Integrand size = 10, antiderivative size = 122 \[ \int \frac {\arccos (a x)^3}{x^2} \, dx=-\frac {\arccos (a x)^3}{x}-6 i a \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )+6 i a \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-6 i a \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-6 a \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+6 a \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right ) \]
-arccos(a*x)^3/x-6*I*a*arccos(a*x)^2*arctan(a*x+I*(-a^2*x^2+1)^(1/2))+6*I* a*arccos(a*x)*polylog(2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))-6*I*a*arccos(a*x)*p olylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))-6*a*polylog(3,-I*(a*x+I*(-a^2*x^2+1 )^(1/2)))+6*a*polylog(3,I*(a*x+I*(-a^2*x^2+1)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.14 \[ \int \frac {\arccos (a x)^3}{x^2} \, dx=-\frac {\arccos (a x)^3}{x}+3 a \left (\arccos (a x)^2 \left (\log \left (1-i e^{i \arccos (a x)}\right )-\log \left (1+i e^{i \arccos (a x)}\right )\right )+2 i \arccos (a x) \left (\operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )-2 \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+2 \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )\right ) \]
-(ArcCos[a*x]^3/x) + 3*a*(ArcCos[a*x]^2*(Log[1 - I*E^(I*ArcCos[a*x])] - Lo g[1 + I*E^(I*ArcCos[a*x])]) + (2*I)*ArcCos[a*x]*(PolyLog[2, (-I)*E^(I*ArcC os[a*x])] - PolyLog[2, I*E^(I*ArcCos[a*x])]) - 2*PolyLog[3, (-I)*E^(I*ArcC os[a*x])] + 2*PolyLog[3, I*E^(I*ArcCos[a*x])])
Time = 0.55 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5139, 5219, 3042, 4669, 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.
\(\displaystyle \int \frac {\arccos (a x)^3}{x^2} \, dx\) |
\(\Big \downarrow \) 5139 |
\(\displaystyle -3 a \int \frac {\arccos (a x)^2}{x \sqrt {1-a^2 x^2}}dx-\frac {\arccos (a x)^3}{x}\) |
\(\Big \downarrow \) 5219 |
\(\displaystyle 3 a \int \frac {\arccos (a x)^2}{a x}d\arccos (a x)-\frac {\arccos (a x)^3}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 a \int \arccos (a x)^2 \csc \left (\arccos (a x)+\frac {\pi }{2}\right )d\arccos (a x)-\frac {\arccos (a x)^3}{x}\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle -\frac {\arccos (a x)^3}{x}+3 a \left (-2 \int \arccos (a x) \log \left (1-i e^{i \arccos (a x)}\right )d\arccos (a x)+2 \int \arccos (a x) \log \left (1+i e^{i \arccos (a x)}\right )d\arccos (a x)-2 i \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {\arccos (a x)^3}{x}+3 a \left (2 \left (i \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )d\arccos (a x)\right )-2 \left (i \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )d\arccos (a x)\right )-2 i \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {\arccos (a x)^3}{x}+3 a \left (2 \left (i \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-\int e^{-i \arccos (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}\right )-2 \left (i \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-\int e^{-i \arccos (a x)} \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}\right )-2 i \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {\arccos (a x)^3}{x}+3 a \left (-2 i \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )+2 \left (i \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )\right )-2 \left (i \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )\right )\right )\) |
-(ArcCos[a*x]^3/x) + 3*a*((-2*I)*ArcCos[a*x]^2*ArcTan[E^(I*ArcCos[a*x])] + 2*(I*ArcCos[a*x]*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - PolyLog[3, (-I)*E^( I*ArcCos[a*x])]) - 2*(I*ArcCos[a*x]*PolyLog[2, I*E^(I*ArcCos[a*x])] - Poly Log[3, I*E^(I*ArcCos[a*x])]))
3.1.28.3.1 Defintions of rubi rules used
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[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(-(c^(m + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[ d + e*x^2]] Subst[Int[(a + b*x)^n*Cos[x]^m, x], x, ArcCos[c*x]], x] /; Fr eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
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]
\[\int \frac {\arccos \left (a x \right )^{3}}{x^{2}}d x\]
\[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{2}} \,d x } \]
\[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{x^{2}}\, dx \]
\[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{2}} \,d x } \]
-(arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3 - 3*a*x*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2/(a^2*x^3 - x), x))/x
\[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^3}{x^2} \,d x \]