3.7.45 \(\int \frac {\cos ^{-1}(x)^2}{x^5} \, dx\) [645]

Optimal. Leaf size=65 \[ -\frac {1}{12 x^2}+\frac {\sqrt {1-x^2} \cos ^{-1}(x)}{6 x^3}+\frac {\sqrt {1-x^2} \cos ^{-1}(x)}{3 x}-\frac {\cos ^{-1}(x)^2}{4 x^4}+\frac {\log (x)}{3} \]

[Out]

-1/12/x^2-1/4*arccos(x)^2/x^4+1/3*ln(x)+1/6*arccos(x)*(-x^2+1)^(1/2)/x^3+1/3*arccos(x)*(-x^2+1)^(1/2)/x

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Rubi [A]
time = 0.07, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4724, 4790, 4772, 29, 30} \begin {gather*} -\frac {\text {ArcCos}(x)^2}{4 x^4}+\frac {\sqrt {1-x^2} \text {ArcCos}(x)}{3 x}+\frac {\sqrt {1-x^2} \text {ArcCos}(x)}{6 x^3}-\frac {1}{12 x^2}+\frac {\log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCos[x]^2/x^5,x]

[Out]

-1/12*1/x^2 + (Sqrt[1 - x^2]*ArcCos[x])/(6*x^3) + (Sqrt[1 - x^2]*ArcCos[x])/(3*x) - ArcCos[x]^2/(4*x^4) + Log[
x]/3

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 30

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

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 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]

Rubi steps

\begin {align*} \int \frac {\cos ^{-1}(x)^2}{x^5} \, dx &=-\frac {\cos ^{-1}(x)^2}{4 x^4}-\frac {1}{2} \int \frac {\cos ^{-1}(x)}{x^4 \sqrt {1-x^2}} \, dx\\ &=\frac {\sqrt {1-x^2} \cos ^{-1}(x)}{6 x^3}-\frac {\cos ^{-1}(x)^2}{4 x^4}+\frac {1}{6} \int \frac {1}{x^3} \, dx-\frac {1}{3} \int \frac {\cos ^{-1}(x)}{x^2 \sqrt {1-x^2}} \, dx\\ &=-\frac {1}{12 x^2}+\frac {\sqrt {1-x^2} \cos ^{-1}(x)}{6 x^3}+\frac {\sqrt {1-x^2} \cos ^{-1}(x)}{3 x}-\frac {\cos ^{-1}(x)^2}{4 x^4}+\frac {1}{3} \int \frac {1}{x} \, dx\\ &=-\frac {1}{12 x^2}+\frac {\sqrt {1-x^2} \cos ^{-1}(x)}{6 x^3}+\frac {\sqrt {1-x^2} \cos ^{-1}(x)}{3 x}-\frac {\cos ^{-1}(x)^2}{4 x^4}+\frac {\log (x)}{3}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 52, normalized size = 0.80 \begin {gather*} -\frac {1}{12 x^2}+\frac {\sqrt {1-x^2} \left (1+2 x^2\right ) \cos ^{-1}(x)}{6 x^3}-\frac {\cos ^{-1}(x)^2}{4 x^4}+\frac {\log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[x]^2/x^5,x]

[Out]

-1/12*1/x^2 + (Sqrt[1 - x^2]*(1 + 2*x^2)*ArcCos[x])/(6*x^3) - ArcCos[x]^2/(4*x^4) + Log[x]/3

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Maple [A]
time = 0.05, size = 52, normalized size = 0.80

method	result	size
default	\(-\frac {1}{12 x^{2}}-\frac {\arccos \left (x \right )^{2}}{4 x^{4}}+\frac {\ln \left (x \right )}{3}+\frac {\arccos \left (x \right ) \sqrt {-x^{2}+1}}{6 x^{3}}+\frac {\arccos \left (x \right ) \sqrt {-x^{2}+1}}{3 x}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(x)^2/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/12/x^2-1/4*arccos(x)^2/x^4+1/3*ln(x)+1/6*arccos(x)*(-x^2+1)^(1/2)/x^3+1/3*arccos(x)*(-x^2+1)^(1/2)/x

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Maxima [A]
time = 1.47, size = 51, normalized size = 0.78 \begin {gather*} \frac {1}{6} \, {\left (\frac {2 \, \sqrt {-x^{2} + 1}}{x} + \frac {\sqrt {-x^{2} + 1}}{x^{3}}\right )} \arccos \left (x\right ) - \frac {1}{12 \, x^{2}} - \frac {\arccos \left (x\right )^{2}}{4 \, x^{4}} + \frac {1}{3} \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(x)^2/x^5,x, algorithm="maxima")

[Out]

1/6*(2*sqrt(-x^2 + 1)/x + sqrt(-x^2 + 1)/x^3)*arccos(x) - 1/12/x^2 - 1/4*arccos(x)^2/x^4 + 1/3*log(x)

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Fricas [A]
time = 0.46, size = 44, normalized size = 0.68 \begin {gather*} \frac {4 \, x^{4} \log \left (x\right ) + 2 \, {\left (2 \, x^{3} + x\right )} \sqrt {-x^{2} + 1} \arccos \left (x\right ) - x^{2} - 3 \, \arccos \left (x\right )^{2}}{12 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(x)^2/x^5,x, algorithm="fricas")

[Out]

1/12*(4*x^4*log(x) + 2*(2*x^3 + x)*sqrt(-x^2 + 1)*arccos(x) - x^2 - 3*arccos(x)^2)/x^4

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acos}^{2}{\left (x \right )}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(x)**2/x**5,x)

[Out]

Integral(acos(x)**2/x**5, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (51) = 102\).
time = 1.14, size = 104, normalized size = 1.60 \begin {gather*} -\frac {1}{48} \, {\left (\frac {x^{3} {\left (\frac {9 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{{\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}} - \frac {9 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}}{x^{3}}\right )} \arccos \left (x\right ) - \frac {2 \, x^{2} + 1}{12 \, x^{2}} - \frac {\arccos \left (x\right )^{2}}{4 \, x^{4}} + \frac {1}{6} \, \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(x)^2/x^5,x, algorithm="giac")

[Out]

-1/48*(x^3*(9*(sqrt(-x^2 + 1) - 1)^2/x^2 + 1)/(sqrt(-x^2 + 1) - 1)^3 - 9*(sqrt(-x^2 + 1) - 1)/x - (sqrt(-x^2 +
1) - 1)^3/x^3)*arccos(x) - 1/12*(2*x^2 + 1)/x^2 - 1/4*arccos(x)^2/x^4 + 1/6*log(x^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {acos}\left (x\right )}^2}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(x)^2/x^5,x)

[Out]

int(acos(x)^2/x^5, x)

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