3.3.23 \(\int \frac {\arcsin (x)}{x^3} \, dx\) [223]

Optimal. Leaf size=28 \[ -\frac {\sqrt {1-x^2}}{2 x}-\frac {\arcsin (x)}{2 x^2} \]

[Out]

-1/2*(-x^2+1)^(1/2)/x-1/2*arcsin(x)/x^2

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4723, 270} \begin {gather*} -\frac {\arcsin (x)}{2 x^2}-\frac {\sqrt {1-x^2}}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSin[x]/x^3,x]

[Out]

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

Rule 270

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 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSin[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 {gather*} \begin {aligned} \text {Integral} &=-\frac {\arcsin (x)}{2 x^2}+\frac {1}{2} \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{2 x}-\frac {\arcsin (x)}{2 x^2}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 23, normalized size = 0.82 \begin {gather*} -\frac {x \sqrt {1-x^2}+\arcsin (x)}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[x]/x^3,x]

[Out]

-1/2*(x*Sqrt[1 - x^2] + ArcSin[x])/x^2

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 23, normalized size = 0.82

method	result	size
default	\(-\frac {\sqrt {-x^{2}+1}}{2 x}-\frac {\arcsin \left (x \right )}{2 x^{2}}\)	\(23\)
parts	\(-\frac {\sqrt {-x^{2}+1}}{2 x}-\frac {\arcsin \left (x \right )}{2 x^{2}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x)/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*(-x^2+1)^(1/2)/x-1/2*arcsin(x)/x^2

________________________________________________________________________________________

Maxima [A]
time = 0.42, size = 22, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{2 \, x} - \frac {\arcsin \left (x\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)/x^3,x, algorithm="maxima")

[Out]

-1/2*sqrt(-x^2 + 1)/x - 1/2*arcsin(x)/x^2

________________________________________________________________________________________

Fricas [A]
time = 0.63, size = 19, normalized size = 0.68 \begin {gather*} -\frac {\sqrt {-x^{2} + 1} x + \arcsin \left (x\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)/x^3,x, algorithm="fricas")

[Out]

-1/2*(sqrt(-x^2 + 1)*x + arcsin(x))/x^2

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.84, size = 37, normalized size = 1.32 \begin {gather*} \frac {\begin {cases} - \frac {i \sqrt {x^{2} - 1}}{x} & \text {for}\: \left |{x^{2}}\right | > 1 \\- \frac {\sqrt {1 - x^{2}}}{x} & \text {otherwise} \end {cases}}{2} - \frac {\operatorname {asin}{\left (x \right )}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(x)/x**3,x)

[Out]

Piecewise((-I*sqrt(x**2 - 1)/x, Abs(x**2) > 1), (-sqrt(1 - x**2)/x, True))/2 - asin(x)/(2*x**2)

________________________________________________________________________________________

Giac [A]
time = 0.46, size = 40, normalized size = 1.43 \begin {gather*} \frac {x}{4 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} - \frac {\sqrt {-x^{2} + 1} - 1}{4 \, x} - \frac {\arcsin \left (x\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)/x^3,x, algorithm="giac")

[Out]

1/4*x/(sqrt(-x^2 + 1) - 1) - 1/4*(sqrt(-x^2 + 1) - 1)/x - 1/2*arcsin(x)/x^2

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\mathrm {asin}\left (x\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(x)/x^3,x)

[Out]

int(asin(x)/x^3, x)

________________________________________________________________________________________

Chatgpt [F] Failed to verify
time = 1.00, size = 19, normalized size = 0.68 \begin {gather*} -\frac {\arcsin \left (x \right )}{2 x}-\frac {1}{2 \sqrt {-x^{2}+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(arcsin(x)/x^3,x)

[Out]

-1/2/x*arcsin(x)-1/2/(-x^2+1)^(1/2)

________________________________________________________________________________________