3.8 \(\int \frac {\cos ^{-1}(a x)}{x^3} \, dx\)

Optimal. Leaf size=34 \[ \frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\cos ^{-1}(a x)}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4628, 264} \[ \frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\cos ^{-1}(a x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sqrt[1 - a^2*x^2])/(2*x) - ArcCos[a*x]/(2*x^2)

Rule 264

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]

Rule 4628

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]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 31, normalized size = 0.91 \[ \frac {a x \sqrt {1-a^2 x^2}-\cos ^{-1}(a x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x*Sqrt[1 - a^2*x^2] - ArcCos[a*x])/(2*x^2)

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fricas [A]  time = 0.51, size = 27, normalized size = 0.79 \[ \frac {\sqrt {-a^{2} x^{2} + 1} a x - \arccos \left (a x\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)/x^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.18, size = 68, normalized size = 2.00 \[ -\frac {1}{4} \, {\left (\frac {a^{4} x}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{x {\left | a \right |}}\right )} a - \frac {\arccos \left (a x\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)/x^3,x, algorithm="giac")

[Out]

-1/4*(a^4*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) - (sqrt(-a^2*x^2 + 1)*abs(a) + a)/(x*abs(a)))*a - 1/2*arc
cos(a*x)/x^2

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maple [A]  time = 0.00, size = 38, normalized size = 1.12 \[ a^{2} \left (-\frac {\arccos \left (a x \right )}{2 a^{2} x^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{2 a x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)/x^3,x)

[Out]

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

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maxima [A]  time = 0.40, size = 28, normalized size = 0.82 \[ \frac {\sqrt {-a^{2} x^{2} + 1} a}{2 \, x} - \frac {\arccos \left (a x\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)/x^3,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {acos}\left (a\,x\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acos(a*x)/x^3, x)

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sympy [C]  time = 1.12, size = 53, normalized size = 1.56 \[ - \frac {a \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{2} - \frac {\operatorname {acos}{\left (a x \right )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/2 - acos(a*x)/(2
*x**2)

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