3.1.20 \(\int \frac {\text {ArcCos}(a x)^2}{x^4} \, dx\) [20]

Optimal. Leaf size=124 \[ -\frac {a^2}{3 x}+\frac {a \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{3 x^2}-\frac {\text {ArcCos}(a x)^2}{3 x^3}-\frac {2}{3} i a^3 \text {ArcCos}(a x) \text {ArcTan}\left (e^{i \text {ArcCos}(a x)}\right )+\frac {1}{3} i a^3 \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )-\frac {1}{3} i a^3 \text {PolyLog}\left (2,i e^{i \text {ArcCos}(a x)}\right ) \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4724, 4790, 4804, 4266, 2317, 2438, 30} \begin {gather*} -\frac {2}{3} i a^3 \text {ArcCos}(a x) \text {ArcTan}\left (e^{i \text {ArcCos}(a x)}\right )+\frac {1}{3} i a^3 \text {Li}_2\left (-i e^{i \text {ArcCos}(a x)}\right )-\frac {1}{3} i a^3 \text {Li}_2\left (i e^{i \text {ArcCos}(a x)}\right )+\frac {a \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{3 x^2}-\frac {a^2}{3 x}-\frac {\text {ArcCos}(a x)^2}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*a^2/x + (a*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(3*x^2) - ArcCos[a*x]^2/(3*x^3) - ((2*I)/3)*a^3*ArcCos[a*x]*Arc
Tan[E^(I*ArcCos[a*x])] + (I/3)*a^3*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (I/3)*a^3*PolyLog[2, I*E^(I*ArcCos[a*x
])]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 4790

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[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] + 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]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[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^4} \, dx &=-\frac {\cos ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} (2 a) \int \frac {\cos ^{-1}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{3 x^2}-\frac {\cos ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^2 \int \frac {1}{x^2} \, dx-\frac {1}{3} a^3 \int \frac {\cos ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a^2}{3 x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{3 x^2}-\frac {\cos ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^3 \text {Subst}\left (\int x \sec (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {a^2}{3 x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{3 x^2}-\frac {\cos ^{-1}(a x)^2}{3 x^3}-\frac {2}{3} i a^3 \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-\frac {1}{3} a^3 \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+\frac {1}{3} a^3 \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {a^2}{3 x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{3 x^2}-\frac {\cos ^{-1}(a x)^2}{3 x^3}-\frac {2}{3} i a^3 \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+\frac {1}{3} \left (i a^3\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )-\frac {1}{3} \left (i a^3\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )\\ &=-\frac {a^2}{3 x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{3 x^2}-\frac {\cos ^{-1}(a x)^2}{3 x^3}-\frac {2}{3} i a^3 \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+\frac {1}{3} i a^3 \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-\frac {1}{3} i a^3 \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.43, size = 152, normalized size = 1.23 \begin {gather*} -\frac {a^2 x^2-a x \sqrt {1-a^2 x^2} \text {ArcCos}(a x)+\text {ArcCos}(a x)^2-a^3 x^3 \text {ArcCos}(a x) \log \left (1-i e^{i \text {ArcCos}(a x)}\right )+a^3 x^3 \text {ArcCos}(a x) \log \left (1+i e^{i \text {ArcCos}(a x)}\right )-i a^3 x^3 \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )+i a^3 x^3 \text {PolyLog}\left (2,i e^{i \text {ArcCos}(a x)}\right )}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.62, size = 166, normalized size = 1.34

method	result	size
derivativedivides	\(a^{3} \left (-\frac {-a x \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )^{2}+a^{2} x^{2}}{3 a^{3} x^{3}}-\frac {\arccos \left (a x \right ) \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}+\frac {\arccos \left (a x \right ) \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}+\frac {i \dilog \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}-\frac {i \dilog \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}\right )\)	\(166\)
default	\(a^{3} \left (-\frac {-a x \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )^{2}+a^{2} x^{2}}{3 a^{3} x^{3}}-\frac {\arccos \left (a x \right ) \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}+\frac {\arccos \left (a x \right ) \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}+\frac {i \dilog \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}-\frac {i \dilog \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}\right )\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/3*(6*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)/(a^2*x^5 -
x^3), x) - arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2)/x^3

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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^4,x, algorithm="fricas")

[Out]

integral(arccos(a*x)^2/x^4, 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^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acos(a*x)**2/x**4, 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^4,x, algorithm="giac")

[Out]

integrate(arccos(a*x)^2/x^4, 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^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acos(a*x)^2/x^4, x)

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