3.1.39 \(\int \frac {\text {ArcCos}(a x)^4}{x^2} \, dx\) [39]

Optimal. Leaf size=176 \[ -\frac {\text {ArcCos}(a x)^4}{x}-8 i a \text {ArcCos}(a x)^3 \text {ArcTan}\left (e^{i \text {ArcCos}(a x)}\right )+12 i a \text {ArcCos}(a x)^2 \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )-12 i a \text {ArcCos}(a x)^2 \text {PolyLog}\left (2,i e^{i \text {ArcCos}(a x)}\right )-24 a \text {ArcCos}(a x) \text {PolyLog}\left (3,-i e^{i \text {ArcCos}(a x)}\right )+24 a \text {ArcCos}(a x) \text {PolyLog}\left (3,i e^{i \text {ArcCos}(a x)}\right )-24 i a \text {PolyLog}\left (4,-i e^{i \text {ArcCos}(a x)}\right )+24 i a \text {PolyLog}\left (4,i e^{i \text {ArcCos}(a x)}\right ) \]

[Out]

-arccos(a*x)^4/x-8*I*a*arccos(a*x)^3*arctan(a*x+I*(-a^2*x^2+1)^(1/2))+12*I*a*arccos(a*x)^2*polylog(2,-I*(a*x+I
*(-a^2*x^2+1)^(1/2)))-12*I*a*arccos(a*x)^2*polylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))-24*a*arccos(a*x)*polylog(3,
-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+24*a*arccos(a*x)*polylog(3,I*(a*x+I*(-a^2*x^2+1)^(1/2)))-24*I*a*polylog(4,-I*(a
*x+I*(-a^2*x^2+1)^(1/2)))+24*I*a*polylog(4,I*(a*x+I*(-a^2*x^2+1)^(1/2)))

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Rubi [A]
time = 0.14, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4724, 4804, 4266, 2611, 6744, 2320, 6724} \begin {gather*} -8 i a \text {ArcCos}(a x)^3 \text {ArcTan}\left (e^{i \text {ArcCos}(a x)}\right )+12 i a \text {ArcCos}(a x)^2 \text {Li}_2\left (-i e^{i \text {ArcCos}(a x)}\right )-12 i a \text {ArcCos}(a x)^2 \text {Li}_2\left (i e^{i \text {ArcCos}(a x)}\right )-24 a \text {ArcCos}(a x) \text {Li}_3\left (-i e^{i \text {ArcCos}(a x)}\right )+24 a \text {ArcCos}(a x) \text {Li}_3\left (i e^{i \text {ArcCos}(a x)}\right )-24 i a \text {Li}_4\left (-i e^{i \text {ArcCos}(a x)}\right )+24 i a \text {Li}_4\left (i e^{i \text {ArcCos}(a x)}\right )-\frac {\text {ArcCos}(a x)^4}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]^4/x^2,x]

[Out]

-(ArcCos[a*x]^4/x) - (8*I)*a*ArcCos[a*x]^3*ArcTan[E^(I*ArcCos[a*x])] + (12*I)*a*ArcCos[a*x]^2*PolyLog[2, (-I)*
E^(I*ArcCos[a*x])] - (12*I)*a*ArcCos[a*x]^2*PolyLog[2, I*E^(I*ArcCos[a*x])] - 24*a*ArcCos[a*x]*PolyLog[3, (-I)
*E^(I*ArcCos[a*x])] + 24*a*ArcCos[a*x]*PolyLog[3, I*E^(I*ArcCos[a*x])] - (24*I)*a*PolyLog[4, (-I)*E^(I*ArcCos[
a*x])] + (24*I)*a*PolyLog[4, I*E^(I*ArcCos[a*x])]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

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] + Dist[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]

Rule 4266

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] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[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]

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCo
s[c*x])^n/(d*(m + 1))), x] + Dist[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]

Rule 4804

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(-(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] /
; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rule 6744

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] - Dist[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]

