Integrand size = 10, antiderivative size = 304 \[ \int \frac {\arccos (a x)^4}{x^4} \, dx=-\frac {2 a^2 \arccos (a x)^2}{x}+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 x^2}-\frac {\arccos (a x)^4}{3 x^3}-8 i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )-\frac {4}{3} i a^3 \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right ) \] Output:
-2*a^2*arccos(a*x)^2/x+2/3*a*(-a^2*x^2+1)^(1/2)*arccos(a*x)^3/x^2-1/3*arcc os(a*x)^4/x^3-8*I*a^3*arccos(a*x)*arctan(a*x+I*(-a^2*x^2+1)^(1/2))-4/3*I*a ^3*arccos(a*x)^3*arctan(a*x+I*(-a^2*x^2+1)^(1/2))+4*I*a^3*polylog(2,-I*(a* x+I*(-a^2*x^2+1)^(1/2)))+2*I*a^3*arccos(a*x)^2*polylog(2,-I*(a*x+I*(-a^2*x ^2+1)^(1/2)))-4*I*a^3*polylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))-2*I*a^3*arcc os(a*x)^2*polylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))-4*a^3*arccos(a*x)*polylo g(3,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+4*a^3*arccos(a*x)*polylog(3,I*(a*x+I*(- a^2*x^2+1)^(1/2)))-4*I*a^3*polylog(4,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+4*I*a^ 3*polylog(4,I*(a*x+I*(-a^2*x^2+1)^(1/2)))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1475\) vs. \(2(304)=608\).
Time = 12.05 (sec) , antiderivative size = 1475, normalized size of antiderivative = 4.85 \[ \int \frac {\arccos (a x)^4}{x^4} \, dx =\text {Too large to display} \] Input:
Integrate[ArcCos[a*x]^4/x^4,x]
Output:
a^3*(-1/6*(ArcCos[a*x]^2*(12 + ArcCos[a*x]^2)) + 4*(ArcCos[a*x]*(Log[1 - I *E^(I*ArcCos[a*x])] - Log[1 + I*E^(I*ArcCos[a*x])]) + I*(PolyLog[2, (-I)*E ^(I*ArcCos[a*x])] - PolyLog[2, I*E^(I*ArcCos[a*x])])) + (2*((Pi^3*Log[Cot[ (Pi/2 - ArcCos[a*x])/2]])/8 + (3*Pi^2*((Pi/2 - ArcCos[a*x])*(Log[1 - E^(I* (Pi/2 - ArcCos[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcCos[a*x]))]) + I*(PolyLog [2, -E^(I*(Pi/2 - ArcCos[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcCos[a*x]))]) ))/4 - (3*Pi*((Pi/2 - ArcCos[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcCos[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcCos[a*x]))]) + (2*I)*(Pi/2 - ArcCos[a*x])*(PolyL og[2, -E^(I*(Pi/2 - ArcCos[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcCos[a*x])) ]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcCos[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcCos[a*x]))])))/2 + 8*((I/64)*(Pi/2 - ArcCos[a*x])^4 + (I/4)*(Pi/2 + (- 1/2*Pi + ArcCos[a*x])/2)^4 - ((Pi/2 - ArcCos[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcCos[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + Arc Cos[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))] + ((3 *I)/8)*(Pi/2 - ArcCos[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcCos[a*x]))] + (3 *Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + Arc Cos[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))] + (I/2) *PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))]))/4 + ((3*I)/2) *(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/...
Time = 1.52 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {5139, 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.
