Optimal. Leaf size=58 \[ -\frac {1}{6} a \sqrt {1-\frac {a^2}{x^2}} x^2+\frac {1}{3} x^3 \sec ^{-1}\left (\frac {x}{a}\right )-\frac {1}{6} a^3 \tanh ^{-1}\left (\sqrt {1-\frac {a^2}{x^2}}\right ) \]
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Rubi [A]
time = 0.03, 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 = {4917, 5328,
272, 44, 65, 214} \begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 214
Rule 272
Rule 4917
Rule 5328
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 \text {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 \text {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 \text {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 = 0.03, size = 61, normalized size = 1.05 \begin {gather*} \frac {1}{3} x^3 \text {ArcCos}\left (\frac {a}{x}\right )-\frac {1}{6} a \left (\sqrt {1-\frac {a^2}{x^2}} x^2+a^2 \log \left (\left (1+\sqrt {1-\frac {a^2}{x^2}}\right ) x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 56, normalized size = 0.97
method | result | size |
derivativedivides | \(-a^{3} \left (-\frac {x^{3} \arccos \left (\frac {a}{x}\right )}{3 a^{3}}+\frac {x^{2} \sqrt {1-\frac {a^{2}}{x^{2}}}}{6 a^{2}}+\frac {\arctanh \left (\frac {1}{\sqrt {1-\frac {a^{2}}{x^{2}}}}\right )}{6}\right )\) | \(56\) |
default | \(-a^{3} \left (-\frac {x^{3} \arccos \left (\frac {a}{x}\right )}{3 a^{3}}+\frac {x^{2} \sqrt {1-\frac {a^{2}}{x^{2}}}}{6 a^{2}}+\frac {\arctanh \left (\frac {1}{\sqrt {1-\frac {a^{2}}{x^{2}}}}\right )}{6}\right )\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 72, normalized size = 1.24 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.54, size = 93, normalized size = 1.60 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.89, size = 95, normalized size = 1.64 \begin {gather*} - \frac {a \left (\begin {cases} \frac {a^{2} \operatorname {acosh}{\left (\frac {x}{a} \right )}}{2} - \frac {a x}{2 \sqrt {-1 + \frac {x^{2}}{a^{2}}}} + \frac {x^{3}}{2 a \sqrt {-1 + \frac {x^{2}}{a^{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 \sqrt {1 - \frac {x^{2}}{a^{2}}}}{2} & \text {otherwise} \end {cases}\right )}{3} + \frac {x^{3} \operatorname {acos}{\left (\frac {a}{x} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 77, normalized size = 1.33 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \int x^2\,\mathrm {acos}\left (\frac {a}{x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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