3.1.27 \(\int \frac {\text {ArcCos}(a x)^3}{x} \, dx\) [27]

Optimal. Leaf size=101 \[ -\frac {1}{4} i \text {ArcCos}(a x)^4+\text {ArcCos}(a x)^3 \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )-\frac {3}{2} i \text {ArcCos}(a x)^2 \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(a x)}\right )+\frac {3}{2} \text {ArcCos}(a x) \text {PolyLog}\left (3,-e^{2 i \text {ArcCos}(a x)}\right )+\frac {3}{4} i \text {PolyLog}\left (4,-e^{2 i \text {ArcCos}(a x)}\right ) \]

[Out]

-1/4*I*arccos(a*x)^4+arccos(a*x)^3*ln(1+(a*x+I*(-a^2*x^2+1)^(1/2))^2)-3/2*I*arccos(a*x)^2*polylog(2,-(a*x+I*(-
a^2*x^2+1)^(1/2))^2)+3/2*arccos(a*x)*polylog(3,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)+3/4*I*polylog(4,-(a*x+I*(-a^2*x^
2+1)^(1/2))^2)

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Rubi [A]
time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4722, 3800, 2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {3}{2} i \text {ArcCos}(a x)^2 \text {Li}_2\left (-e^{2 i \text {ArcCos}(a x)}\right )+\frac {3}{2} \text {ArcCos}(a x) \text {Li}_3\left (-e^{2 i \text {ArcCos}(a x)}\right )+\frac {3}{4} i \text {Li}_4\left (-e^{2 i \text {ArcCos}(a x)}\right )-\frac {1}{4} i \text {ArcCos}(a x)^4+\text {ArcCos}(a x)^3 \log \left (1+e^{2 i \text {ArcCos}(a x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]^3/x,x]

[Out]

(-1/4*I)*ArcCos[a*x]^4 + ArcCos[a*x]^3*Log[1 + E^((2*I)*ArcCos[a*x])] - ((3*I)/2)*ArcCos[a*x]^2*PolyLog[2, -E^
((2*I)*ArcCos[a*x])] + (3*ArcCos[a*x]*PolyLog[3, -E^((2*I)*ArcCos[a*x])])/2 + ((3*I)/4)*PolyLog[4, -E^((2*I)*A
rcCos[a*x])]

Rule 2221

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]
- Dist[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]

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 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x
] - Dist[2*I, Int[(c + d*x)^m*(E^(2*I*(e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 4722

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcCos[c
*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

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

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Mathematica [A]
time = 0.02, size = 101, normalized size = 1.00 \begin {gather*} -\frac {1}{4} i \text {ArcCos}(a x)^4+\text {ArcCos}(a x)^3 \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )-\frac {3}{2} i \text {ArcCos}(a x)^2 \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(a x)}\right )+\frac {3}{2} \text {ArcCos}(a x) \text {PolyLog}\left (3,-e^{2 i \text {ArcCos}(a x)}\right )+\frac {3}{4} i \text {PolyLog}\left (4,-e^{2 i \text {ArcCos}(a x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a*x]^3/x,x]

[Out]

(-1/4*I)*ArcCos[a*x]^4 + ArcCos[a*x]^3*Log[1 + E^((2*I)*ArcCos[a*x])] - ((3*I)/2)*ArcCos[a*x]^2*PolyLog[2, -E^
((2*I)*ArcCos[a*x])] + (3*ArcCos[a*x]*PolyLog[3, -E^((2*I)*ArcCos[a*x])])/2 + ((3*I)/4)*PolyLog[4, -E^((2*I)*A
rcCos[a*x])]

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Maple [A]
time = 0.14, size = 135, normalized size = 1.34

method	result	size
derivativedivides	\(-\frac {i \arccos \left (a x \right )^{4}}{4}+\arccos \left (a x \right )^{3} \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \arccos \left (a x \right )^{2} \polylog \left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 \arccos \left (a x \right ) \polylog \left (3, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 i \polylog \left (4, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{4}\)	\(135\)
default	\(-\frac {i \arccos \left (a x \right )^{4}}{4}+\arccos \left (a x \right )^{3} \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \arccos \left (a x \right )^{2} \polylog \left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 \arccos \left (a x \right ) \polylog \left (3, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 i \polylog \left (4, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{4}\)	\(135\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)^3/x,x,method=_RETURNVERBOSE)

[Out]

-1/4*I*arccos(a*x)^4+arccos(a*x)^3*ln(1+(a*x+I*(-a^2*x^2+1)^(1/2))^2)-3/2*I*arccos(a*x)^2*polylog(2,-(a*x+I*(-
a^2*x^2+1)^(1/2))^2)+3/2*arccos(a*x)*polylog(3,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)+3/4*I*polylog(4,-(a*x+I*(-a^2*x^
2+1)^(1/2))^2)

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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)^3/x,x, algorithm="maxima")

[Out]

integrate(arccos(a*x)^3/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)^3/x,x, algorithm="fricas")

[Out]

integral(arccos(a*x)^3/x, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x)**3/x,x)

[Out]

Integral(acos(a*x)**3/x, 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)^3/x,x, algorithm="giac")

[Out]

integrate(arccos(a*x)^3/x, 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 )}^3}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(a*x)^3/x,x)

[Out]

int(acos(a*x)^3/x, x)

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