Integrand size = 24, antiderivative size = 264 \[ \int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=-\frac {3 a \arccos (a x)^2}{2 x}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}-6 a^2 \arccos (a x) \text {arctanh}\left (e^{i \arccos (a x)}\right )-a^2 \arccos (a x)^3 \text {arctanh}\left (e^{i \arccos (a x)}\right )+3 i a^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )+\frac {3}{2} i a^2 \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-3 i a^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-\frac {3}{2} i a^2 \arccos (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-3 a^2 \arccos (a x) \operatorname {PolyLog}\left (3,-e^{i \arccos (a x)}\right )+3 a^2 \arccos (a x) \operatorname {PolyLog}\left (3,e^{i \arccos (a x)}\right )-3 i a^2 \operatorname {PolyLog}\left (4,-e^{i \arccos (a x)}\right )+3 i a^2 \operatorname {PolyLog}\left (4,e^{i \arccos (a x)}\right ) \] Output:
-3/2*a*arccos(a*x)^2/x-1/2*(-a^2*x^2+1)^(1/2)*arccos(a*x)^3/x^2-6*a^2*arcc os(a*x)*arctanh(a*x+I*(-a^2*x^2+1)^(1/2))-a^2*arccos(a*x)^3*arctanh(a*x+I* (-a^2*x^2+1)^(1/2))+3*I*a^2*polylog(2,-a*x-I*(-a^2*x^2+1)^(1/2))+3/2*I*a^2 *arccos(a*x)^2*polylog(2,-a*x-I*(-a^2*x^2+1)^(1/2))-3*I*a^2*polylog(2,a*x+ I*(-a^2*x^2+1)^(1/2))-3/2*I*a^2*arccos(a*x)^2*polylog(2,a*x+I*(-a^2*x^2+1) ^(1/2))-3*a^2*arccos(a*x)*polylog(3,-a*x-I*(-a^2*x^2+1)^(1/2))+3*a^2*arcco s(a*x)*polylog(3,a*x+I*(-a^2*x^2+1)^(1/2))-3*I*a^2*polylog(4,-a*x-I*(-a^2* x^2+1)^(1/2))+3*I*a^2*polylog(4,a*x+I*(-a^2*x^2+1)^(1/2))
Time = 0.55 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.93 \[ \int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {1}{2} \left (\frac {3 a \arccos (a x)^2}{x}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{x^2}+12 i a^2 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+2 i a^2 \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )-3 i a^2 \left (2+\arccos (a x)^2\right ) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+3 i a^2 \left (2+\arccos (a x)^2\right ) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )+6 a^2 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )-6 a^2 \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )+6 i a^2 \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )-6 i a^2 \operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right )\right ) \] Input:
Integrate[ArcCos[a*x]^3/(x^3*Sqrt[1 - a^2*x^2]),x]
Output:
((3*a*ArcCos[a*x]^2)/x - (Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/x^2 + (12*I)*a^ 2*ArcCos[a*x]*ArcTan[E^(I*ArcCos[a*x])] + (2*I)*a^2*ArcCos[a*x]^3*ArcTan[E ^(I*ArcCos[a*x])] - (3*I)*a^2*(2 + ArcCos[a*x]^2)*PolyLog[2, (-I)*E^(I*Arc Cos[a*x])] + (3*I)*a^2*(2 + ArcCos[a*x]^2)*PolyLog[2, I*E^(I*ArcCos[a*x])] + 6*a^2*ArcCos[a*x]*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] - 6*a^2*ArcCos[a*x ]*PolyLog[3, I*E^(I*ArcCos[a*x])] + (6*I)*a^2*PolyLog[4, (-I)*E^(I*ArcCos[ a*x])] - (6*I)*a^2*PolyLog[4, I*E^(I*ArcCos[a*x])])/2
Time = 1.41 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5205, 5139, 5219, 3042, 4669, 2715, 2838, 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.