\(\displaystyle \int \frac {\arccos (a x)^4}{x^4} \, dx\) |
\(\Big \downarrow \) 5139 |
\(\displaystyle -\frac {4}{3} a \int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\arccos (a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 5205 |
\(\displaystyle -\frac {4}{3} a \left (\frac {1}{2} a^2 \int \frac {\arccos (a x)^3}{x \sqrt {1-a^2 x^2}}dx-\frac {3}{2} a \int \frac {\arccos (a x)^2}{x^2}dx-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}\right )-\frac {\arccos (a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 5139 |
\(\displaystyle -\frac {4}{3} a \left (-\frac {3}{2} a \left (-2 a \int \frac {\arccos (a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\arccos (a x)^2}{x}\right )+\frac {1}{2} a^2 \int \frac {\arccos (a x)^3}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}\right )-\frac {\arccos (a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 5219 |
\(\displaystyle -\frac {4}{3} a \left (-\frac {1}{2} a^2 \int \frac {\arccos (a x)^3}{a x}d\arccos (a x)-\frac {3}{2} a \left (2 a \int \frac {\arccos (a x)}{a x}d\arccos (a x)-\frac {\arccos (a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}\right )-\frac {\arccos (a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {4}{3} a \left (-\frac {1}{2} a^2 \int \arccos (a x)^3 \csc \left (\arccos (a x)+\frac {\pi }{2}\right )d\arccos (a x)-\frac {3}{2} a \left (2 a \int \arccos (a x) \csc \left (\arccos (a x)+\frac {\pi }{2}\right )d\arccos (a x)-\frac {\arccos (a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}\right )-\frac {\arccos (a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle -\frac {\arccos (a x)^4}{3 x^3}-\frac {4}{3} a \left (-\frac {1}{2} a^2 \left (-3 \int \arccos (a x)^2 \log \left (1-i e^{i \arccos (a x)}\right )d\arccos (a x)+3 \int \arccos (a x)^2 \log \left (1+i e^{i \arccos (a x)}\right )d\arccos (a x)-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )\right )-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (-\int \log \left (1-i e^{i \arccos (a x)}\right )d\arccos (a x)+\int \log \left (1+i e^{i \arccos (a x)}\right )d\arccos (a x)-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {\arccos (a x)^4}{3 x^3}-\frac {4}{3} a \left (-\frac {1}{2} a^2 \left (-3 \int \arccos (a x)^2 \log \left (1-i e^{i \arccos (a x)}\right )d\arccos (a x)+3 \int \arccos (a x)^2 \log \left (1+i e^{i \arccos (a x)}\right )d\arccos (a x)-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )\right )-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (i \int e^{-i \arccos (a x)} \log \left (1-i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-i \int e^{-i \arccos (a x)} \log \left (1+i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {\arccos (a x)^4}{3 x^3}-\frac {4}{3} a \left (-\frac {1}{2} a^2 \left (-3 \int \arccos (a x)^2 \log \left (1-i e^{i \arccos (a x)}\right )d\arccos (a x)+3 \int \arccos (a x)^2 \log \left (1+i e^{i \arccos (a x)}\right )d\arccos (a x)-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )\right )\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {\arccos (a x)^4}{3 x^3}-\frac {4}{3} a \left (-\frac {1}{2} a^2 \left (3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )d\arccos (a x)\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )d\arccos (a x)\right )-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )\right )\right )\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle -\frac {\arccos (a x)^4}{3 x^3}-\frac {4}{3} a \left (-\frac {1}{2} a^2 \left (3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )d\arccos (a x)-i \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )\right )\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )d\arccos (a x)-i \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )\right )\right )-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )\right )\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {\arccos (a x)^4}{3 x^3}-\frac {4}{3} a \left (-\frac {1}{2} a^2 \left (3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i \left (\int e^{-i \arccos (a x)} \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-i \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )\right )\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-2 i \left (\int e^{-i \arccos (a x)} \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-i \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )\right )\right )-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )\right )\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {\arccos (a x)^4}{3 x^3}-\frac {4}{3} a \left (-\frac {1}{2} a^2 \left (-2 i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )-i \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )\right )\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right )-i \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}-\frac {3}{2} a \left (-\frac {\arccos (a x)^2}{x}+2 a \left (-2 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )\right )\right )\) |
Input:
Int[ArcCos[a*x]^4/x^4,x]
Output:
-1/3*ArcCos[a*x]^4/x^3 - (4*a*(-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*ArcCos[a*x])] - I*PolyLog[2, I*E^(I*ArcCos[a *x])])))/2 - (a^2*((-2*I)*ArcCos[a*x]^3*ArcTan[E^(I*ArcCos[a*x])] + 3*(I*A rcCos[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))/3
