3.4.65 \(\int \frac {\text {ArcSin}(\sqrt {x})}{x^2} \, dx\) [365]

Optimal. Leaf size=28 \[ -\frac {\sqrt {1-x}}{\sqrt {x}}-\frac {\text {ArcSin}\left (\sqrt {x}\right )}{x} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4926, 12, 37} \begin {gather*} -\frac {\text {ArcSin}\left (\sqrt {x}\right )}{x}-\frac {\sqrt {1-x}}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSin[Sqrt[x]]/x^2,x]

[Out]

-(Sqrt[1 - x]/Sqrt[x]) - ArcSin[Sqrt[x]]/x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 4926

Int[((a_.) + ArcSin[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcSin[
u])/(d*(m + 1))), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/Sqrt[1 - u^2]), x]
, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(
m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rubi steps

\begin {align*} \int \frac {\sin ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx &=-\frac {\sin ^{-1}\left (\sqrt {x}\right )}{x}+\int \frac {1}{2 \sqrt {1-x} x^{3/2}} \, dx\\ &=-\frac {\sin ^{-1}\left (\sqrt {x}\right )}{x}+\frac {1}{2} \int \frac {1}{\sqrt {1-x} x^{3/2}} \, dx\\ &=-\frac {\sqrt {1-x}}{\sqrt {x}}-\frac {\sin ^{-1}\left (\sqrt {x}\right )}{x}\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[Sqrt[x]]/x^2,x]

[Out]

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

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Maple [A]
time = 0.00, size = 23, normalized size = 0.82

method	result	size
derivativedivides	\(-\frac {\arcsin \left (\sqrt {x}\right )}{x}-\frac {\sqrt {1-x}}{\sqrt {x}}\)	\(23\)
default	\(-\frac {\arcsin \left (\sqrt {x}\right )}{x}-\frac {\sqrt {1-x}}{\sqrt {x}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x^(1/2))/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.47, size = 22, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {-x + 1}}{\sqrt {x}} - \frac {\arcsin \left (\sqrt {x}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x^(1/2))/x^2,x, algorithm="maxima")

[Out]

-sqrt(-x + 1)/sqrt(x) - arcsin(sqrt(x))/x

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Fricas [A]
time = 2.23, size = 21, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {x} \sqrt {-x + 1} + \arcsin \left (\sqrt {x}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x^(1/2))/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 1.87, size = 42, normalized size = 1.50 \begin {gather*} \frac {\begin {cases} - \frac {2 i \sqrt {x - 1}}{\sqrt {x}} & \text {for}\: \left |{x}\right | > 1 \\- \frac {2 \sqrt {1 - x}}{\sqrt {x}} & \text {otherwise} \end {cases}}{2} - \frac {\operatorname {asin}{\left (\sqrt {x} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(x**(1/2))/x**2,x)

[Out]

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

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Giac [A]
time = 0.40, size = 40, normalized size = 1.43 \begin {gather*} -\frac {\sqrt {-x + 1} - 1}{2 \, \sqrt {x}} - \frac {\arcsin \left (\sqrt {x}\right )}{x} + \frac {\sqrt {x}}{2 \, {\left (\sqrt {-x + 1} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x^(1/2))/x^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\mathrm {asin}\left (\sqrt {x}\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(x^(1/2))/x^2,x)

[Out]

int(asin(x^(1/2))/x^2, x)

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