\(\displaystyle \int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 5205 |
\(\displaystyle \frac {1}{2} a^2 \int \frac {\arccos (a x)^3}{x \sqrt {1-a^2 x^2}}dx-\frac {3}{2} a \int \frac {\arccos (a x)^2}{x^2}dx-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 5139 |
\(\displaystyle -\frac {3}{2} a \left (-2 a \int \frac {\arccos (a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\arccos (a x)^2}{x}\right )+\frac {1}{2} a^2 \int \frac {\arccos (a x)^3}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 5219 |
\(\displaystyle -\frac {1}{2} a^2 \int \frac {\arccos (a x)^3}{a x}d\arccos (a x)-\frac {3}{2} a \left (2 a \int \frac {\arccos (a x)}{a x}d\arccos (a x)-\frac {\arccos (a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{2} a^2 \int \arccos (a x)^3 \csc \left (\arccos (a x)+\frac {\pi }{2}\right )d\arccos (a x)-\frac {3}{2} a \left (2 a \int \arccos (a x) \csc \left (\arccos (a x)+\frac {\pi }{2}\right )d\arccos (a x)-\frac {\arccos (a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle -\frac {1}{2} a^2 \left (-3 \int \arccos (a x)^2 \log \left (1-i e^{i \arccos (a x)}\right )d\arccos (a x)+3 \int \arccos (a x)^2 \log \left (1+i e^{i \arccos (a x)}\right )d\arccos (a x)-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )\right )-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (-\int \log \left (1-i e^{i \arccos (a x)}\right )d\arccos (a x)+\int \log \left (1+i e^{i \arccos (a x)}\right )d\arccos (a x)-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {1}{2} a^2 \left (-3 \int \arccos (a x)^2 \log \left (1-i e^{i \arccos (a x)}\right )d\arccos (a x)+3 \int \arccos (a x)^2 \log \left (1+i e^{i \arccos (a x)}\right )d\arccos (a x)-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )\right )-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (i \int e^{-i \arccos (a x)} \log \left (1-i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-i \int e^{-i \arccos (a x)} \log \left (1+i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {1}{2} a^2 \left (-3 \int \arccos (a x)^2 \log \left (1-i e^{i \arccos (a x)}\right )d\arccos (a x)+3 \int \arccos (a x)^2 \log \left (1+i e^{i \arccos (a x)}\right )d\arccos (a x)-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {1}{2} a^2 \left (3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )d\arccos (a x)\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )d\arccos (a x)\right )-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )\right )\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle -\frac {1}{2} a^2 \left (3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )d\arccos (a x)-i \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )\right )\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )d\arccos (a x)-i \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )\right )\right )-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {1}{2} a^2 \left (3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i \left (\int e^{-i \arccos (a x)} \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-i \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )\right )\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-2 i \left (\int e^{-i \arccos (a x)} \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-i \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )\right )\right )-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {1}{2} a^2 \left (-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )-i \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )\right )\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right )-i \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )\right )\) |
Input:
Int[ArcCos[a*x]^3/(x^3*Sqrt[1 - a^2*x^2]),x]
Output:
-1/2*(Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/x^2 - (3*a*(-(ArcCos[a*x]^2/x) + 2* a*((-2*I)*ArcCos[a*x]*ArcTan[E^(I*ArcCos[a*x])] + I*PolyLog[2, (-I)*E^(I*A rcCos[a*x])] - I*PolyLog[2, I*E^(I*ArcCos[a*x])])))/2 - (a^2*((-2*I)*ArcCo s[a*x]^3*ArcTan[E^(I*ArcCos[a*x])] + 3*(I*ArcCos[a*x]^2*PolyLog[2, (-I)*E^ (I*ArcCos[a*x])] - (2*I)*((-I)*ArcCos[a*x]*PolyLog[3, (-I)*E^(I*ArcCos[a*x ])] + PolyLog[4, (-I)*E^(I*ArcCos[a*x])])) - 3*(I*ArcCos[a*x]^2*PolyLog[2, I*E^(I*ArcCos[a*x])] - (2*I)*((-I)*ArcCos[a*x]*PolyLog[3, I*E^(I*ArcCos[a *x])] + PolyLog[4, I*E^(I*ArcCos[a*x])]))))/2
