Optimal. Leaf size=47 \[ \frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}-\frac {1}{4} a^2 \text {ArcSin}\left (\frac {x}{a}\right )+\frac {1}{2} x^2 \text {ArcSin}\left (\frac {x}{a}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5373, 4723, 327,
222} \begin {gather*} -\frac {1}{4} a^2 \text {ArcSin}\left (\frac {x}{a}\right )+\frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{2} x^2 \text {ArcSin}\left (\frac {x}{a}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 327
Rule 4723
Rule 5373
Rubi steps
\begin {align*} \int x \csc ^{-1}\left (\frac {a}{x}\right ) \, dx &=\int x \sin ^{-1}\left (\frac {x}{a}\right ) \, dx\\ &=\frac {1}{2} x^2 \sin ^{-1}\left (\frac {x}{a}\right )-\frac {\int \frac {x^2}{\sqrt {1-\frac {x^2}{a^2}}} \, dx}{2 a}\\ &=\frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{2} x^2 \sin ^{-1}\left (\frac {x}{a}\right )-\frac {1}{4} a \int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx\\ &=\frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}-\frac {1}{4} a^2 \sin ^{-1}\left (\frac {x}{a}\right )+\frac {1}{2} x^2 \sin ^{-1}\left (\frac {x}{a}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 44, normalized size = 0.94 \begin {gather*} \frac {1}{4} \left (a x \sqrt {1-\frac {x^2}{a^2}}+2 x^2 \csc ^{-1}\left (\frac {a}{x}\right )-a^2 \text {ArcSin}\left (\frac {x}{a}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(89\) vs.
\(2(39)=78\).
time = 0.49, size = 90, normalized size = 1.91
method | result | size |
derivativedivides | \(-a^{2} \left (-\frac {x^{2} \mathrm {arccsc}\left (\frac {a}{x}\right )}{2 a^{2}}-\frac {\sqrt {\frac {a^{2}}{x^{2}}-1}\, \left (-\frac {\arctan \left (\frac {1}{\sqrt {\frac {a^{2}}{x^{2}}-1}}\right ) a^{2}}{x^{2}}+\sqrt {\frac {a^{2}}{x^{2}}-1}\right ) x^{3}}{4 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{3}}\right )\) | \(90\) |
default | \(-a^{2} \left (-\frac {x^{2} \mathrm {arccsc}\left (\frac {a}{x}\right )}{2 a^{2}}-\frac {\sqrt {\frac {a^{2}}{x^{2}}-1}\, \left (-\frac {\arctan \left (\frac {1}{\sqrt {\frac {a^{2}}{x^{2}}-1}}\right ) a^{2}}{x^{2}}+\sqrt {\frac {a^{2}}{x^{2}}-1}\right ) x^{3}}{4 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{3}}\right )\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 46, normalized size = 0.98 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {arccsc}\left (\frac {a}{x}\right ) - \frac {a^{3} \arcsin \left (\frac {x}{a}\right ) - a^{2} x \sqrt {-\frac {x^{2}}{a^{2}} + 1}}{4 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 38, normalized size = 0.81 \begin {gather*} \frac {1}{4} \, x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} - \frac {1}{4} \, {\left (a^{2} - 2 \, x^{2}\right )} \operatorname {arccsc}\left (\frac {a}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 41, normalized size = 0.87 \begin {gather*} \begin {cases} - \frac {a^{2} \operatorname {acsc}{\left (\frac {a}{x} \right )}}{4} + \frac {a x \sqrt {1 - \frac {x^{2}}{a^{2}}}}{4} + \frac {x^{2} \operatorname {acsc}{\left (\frac {a}{x} \right )}}{2} & \text {for}\: a \neq 0 \\\tilde {\infty } x^{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 48, normalized size = 1.02 \begin {gather*} \frac {1}{2} \, a^{2} {\left (\frac {x^{2}}{a^{2}} - 1\right )} \arcsin \left (\frac {x}{a}\right ) + \frac {1}{4} \, a^{2} \arcsin \left (\frac {x}{a}\right ) + \frac {1}{4} \, a x \sqrt {-\frac {x^{2}}{a^{2}} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.61, size = 38, normalized size = 0.81 \begin {gather*} \frac {a^2\,\mathrm {asin}\left (\frac {x}{a}\right )\,\left (\frac {2\,x^2}{a^2}-1\right )}{4}+\frac {a\,x\,\sqrt {1-\frac {x^2}{a^2}}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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