3.1.40 \(\int \frac {\text {ArcCos}(a x)^4}{x^3} \, dx\) [40]

Optimal. Leaf size=121 \[ -2 i a^2 \text {ArcCos}(a x)^3+\frac {2 a \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^3}{x}-\frac {\text {ArcCos}(a x)^4}{2 x^2}+6 a^2 \text {ArcCos}(a x)^2 \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )-6 i a^2 \text {ArcCos}(a x) \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(a x)}\right )+3 a^2 \text {PolyLog}\left (3,-e^{2 i \text {ArcCos}(a x)}\right ) \]

[Out]

-2*I*a^2*arccos(a*x)^3-1/2*arccos(a*x)^4/x^2+6*a^2*arccos(a*x)^2*ln(1+(a*x+I*(-a^2*x^2+1)^(1/2))^2)-6*I*a^2*ar
ccos(a*x)*polylog(2,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)+3*a^2*polylog(3,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)+2*a*arccos(a
*x)^3*(-a^2*x^2+1)^(1/2)/x

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Rubi [A]
time = 0.14, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4724, 4772, 4722, 3800, 2221, 2611, 2320, 6724} \begin {gather*} -6 i a^2 \text {ArcCos}(a x) \text {Li}_2\left (-e^{2 i \text {ArcCos}(a x)}\right )+3 a^2 \text {Li}_3\left (-e^{2 i \text {ArcCos}(a x)}\right )+\frac {2 a \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^3}{x}-2 i a^2 \text {ArcCos}(a x)^3+6 a^2 \text {ArcCos}(a x)^2 \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )-\frac {\text {ArcCos}(a x)^4}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]^4/x^3,x]

[Out]

(-2*I)*a^2*ArcCos[a*x]^3 + (2*a*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/x - ArcCos[a*x]^4/(2*x^2) + 6*a^2*ArcCos[a*x]
^2*Log[1 + E^((2*I)*ArcCos[a*x])] - (6*I)*a^2*ArcCos[a*x]*PolyLog[2, -E^((2*I)*ArcCos[a*x])] + 3*a^2*PolyLog[3
, -E^((2*I)*ArcCos[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 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 4772

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] + Dist[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, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]

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]

Rubi steps

\begin {align*} \int \frac {\cos ^{-1}(a x)^4}{x^3} \, dx &=-\frac {\cos ^{-1}(a x)^4}{2 x^2}-(2 a) \int \frac {\cos ^{-1}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {2 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{x}-\frac {\cos ^{-1}(a x)^4}{2 x^2}+\left (6 a^2\right ) \int \frac {\cos ^{-1}(a x)^2}{x} \, dx\\ &=\frac {2 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{x}-\frac {\cos ^{-1}(a x)^4}{2 x^2}-\left (6 a^2\right ) \text {Subst}\left (\int x^2 \tan (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-2 i a^2 \cos ^{-1}(a x)^3+\frac {2 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{x}-\frac {\cos ^{-1}(a x)^4}{2 x^2}+\left (12 i a^2\right ) \text {Subst}\left (\int \frac {e^{2 i x} x^2}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(a x)\right )\\ &=-2 i a^2 \cos ^{-1}(a x)^3+\frac {2 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{x}-\frac {\cos ^{-1}(a x)^4}{2 x^2}+6 a^2 \cos ^{-1}(a x)^2 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-\left (12 a^2\right ) \text {Subst}\left (\int x \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-2 i a^2 \cos ^{-1}(a x)^3+\frac {2 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{x}-\frac {\cos ^{-1}(a x)^4}{2 x^2}+6 a^2 \cos ^{-1}(a x)^2 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-6 i a^2 \cos ^{-1}(a x) \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )+\left (6 i a^2\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-2 i a^2 \cos ^{-1}(a x)^3+\frac {2 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{x}-\frac {\cos ^{-1}(a x)^4}{2 x^2}+6 a^2 \cos ^{-1}(a x)^2 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-6 i a^2 \cos ^{-1}(a x) \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )+\left (3 a^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \cos ^{-1}(a x)}\right )\\ &=-2 i a^2 \cos ^{-1}(a x)^3+\frac {2 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{x}-\frac {\cos ^{-1}(a x)^4}{2 x^2}+6 a^2 \cos ^{-1}(a x)^2 \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-6 i a^2 \cos ^{-1}(a x) \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )+3 a^2 \text {Li}_3\left (-e^{2 i \cos ^{-1}(a x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.27, size = 115, normalized size = 0.95 \begin {gather*} -\frac {\text {ArcCos}(a x)^4}{2 x^2}-a^2 \left (-2 \text {ArcCos}(a x)^2 \left (-i \text {ArcCos}(a x)+\frac {\sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{a x}+3 \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )\right )+6 i \text {ArcCos}(a x) \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(a x)}\right )-3 \text {PolyLog}\left (3,-e^{2 i \text {ArcCos}(a x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a*x]^4/x^3,x]

[Out]

-1/2*ArcCos[a*x]^4/x^2 - a^2*(-2*ArcCos[a*x]^2*((-I)*ArcCos[a*x] + (Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(a*x) + 3*L
og[1 + E^((2*I)*ArcCos[a*x])]) + (6*I)*ArcCos[a*x]*PolyLog[2, -E^((2*I)*ArcCos[a*x])] - 3*PolyLog[3, -E^((2*I)
*ArcCos[a*x])])

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Maple [A]
time = 0.36, size = 150, normalized size = 1.24

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\arccos \left (a x \right )^{3} \left (-4 i a^{2} x^{2}-4 a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{2 a^{2} x^{2}}-4 i \arccos \left (a x \right )^{3}+6 \arccos \left (a x \right )^{2} \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-6 i \arccos \left (a x \right ) \polylog \left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+3 \polylog \left (3, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )\right )\)	\(150\)
default	\(a^{2} \left (-\frac {\arccos \left (a x \right )^{3} \left (-4 i a^{2} x^{2}-4 a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{2 a^{2} x^{2}}-4 i \arccos \left (a x \right )^{3}+6 \arccos \left (a x \right )^{2} \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-6 i \arccos \left (a x \right ) \polylog \left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+3 \polylog \left (3, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )\right )\)	\(150\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)^4/x^3,x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/2*arccos(a*x)^3*(-4*I*a^2*x^2-4*a*x*(-a^2*x^2+1)^(1/2)+arccos(a*x))/a^2/x^2-4*I*arccos(a*x)^3+6*arccos
(a*x)^2*ln(1+(a*x+I*(-a^2*x^2+1)^(1/2))^2)-6*I*arccos(a*x)*polylog(2,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)+3*polylog(
3,-(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)^4/x^3,x, algorithm="maxima")

[Out]

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

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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^3,x, algorithm="fricas")

[Out]

integral(arccos(a*x)^4/x^3, 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^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x)**4/x**3,x)

[Out]

Integral(acos(a*x)**4/x**3, 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^3,x, algorithm="giac")

[Out]

integrate(arccos(a*x)^4/x^3, 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^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(a*x)^4/x^3,x)

[Out]

int(acos(a*x)^4/x^3, x)

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