Optimal. Leaf size=25 \[ a \sqrt {1-\frac {x^2}{a^2}}+x \sin ^{-1}\left (\frac {x}{a}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5265, 4619, 261} \[ a \sqrt {1-\frac {x^2}{a^2}}+x \sin ^{-1}\left (\frac {x}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 261
Rule 4619
Rule 5265
Rubi steps
\begin {align*} \int \csc ^{-1}\left (\frac {a}{x}\right ) \, dx &=\int \sin ^{-1}\left (\frac {x}{a}\right ) \, dx\\ &=x \sin ^{-1}\left (\frac {x}{a}\right )-\frac {\int \frac {x}{\sqrt {1-\frac {x^2}{a^2}}} \, dx}{a}\\ &=a \sqrt {1-\frac {x^2}{a^2}}+x \sin ^{-1}\left (\frac {x}{a}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ a \sqrt {1-\frac {x^2}{a^2}}+x \csc ^{-1}\left (\frac {a}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 26, normalized size = 1.04 \[ x \operatorname {arccsc}\left (\frac {a}{x}\right ) + x \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 26, normalized size = 1.04 \[ a {\left (\frac {x \arcsin \left (\frac {x}{a}\right )}{a} + \sqrt {-\frac {x^{2}}{a^{2}} + 1}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 52, normalized size = 2.08 \[ -a \left (-\frac {\mathrm {arccsc}\left (\frac {a}{x}\right ) x}{a}-\frac {x^{2} \left (-1+\frac {a^{2}}{x^{2}}\right )}{\sqrt {\frac {\left (-1+\frac {a^{2}}{x^{2}}\right ) x^{2}}{a^{2}}}\, a^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 23, normalized size = 0.92 \[ x \operatorname {arccsc}\left (\frac {a}{x}\right ) + a \sqrt {-\frac {x^{2}}{a^{2}} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 23, normalized size = 0.92 \[ a\,\sqrt {1-\frac {x^2}{a^2}}+x\,\mathrm {asin}\left (\frac {x}{a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 22, normalized size = 0.88 \[ \begin {cases} a \sqrt {1 - \frac {x^{2}}{a^{2}}} + x \operatorname {acsc}{\left (\frac {a}{x} \right )} & \text {for}\: a \neq 0 \\\tilde {\infty } x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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