3.1.18 \(\int \frac {\text {ArcCos}(a x)^2}{x^2} \, dx\) [18]

Optimal. Leaf size=74 \[ -\frac {\text {ArcCos}(a x)^2}{x}-4 i a \text {ArcCos}(a x) \text {ArcTan}\left (e^{i \text {ArcCos}(a x)}\right )+2 i a \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )-2 i a \text {PolyLog}\left (2,i e^{i \text {ArcCos}(a x)}\right ) \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4724, 4804, 4266, 2317, 2438} \begin {gather*} -4 i a \text {ArcCos}(a x) \text {ArcTan}\left (e^{i \text {ArcCos}(a x)}\right )+2 i a \text {Li}_2\left (-i e^{i \text {ArcCos}(a x)}\right )-2 i a \text {Li}_2\left (i e^{i \text {ArcCos}(a x)}\right )-\frac {\text {ArcCos}(a x)^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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]

Rule 2438

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]

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]

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 98, normalized size = 1.32 \begin {gather*} -\frac {\text {ArcCos}(a x) \left (\text {ArcCos}(a x)+2 a x \left (-\log \left (1-i e^{i \text {ArcCos}(a x)}\right )+\log \left (1+i e^{i \text {ArcCos}(a x)}\right )\right )\right )}{x}+2 i a \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )-2 i a \text {PolyLog}\left (2,i e^{i \text {ArcCos}(a x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.10, size = 136, normalized size = 1.84

method	result	size
derivativedivides	\(a \left (-\frac {\arccos \left (a x \right )^{2}}{a x}-2 \arccos \left (a x \right ) \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+2 \arccos \left (a x \right ) \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+2 i \dilog \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-2 i \dilog \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )\right )\)	\(136\)
default	\(a \left (-\frac {\arccos \left (a x \right )^{2}}{a x}-2 \arccos \left (a x \right ) \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+2 \arccos \left (a x \right ) \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+2 i \dilog \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-2 i \dilog \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )\right )\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)^2/x^2,x,method=_RETURNVERBOSE)

[Out]

a*(-arccos(a*x)^2/a/x-2*arccos(a*x)*ln(1+I*(a*x+I*(-a^2*x^2+1)^(1/2)))+2*arccos(a*x)*ln(1-I*(a*x+I*(-a^2*x^2+1
)^(1/2)))+2*I*dilog(1+I*(a*x+I*(-a^2*x^2+1)^(1/2)))-2*I*dilog(1-I*(a*x+I*(-a^2*x^2+1)^(1/2))))

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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)^2/x^2,x, algorithm="maxima")

[Out]

(2*a*x*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)/(a^2*x^3 - x), x) - a
rctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2)/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)^2/x^2,x, algorithm="fricas")

[Out]

integral(arccos(a*x)^2/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acos(a*x)**2/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)^2/x^2,x, algorithm="giac")

[Out]

integrate(arccos(a*x)^2/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 )}^2}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acos(a*x)^2/x^2, x)

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