a^2 x^2} \arccos (a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \arccos (a x)^2 \log \left (1-e^{2 i \arccos (a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {3 i \sqrt {1-a^2 x^2} \arccos (a x) \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (3,e^{2 i \arccos (a x)}\right )}{2 a c \sqrt {c-a^2 c x^2}} \] Output:
x*arccos(a*x)^3/c/(-a^2*c*x^2+c)^(1/2)+I*(-a^2*x^2+1)^(1/2)*arccos(a*x)^3/ a/c/(-a^2*c*x^2+c)^(1/2)-3*(-a^2*x^2+1)^(1/2)*arccos(a*x)^2*ln(1-(a*x+I*(- a^2*x^2+1)^(1/2))^2)/a/c/(-a^2*c*x^2+c)^(1/2)+3*I*(-a^2*x^2+1)^(1/2)*arcco s(a*x)*polylog(2,(a*x+I*(-a^2*x^2+1)^(1/2))^2)/a/c/(-a^2*c*x^2+c)^(1/2)-3/ 2*(-a^2*x^2+1)^(1/2)*polylog(3,(a*x+I*(-a^2*x^2+1)^(1/2))^2)/a/c/(-a^2*c*x ^2+c)^(1/2)
Time = 0.23 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.77 \[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {-i \pi ^3 \sqrt {1-a^2 x^2}-8 a x \arccos (a x)^3+8 i \sqrt {1-a^2 x^2} \arccos (a x)^3+24 \sqrt {1-a^2 x^2} \arccos (a x)^2 \log \left (1-e^{-2 i \arccos (a x)}\right )+24 i \sqrt {1-a^2 x^2} \arccos (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arccos (a x)}\right )+12 \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (3,e^{-2 i \arccos (a x)}\right )}{8 a c \sqrt {c \left (1-a^2 x^2\right )}} \] Input:
Integrate[ArcCos[a*x]^3/(c - a^2*c*x^2)^(3/2),x]
Output:
-1/8*((-I)*Pi^3*Sqrt[1 - a^2*x^2] - 8*a*x*ArcCos[a*x]^3 + (8*I)*Sqrt[1 - a ^2*x^2]*ArcCos[a*x]^3 + 24*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2*Log[1 - E^((-2* I)*ArcCos[a*x])] + (24*I)*Sqrt[1 - a^2*x^2]*ArcCos[a*x]*PolyLog[2, E^((-2* I)*ArcCos[a*x])] + 12*Sqrt[1 - a^2*x^2]*PolyLog[3, E^((-2*I)*ArcCos[a*x])] )/(a*c*Sqrt[c*(1 - a^2*x^2)])
Time = 0.75 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.64, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5161, 5181, 3042, 25, 4200, 25, 2620, 3011, 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.
\(\displaystyle \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 5161 |
\(\displaystyle \frac {3 a \sqrt {1-a^2 x^2} \int \frac {x \arccos (a x)^2}{1-a^2 x^2}dx}{c \sqrt {c-a^2 c x^2}}+\frac {x \arccos (a x)^3}{c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 5181 |
\(\displaystyle \frac {x \arccos (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \int \frac {a x \arccos (a x)^2}{\sqrt {1-a^2 x^2}}d\arccos (a x)}{a c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x \arccos (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \int -\arccos (a x)^2 \tan \left (\arccos (a x)+\frac {\pi }{2}\right )d\arccos (a x)}{a c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3 \sqrt {1-a^2 x^2} \int \arccos (a x)^2 \tan \left (\arccos (a x)+\frac {\pi }{2}\right )d\arccos (a x)}{a c \sqrt {c-a^2 c x^2}}+\frac {x \arccos (a x)^3}{c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 4200 |
\(\displaystyle \frac {x \arccos (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (2 i \int -\frac {e^{2 i \arccos (a x)} \arccos (a x)^2}{1-e^{2 i \arccos (a x)}}d\arccos (a x)-\frac {1}{3} i \arccos (a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {x \arccos (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (-2 i \int \frac {e^{2 i \arccos (a x)} \arccos (a x)^2}{1-e^{2 i \arccos (a x)}}d\arccos (a x)-\frac {1}{3} i \arccos (a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {x \arccos (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (-2 i \left (\frac {1}{2} i \arccos (a x)^2 \log \left (1-e^{2 i \arccos (a x)}\right )-i \int \arccos (a x) \log \left (1-e^{2 i \arccos (a x)}\right )d\arccos (a x)\right )-\frac {1}{3} i \arccos (a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {x \arccos (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (-2 i \left (\frac {1}{2} i \arccos (a x)^2 \log \left (1-e^{2 i \arccos (a x)}\right )-i \left (\frac {1}{2} i \arccos (a x) \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )-\frac {1}{2} i \int \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )d\arccos (a x)\right )\right )-\frac {1}{3} i \arccos (a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {x \arccos (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (-2 i \left (\frac {1}{2} i \arccos (a x)^2 \log \left (1-e^{2 i \arccos (a x)}\right )-i \left (\frac {1}{2} i \arccos (a x) \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )-\frac {1}{4} \int e^{-2 i \arccos (a x)} \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )de^{2 i \arccos (a x)}\right )\right )-\frac {1}{3} i \arccos (a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {x \arccos (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (-2 i \left (\frac {1}{2} i \arccos (a x)^2 \log \left (1-e^{2 i \arccos (a x)}\right )-i \left (\frac {1}{2} i \arccos (a x) \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )-\frac {1}{4} \operatorname {PolyLog}\left (3,e^{2 i \arccos (a x)}\right )\right )\right )-\frac {1}{3} i \arccos (a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\) |
Input:
Int[ArcCos[a*x]^3/(c - a^2*c*x^2)^(3/2),x]
Output:
(x*ArcCos[a*x]^3)/(c*Sqrt[c - a^2*c*x^2]) - (3*Sqrt[1 - a^2*x^2]*((-1/3*I) *ArcCos[a*x]^3 - (2*I)*((I/2)*ArcCos[a*x]^2*Log[1 - E^((2*I)*ArcCos[a*x])] - I*((I/2)*ArcCos[a*x]*PolyLog[2, E^((2*I)*ArcCos[a*x])] - PolyLog[3, E^( (2*I)*ArcCos[a*x])]/4))))/(a*c*Sqrt[c - a^2*c*x^2])
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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^ m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] , x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[x*((a + b*ArcCos[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp[b *c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcCos[c*x ])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[1/e Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[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]
Time = 0.55 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.18
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-i \sqrt {-a^{2} x^{2}+1}+a x \right ) \arccos \left (a x \right )^{3}}{c^{2} a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \arccos \left (a x \right )^{3}-3 \arccos \left (a x \right )^{2} \ln \left (1+a x +i \sqrt {-a^{2} x^{2}+1}\right )-3 \arccos \left (a x \right )^{2} \ln \left (1-a x -i \sqrt {-a^{2} x^{2}+1}\right )+6 i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -a x -i \sqrt {-a^{2} x^{2}+1}\right )+6 i \arccos \left (a x \right ) \operatorname {polylog}\left (2, a x +i \sqrt {-a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -a x -i \sqrt {-a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}\) | \(278\) |
Input:
int(arccos(a*x)^3/(-a^2*c*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-(-c*(a^2*x^2-1))^(1/2)*(-I*(-a^2*x^2+1)^(1/2)+a*x)*arccos(a*x)^3/c^2/a/(a ^2*x^2-1)-(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)*(2*I*arccos(a*x)^3-3*a rccos(a*x)^2*ln(1+a*x+I*(-a^2*x^2+1)^(1/2))-3*arccos(a*x)^2*ln(1-a*x-I*(-a ^2*x^2+1)^(1/2))+6*I*arccos(a*x)*polylog(2,-a*x-I*(-a^2*x^2+1)^(1/2))+6*I* arccos(a*x)*polylog(2,a*x+I*(-a^2*x^2+1)^(1/2))-6*polylog(3,-a*x-I*(-a^2*x ^2+1)^(1/2))-6*polylog(3,a*x+I*(-a^2*x^2+1)^(1/2)))/c^2/a/(a^2*x^2-1)
\[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(arccos(a*x)^3/(-a^2*c*x^2+c)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-a^2*c*x^2 + c)*arccos(a*x)^3/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)
\[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(acos(a*x)**3/(-a**2*c*x**2+c)**(3/2),x)
Output:
Integral(acos(a*x)**3/(-c*(a*x - 1)*(a*x + 1))**(3/2), x)
\[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(arccos(a*x)^3/(-a^2*c*x^2+c)^(3/2),x, algorithm="maxima")
Output:
integrate(arccos(a*x)^3/(-a^2*c*x^2 + c)^(3/2), x)
\[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(arccos(a*x)^3/(-a^2*c*x^2+c)^(3/2),x, algorithm="giac")
Output:
integrate(arccos(a*x)^3/(-a^2*c*x^2 + c)^(3/2), x)
Timed out. \[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \] Input:
int(acos(a*x)^3/(c - a^2*c*x^2)^(3/2),x)
Output:
int(acos(a*x)^3/(c - a^2*c*x^2)^(3/2), x)
\[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {\int \frac {\mathit {acos} \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}d x}{\sqrt {c}\, c} \] Input:
int(acos(a*x)^3/(-a^2*c*x^2+c)^(3/2),x)
Output:
( - int(acos(a*x)**3/(sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 + 1)),x))/(sqrt(c)*c)