Rubi steps

\begin {align*} \int \frac {\cos ^{-1}(a x)^4}{x^2} \, dx &=-\frac {\cos ^{-1}(a x)^4}{x}-(4 a) \int \frac {\cos ^{-1}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\cos ^{-1}(a x)^4}{x}+(4 a) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-(12 a) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+(12 a) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-(24 i a) \text {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+(24 i a) \text {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-24 a \cos ^{-1}(a x) \text {Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+24 a \cos ^{-1}(a x) \text {Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )+(24 a) \text {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )-(24 a) \text {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-24 a \cos ^{-1}(a x) \text {Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+24 a \cos ^{-1}(a x) \text {Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )-(24 i a) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )+(24 i a) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )\\ &=-\frac {\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-24 a \cos ^{-1}(a x) \text {Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+24 a \cos ^{-1}(a x) \text {Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )-24 i a \text {Li}_4\left (-i e^{i \cos ^{-1}(a x)}\right )+24 i a \text {Li}_4\left (i e^{i \cos ^{-1}(a x)}\right )\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(549\) vs. \(2(176)=352\).
time = 0.75, size = 549, normalized size = 3.12 \begin {gather*} a \left (-\frac {7 i \pi ^4}{16}-\frac {1}{2} i \pi ^3 \text {ArcCos}(a x)+\frac {3}{2} i \pi ^2 \text {ArcCos}(a x)^2-2 i \pi \text {ArcCos}(a x)^3+i \text {ArcCos}(a x)^4-\frac {\text {ArcCos}(a x)^4}{a x}+3 \pi ^2 \text {ArcCos}(a x) \log \left (1-i e^{-i \text {ArcCos}(a x)}\right )-6 \pi \text {ArcCos}(a x)^2 \log \left (1-i e^{-i \text {ArcCos}(a x)}\right )-\frac {1}{2} \pi ^3 \log \left (1+i e^{-i \text {ArcCos}(a x)}\right )+4 \text {ArcCos}(a x)^3 \log \left (1+i e^{-i \text {ArcCos}(a x)}\right )+\frac {1}{2} \pi ^3 \log \left (1+i e^{i \text {ArcCos}(a x)}\right )-3 \pi ^2 \text {ArcCos}(a x) \log \left (1+i e^{i \text {ArcCos}(a x)}\right )+6 \pi \text {ArcCos}(a x)^2 \log \left (1+i e^{i \text {ArcCos}(a x)}\right )-4 \text {ArcCos}(a x)^3 \log \left (1+i e^{i \text {ArcCos}(a x)}\right )+\frac {1}{2} \pi ^3 \log \left (\tan \left (\frac {1}{4} (\pi +2 \text {ArcCos}(a x))\right )\right )+12 i \text {ArcCos}(a x)^2 \text {PolyLog}\left (2,-i e^{-i \text {ArcCos}(a x)}\right )+3 i \pi (\pi -4 \text {ArcCos}(a x)) \text {PolyLog}\left (2,i e^{-i \text {ArcCos}(a x)}\right )+3 i \pi ^2 \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )-12 i \pi \text {ArcCos}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )+12 i \text {ArcCos}(a x)^2 \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )+24 \text {ArcCos}(a x) \text {PolyLog}\left (3,-i e^{-i \text {ArcCos}(a x)}\right )-12 \pi \text {PolyLog}\left (3,i e^{-i \text {ArcCos}(a x)}\right )+12 \pi \text {PolyLog}\left (3,-i e^{i \text {ArcCos}(a x)}\right )-24 \text {ArcCos}(a x) \text {PolyLog}\left (3,-i e^{i \text {ArcCos}(a x)}\right )-24 i \text {PolyLog}\left (4,-i e^{-i \text {ArcCos}(a x)}\right )-24 i \text {PolyLog}\left (4,-i e^{i \text {ArcCos}(a x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a*x]^4/x^2,x]

[Out]

a*(((-7*I)/16)*Pi^4 - (I/2)*Pi^3*ArcCos[a*x] + ((3*I)/2)*Pi^2*ArcCos[a*x]^2 - (2*I)*Pi*ArcCos[a*x]^3 + I*ArcCo
s[a*x]^4 - ArcCos[a*x]^4/(a*x) + 3*Pi^2*ArcCos[a*x]*Log[1 - I/E^(I*ArcCos[a*x])] - 6*Pi*ArcCos[a*x]^2*Log[1 -
I/E^(I*ArcCos[a*x])] - (Pi^3*Log[1 + I/E^(I*ArcCos[a*x])])/2 + 4*ArcCos[a*x]^3*Log[1 + I/E^(I*ArcCos[a*x])] +
(Pi^3*Log[1 + I*E^(I*ArcCos[a*x])])/2 - 3*Pi^2*ArcCos[a*x]*Log[1 + I*E^(I*ArcCos[a*x])] + 6*Pi*ArcCos[a*x]^2*L
og[1 + I*E^(I*ArcCos[a*x])] - 4*ArcCos[a*x]^3*Log[1 + I*E^(I*ArcCos[a*x])] + (Pi^3*Log[Tan[(Pi + 2*ArcCos[a*x]
)/4]])/2 + (12*I)*ArcCos[a*x]^2*PolyLog[2, (-I)/E^(I*ArcCos[a*x])] + (3*I)*Pi*(Pi - 4*ArcCos[a*x])*PolyLog[2,
I/E^(I*ArcCos[a*x])] + (3*I)*Pi^2*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (12*I)*Pi*ArcCos[a*x]*PolyLog[2, (-I)*E
^(I*ArcCos[a*x])] + (12*I)*ArcCos[a*x]^2*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] + 24*ArcCos[a*x]*PolyLog[3, (-I)/E
^(I*ArcCos[a*x])] - 12*Pi*PolyLog[3, I/E^(I*ArcCos[a*x])] + 12*Pi*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] - 24*ArcC
os[a*x]*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] - (24*I)*PolyLog[4, (-I)/E^(I*ArcCos[a*x])] - (24*I)*PolyLog[4, (-I
)*E^(I*ArcCos[a*x])])

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Maple [F]
time = 0.30, size = 0, normalized size = 0.00 \[\int \frac {\arccos \left (a x \right )^{4}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)^4/x^2,x)

[Out]

int(arccos(a*x)^4/x^2,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^4/x^2,x, algorithm="maxima")

[Out]

-(arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^4 - 4*a*x*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(sqrt(a*x
+ 1)*sqrt(-a*x + 1), a*x)^3/(a^2*x^3 - x), x))/x

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^4/x^2,x, algorithm="fricas")

[Out]

integral(arccos(a*x)^4/x^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acos}^{4}{\left (a x \right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x)**4/x**2,x)

[Out]

Integral(acos(a*x)**4/x**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^4/x^2,x, algorithm="giac")

[Out]

integrate(arccos(a*x)^4/x^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acos}\left (a\,x\right )}^4}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(a*x)^4/x^2,x)

[Out]

int(acos(a*x)^4/x^2, x)

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