Optimal. Leaf size=58 \[ -\frac {1}{6} a x^2 \sqrt {1-\frac {a^2}{x^2}}-\frac {1}{6} a^3 \tanh ^{-1}\left (\sqrt {1-\frac {a^2}{x^2}}\right )+\frac {1}{3} x^3 \sec ^{-1}\left (\frac {x}{a}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4833, 5220, 266, 51, 63, 208} \[ -\frac {1}{6} a x^2 \sqrt {1-\frac {a^2}{x^2}}-\frac {1}{6} a^3 \tanh ^{-1}\left (\sqrt {1-\frac {a^2}{x^2}}\right )+\frac {1}{3} x^3 \sec ^{-1}\left (\frac {x}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rule 4833
Rule 5220
Rubi steps
\begin {align*} \int x^2 \cos ^{-1}\left (\frac {a}{x}\right ) \, dx &=\int x^2 \sec ^{-1}\left (\frac {x}{a}\right ) \, dx\\ &=\frac {1}{3} x^3 \sec ^{-1}\left (\frac {x}{a}\right )-\frac {1}{3} a \int \frac {x}{\sqrt {1-\frac {a^2}{x^2}}} \, dx\\ &=\frac {1}{3} x^3 \sec ^{-1}\left (\frac {x}{a}\right )+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-a^2 x}} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{6} a \sqrt {1-\frac {a^2}{x^2}} x^2+\frac {1}{3} x^3 \sec ^{-1}\left (\frac {x}{a}\right )+\frac {1}{12} a^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{6} a \sqrt {1-\frac {a^2}{x^2}} x^2+\frac {1}{3} x^3 \sec ^{-1}\left (\frac {x}{a}\right )-\frac {1}{6} a \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-\frac {a^2}{x^2}}\right )\\ &=-\frac {1}{6} a \sqrt {1-\frac {a^2}{x^2}} x^2+\frac {1}{3} x^3 \sec ^{-1}\left (\frac {x}{a}\right )-\frac {1}{6} a^3 \tanh ^{-1}\left (\sqrt {1-\frac {a^2}{x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 61, normalized size = 1.05 \[ \frac {1}{3} x^3 \cos ^{-1}\left (\frac {a}{x}\right )-\frac {1}{6} a \left (x^2 \sqrt {1-\frac {a^2}{x^2}}+a^2 \log \left (x \left (\sqrt {1-\frac {a^2}{x^2}}+1\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 93, normalized size = 1.60 \[ \frac {1}{6} \, a^{3} \log \left (x \sqrt {-\frac {a^{2} - x^{2}}{x^{2}}} - x\right ) - \frac {1}{6} \, a x^{2} \sqrt {-\frac {a^{2} - x^{2}}{x^{2}}} + \frac {1}{3} \, {\left (x^{3} - 1\right )} \arccos \left (\frac {a}{x}\right ) + \frac {2}{3} \, \arctan \left (\frac {x \sqrt {-\frac {a^{2} - x^{2}}{x^{2}}} - x}{a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 77, normalized size = 1.33 \[ -\frac {a^{4} {\left (\frac {2 \, x^{2} \sqrt {-\frac {a^{2}}{x^{2}} + 1}}{a^{2}} + \log \left (\sqrt {-\frac {a^{2}}{x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {a^{2}}{x^{2}} + 1} + 1\right )\right )} - 4 \, a x^{3} \arccos \left (\frac {a}{x}\right )}{12 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 56, normalized size = 0.97 \[ -a^{3} \left (-\frac {x^{3} \arccos \left (\frac {a}{x}\right )}{3 a^{3}}+\frac {x^{2} \sqrt {1-\frac {a^{2}}{x^{2}}}}{6 a^{2}}+\frac {\arctanh \left (\frac {1}{\sqrt {1-\frac {a^{2}}{x^{2}}}}\right )}{6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 72, normalized size = 1.24 \[ \frac {1}{3} \, x^{3} \arccos \left (\frac {a}{x}\right ) - \frac {1}{12} \, {\left (a^{2} \log \left (\sqrt {-\frac {a^{2}}{x^{2}} + 1} + 1\right ) - a^{2} \log \left (\sqrt {-\frac {a^{2}}{x^{2}} + 1} - 1\right ) + 2 \, x^{2} \sqrt {-\frac {a^{2}}{x^{2}} + 1}\right )} a \]
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 \[ \int x^2\,\mathrm {acos}\left (\frac {a}{x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.87, size = 97, normalized size = 1.67 \[ - \frac {a \left (\begin {cases} \frac {a^{2} \operatorname {acosh}{\left (\frac {x}{a} \right )}}{2} + \frac {a x \sqrt {-1 + \frac {x^{2}}{a^{2}}}}{2} & \text {for}\: \left |{\frac {x^{2}}{a^{2}}}\right | > 1 \\- \frac {i a^{2} \operatorname {asin}{\left (\frac {x}{a} \right )}}{2} + \frac {i a x}{2 \sqrt {1 - \frac {x^{2}}{a^{2}}}} - \frac {i x^{3}}{2 a \sqrt {1 - \frac {x^{2}}{a^{2}}}} & \text {otherwise} \end {cases}\right )}{3} + \frac {x^{3} \operatorname {acos}{\left (\frac {a}{x} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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