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
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_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcCos[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) ) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Simp[b* c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*( 1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[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[((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]
Time = 0.50 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.66
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{2} \left (a^{2} x^{2} \arccos \left (a x \right )+3 \sqrt {-a^{2} x^{2}+1}\, a x -\arccos \left (a x \right )\right )}{2 x^{2} \left (a^{2} x^{2}-1\right )}-\frac {a^{2} \left (\arccos \left (a x \right )^{3} \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-\arccos \left (a x \right )^{3} \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-3 i \arccos \left (a x \right )^{2} \operatorname {polylog}\left (2, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+3 i \arccos \left (a x \right )^{2} \operatorname {polylog}\left (2, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+6 \arccos \left (a x \right ) \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+6 \arccos \left (a x \right ) \operatorname {polylog}\left (3, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-6 \arccos \left (a x \right ) \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-6 \arccos \left (a x \right ) \operatorname {polylog}\left (3, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+6 i \operatorname {polylog}\left (4, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+6 i \operatorname {dilog}\left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-6 i \operatorname {dilog}\left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-6 i \operatorname {polylog}\left (4, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )\right )}{2}\) | \(439\) |
Input:
int(arccos(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*(-a^2*x^2+1)^(1/2)/x^2/(a^2*x^2-1)*arccos(a*x)^2*(a^2*x^2*arccos(a*x) +3*(-a^2*x^2+1)^(1/2)*a*x-arccos(a*x))-1/2*a^2*(arccos(a*x)^3*ln(1-I*(a*x+ I*(-a^2*x^2+1)^(1/2)))-arccos(a*x)^3*ln(1+I*(a*x+I*(-a^2*x^2+1)^(1/2)))-3* I*arccos(a*x)^2*polylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))+3*I*arccos(a*x)^2* polylog(2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+6*arccos(a*x)*ln(1-I*(a*x+I*(-a^2 *x^2+1)^(1/2)))+6*arccos(a*x)*polylog(3,I*(a*x+I*(-a^2*x^2+1)^(1/2)))-6*ar ccos(a*x)*ln(1+I*(a*x+I*(-a^2*x^2+1)^(1/2)))-6*arccos(a*x)*polylog(3,-I*(a *x+I*(-a^2*x^2+1)^(1/2)))+6*I*polylog(4,I*(a*x+I*(-a^2*x^2+1)^(1/2)))+6*I* dilog(1+I*(a*x+I*(-a^2*x^2+1)^(1/2)))-6*I*dilog(1-I*(a*x+I*(-a^2*x^2+1)^(1 /2)))-6*I*polylog(4,-I*(a*x+I*(-a^2*x^2+1)^(1/2))))
\[ \int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:
integrate(arccos(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-a^2*x^2 + 1)*arccos(a*x)^3/(a^2*x^5 - x^3), x)
\[ \int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:
integrate(acos(a*x)**3/x**3/(-a**2*x**2+1)**(1/2),x)
Output:
Integral(acos(a*x)**3/(x**3*sqrt(-(a*x - 1)*(a*x + 1))), x)
\[ \int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:
integrate(arccos(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
integrate(arccos(a*x)^3/(sqrt(-a^2*x^2 + 1)*x^3), x)
\[ \int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:
integrate(arccos(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
integrate(arccos(a*x)^3/(sqrt(-a^2*x^2 + 1)*x^3), x)
Timed out. \[ \int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^3}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \] Input:
int(acos(a*x)^3/(x^3*(1 - a^2*x^2)^(1/2)),x)
Output:
int(acos(a*x)^3/(x^3*(1 - a^2*x^2)^(1/2)), x)
\[ \int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acos} \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}\, x^{3}}d x \] Input:
int(acos(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x)
Output:
int(acos(a*x)**3/(sqrt( - a**2*x**2 + 1)*x**3),x)