Optimal. Leaf size=56 \[ -\frac {1}{9} a^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}+\frac {1}{3} a^3 \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{3} x^3 \sin ^{-1}\left (\frac {x}{a}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5265, 4627, 266, 43} \[ -\frac {1}{9} a^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}+\frac {1}{3} a^3 \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{3} x^3 \sin ^{-1}\left (\frac {x}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 4627
Rule 5265
Rubi steps
\begin {align*} \int x^2 \csc ^{-1}\left (\frac {a}{x}\right ) \, dx &=\int x^2 \sin ^{-1}\left (\frac {x}{a}\right ) \, dx\\ &=\frac {1}{3} x^3 \sin ^{-1}\left (\frac {x}{a}\right )-\frac {\int \frac {x^3}{\sqrt {1-\frac {x^2}{a^2}}} \, dx}{3 a}\\ &=\frac {1}{3} x^3 \sin ^{-1}\left (\frac {x}{a}\right )-\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {1-\frac {x}{a^2}}} \, dx,x,x^2\right )}{6 a}\\ &=\frac {1}{3} x^3 \sin ^{-1}\left (\frac {x}{a}\right )-\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{\sqrt {1-\frac {x}{a^2}}}-a^2 \sqrt {1-\frac {x}{a^2}}\right ) \, dx,x,x^2\right )}{6 a}\\ &=\frac {1}{3} a^3 \sqrt {1-\frac {x^2}{a^2}}-\frac {1}{9} a^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}+\frac {1}{3} x^3 \sin ^{-1}\left (\frac {x}{a}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.75 \[ \frac {1}{9} a \left (2 a^2+x^2\right ) \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{3} x^3 \csc ^{-1}\left (\frac {a}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 39, normalized size = 0.70 \[ \frac {1}{3} \, x^{3} \operatorname {arccsc}\left (\frac {a}{x}\right ) + \frac {1}{9} \, {\left (2 \, a^{2} x + x^{3}\right )} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 68, normalized size = 1.21 \[ \frac {1}{3} \, a^{2} x {\left (\frac {x^{2}}{a^{2}} - 1\right )} \arcsin \left (\frac {x}{a}\right ) - \frac {1}{9} \, a^{3} {\left (-\frac {x^{2}}{a^{2}} + 1\right )}^{\frac {3}{2}} + \frac {1}{3} \, a^{2} x \arcsin \left (\frac {x}{a}\right ) + \frac {1}{3} \, a^{3} \sqrt {-\frac {x^{2}}{a^{2}} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 66, normalized size = 1.18 \[ -a^{3} \left (-\frac {x^{3} \mathrm {arccsc}\left (\frac {a}{x}\right )}{3 a^{3}}-\frac {\left (-1+\frac {a^{2}}{x^{2}}\right ) \left (\frac {2 a^{2}}{x^{2}}+1\right ) x^{4}}{9 \sqrt {\frac {\left (-1+\frac {a^{2}}{x^{2}}\right ) x^{2}}{a^{2}}}\, a^{4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 54, normalized size = 0.96 \[ \frac {1}{3} \, x^{3} \operatorname {arccsc}\left (\frac {a}{x}\right ) + \frac {2 \, a^{4} \sqrt {-\frac {x^{2}}{a^{2}} + 1} + a^{2} x^{2} \sqrt {-\frac {x^{2}}{a^{2}} + 1}}{9 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \left \{\begin {array}{cl} \frac {x^3\,\mathrm {asin}\left (\frac {x}{a}\right )}{3}+\frac {\sqrt {a^2-x^2}\,\left (2\,a^2+x^2\right )}{9} & \text {\ if\ \ }0<a\\ \int x^2\,\mathrm {asin}\left (\frac {x}{a}\right ) \,d x & \text {\ if\ \ }\neg 0<a \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 51, normalized size = 0.91 \[ \begin {cases} \frac {2 a^{3} \sqrt {1 - \frac {x^{2}}{a^{2}}}}{9} + \frac {a x^{2} \sqrt {1 - \frac {x^{2}}{a^{2}}}}{9} + \frac {x^{3} \operatorname {acsc}{\left (\frac {a}{x} \right )}}{3} & \text {for}\: a \neq 0 \\\tilde {\infty } x^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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