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 = 0.61 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.38
method | result | size |
derivativedivides | \(a^{3} \left (-\frac {\arccos \left (a x \right )^{2} \left (-2 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arccos \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}+\frac {2 \arccos \left (a x \right )^{3} \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}-2 i \operatorname {polylog}\left (2, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right ) \arccos \left (a x \right )^{2}+4 \arccos \left (a x \right ) \operatorname {polylog}\left (3, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+4 i \operatorname {polylog}\left (4, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-\frac {2 \arccos \left (a x \right )^{3} \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}+2 i \operatorname {polylog}\left (2, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right ) \arccos \left (a x \right )^{2}-4 \arccos \left (a x \right ) \operatorname {polylog}\left (3, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-4 i \operatorname {polylog}\left (4, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-4 \arccos \left (a x \right ) \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+4 \arccos \left (a x \right ) \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+4 i \operatorname {dilog}\left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-4 i \operatorname {dilog}\left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )\right )\) | \(419\) |
default | \(a^{3} \left (-\frac {\arccos \left (a x \right )^{2} \left (-2 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arccos \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}+\frac {2 \arccos \left (a x \right )^{3} \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}-2 i \operatorname {polylog}\left (2, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right ) \arccos \left (a x \right )^{2}+4 \arccos \left (a x \right ) \operatorname {polylog}\left (3, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+4 i \operatorname {polylog}\left (4, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-\frac {2 \arccos \left (a x \right )^{3} \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}+2 i \operatorname {polylog}\left (2, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right ) \arccos \left (a x \right )^{2}-4 \arccos \left (a x \right ) \operatorname {polylog}\left (3, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-4 i \operatorname {polylog}\left (4, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-4 \arccos \left (a x \right ) \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+4 \arccos \left (a x \right ) \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+4 i \operatorname {dilog}\left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-4 i \operatorname {dilog}\left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )\right )\) | \(419\) |
Input:
int(arccos(a*x)^4/x^4,x,method=_RETURNVERBOSE)
Output:
a^3*(-1/3/a^3/x^3*arccos(a*x)^2*(-2*arccos(a*x)*(-a^2*x^2+1)^(1/2)*a*x+arc cos(a*x)^2+6*a^2*x^2)+2/3*arccos(a*x)^3*ln(1-I*(a*x+I*(-a^2*x^2+1)^(1/2))) -2*I*polylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))*arccos(a*x)^2+4*arccos(a*x)*p olylog(3,I*(a*x+I*(-a^2*x^2+1)^(1/2)))+4*I*polylog(4,I*(a*x+I*(-a^2*x^2+1) ^(1/2)))-2/3*arccos(a*x)^3*ln(1+I*(a*x+I*(-a^2*x^2+1)^(1/2)))+2*I*polylog( 2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))*arccos(a*x)^2-4*arccos(a*x)*polylog(3,-I* (a*x+I*(-a^2*x^2+1)^(1/2)))-4*I*polylog(4,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))-4 *arccos(a*x)*ln(1+I*(a*x+I*(-a^2*x^2+1)^(1/2)))+4*arccos(a*x)*ln(1-I*(a*x+ I*(-a^2*x^2+1)^(1/2)))+4*I*dilog(1+I*(a*x+I*(-a^2*x^2+1)^(1/2)))-4*I*dilog (1-I*(a*x+I*(-a^2*x^2+1)^(1/2))))
\[ \int \frac {\arccos (a x)^4}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{4}} \,d x } \] Input:
integrate(arccos(a*x)^4/x^4,x, algorithm="fricas")
Output:
integral(arccos(a*x)^4/x^4, x)
\[ \int \frac {\arccos (a x)^4}{x^4} \, dx=\int \frac {\operatorname {acos}^{4}{\left (a x \right )}}{x^{4}}\, dx \] Input:
integrate(acos(a*x)**4/x**4,x)
Output:
Integral(acos(a*x)**4/x**4, x)
\[ \int \frac {\arccos (a x)^4}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{4}} \,d x } \] Input:
integrate(arccos(a*x)^4/x^4,x, algorithm="maxima")
Output:
1/3*(12*a*x^3*integrate(1/3*sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3/(a^2*x^5 - x^3), x) - arctan2(sqrt(a*x + 1)*sq rt(-a*x + 1), a*x)^4)/x^3
\[ \int \frac {\arccos (a x)^4}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{4}} \,d x } \] Input:
integrate(arccos(a*x)^4/x^4,x, algorithm="giac")
Output:
integrate(arccos(a*x)^4/x^4, x)
Timed out. \[ \int \frac {\arccos (a x)^4}{x^4} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^4}{x^4} \,d x \] Input:
int(acos(a*x)^4/x^4,x)
Output:
int(acos(a*x)^4/x^4, x)
\[ \int \frac {\arccos (a x)^4}{x^4} \, dx=\int \frac {\mathit {acos} \left (a x \right )^{4}}{x^{4}}d x \] Input:
int(acos(a*x)^4/x^4,x)
Output:
int(acos(a*x)**4/x**4,x)