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 = 0.0097707, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 5265
Rule 4619
Rule 261
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*}
Mathematica [A] time = 0.0116146, size = 25, normalized size = 1. \[ a \sqrt{1-\frac{x^2}{a^2}}+x \csc ^{-1}\left (\frac{a}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.182, size = 52, normalized size = 2.1 \begin{align*} -a \left ( -{\frac{x}{a}{\rm arccsc} \left ({\frac{a}{x}}\right )}-{\frac{{x}^{2}}{{a}^{2}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ){\frac{1}{\sqrt{{\frac{{x}^{2}}{{a}^{2}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ) }}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00991, size = 31, normalized size = 1.24 \begin{align*} x \operatorname{arccsc}\left (\frac{a}{x}\right ) + a \sqrt{-\frac{x^{2}}{a^{2}} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.97449, size = 55, normalized size = 2.2 \begin{align*} x \operatorname{arccsc}\left (\frac{a}{x}\right ) + x \sqrt{\frac{a^{2} - x^{2}}{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.221764, size = 22, normalized size = 0.88 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08226, size = 35, normalized size = 1.4 \begin{align*} a{\left (\frac{x \arcsin \left (\frac{x}{a}\right )}{a} + \sqrt{-\frac{x^{2}}{a^{2}} + 1}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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