Optimal. Leaf size=28 \[ -\frac {\sqrt {1-x^2}}{2 x}-\frac {\arcsin (x)}{2 x^2} \]
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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.
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Rule 270
Rule 4723
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*}
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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