∫ ArcCos(ax)_________

Optimal. Leaf size=58 \[ \frac {a \sqrt {1-a^2 x^2}}{12 x^3}+\frac {a^3 \sqrt {1-a^2 x^2}}{6 x}-\frac {\text {ArcCos}(a x)}{4 x^4} \]

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

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Rubi [A]
time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4724, 277, 270} \begin {gather*} \frac {a \sqrt {1-a^2 x^2}}{12 x^3}+\frac {a^3 \sqrt {1-a^2 x^2}}{6 x}-\frac {\text {ArcCos}(a x)}{4 x^4} \end {gather*}

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

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*

c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]

- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]

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 -

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

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Mathematica [A]
time = 0.02, size = 41, normalized size = 0.71 \begin {gather*} \frac {a x \sqrt {1-a^2 x^2} \left (1+2 a^2 x^2\right )-3 \text {ArcCos}(a x)}{12 x^4} \end {gather*}

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method	result	size

derivativedivides	\(a^{4} \left (-\frac {\arccos \left (a x \right )}{4 a^{4} x^{4}}+\frac {\sqrt {-a^{2} x^{2}+1}}{12 a^{3} x^{3}}+\frac {\sqrt {-a^{2} x^{2}+1}}{6 a x}\right )\)	\(58\)
default	\(a^{4} \left (-\frac {\arccos \left (a x \right )}{4 a^{4} x^{4}}+\frac {\sqrt {-a^{2} x^{2}+1}}{12 a^{3} x^{3}}+\frac {\sqrt {-a^{2} x^{2}+1}}{6 a x}\right )\)	\(58\)

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

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Maxima [A]
time = 0.47, size = 50, normalized size = 0.86 \begin {gather*} \frac {1}{12} \, {\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x} + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{3}}\right )} a - \frac {\arccos \left (a x\right )}{4 \, x^{4}} \end {gather*}

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

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Fricas [A]
time = 3.18, size = 37, normalized size = 0.64 \begin {gather*} \frac {{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt {-a^{2} x^{2} + 1} - 3 \, \arccos \left (a x\right )}{12 \, x^{4}} \end {gather*}

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Sympy [C] Result contains complex when optimal does not.
time = 1.07, size = 102, normalized size = 1.76 \begin {gather*} - \frac {a \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{4} - \frac {\operatorname {acos}{\left (a x \right )}}{4 x^{4}} \end {gather*}

-a*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a

**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))/4 - acos(a*x)/(4*x**4)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (48) = 96\).
time = 0.42, size = 130, normalized size = 2.24 \begin {gather*} -\frac {1}{96} \, {\left (\frac {{\left (a^{4} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}}\right )} a^{6} x^{3}}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {\frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4}}{x} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{x^{3}}}{a^{2} {\left | a \right |}}\right )} a - \frac {\arccos \left (a x\right )}{4 \, x^{4}} \end {gather*}

-1/96*((a^4 + 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/x^2)*a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*abs(a)) - (9

*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4/x + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/x^3)/(a^2*abs(a)))*a - 1/4*arccos(a

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

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3.1.10 \(\int \frac {\text {ArcCos}(a x)}{x^5} \, dx\) [10]