Optimal. Leaf size=29 \[ -\frac {\cos ^{-1}\left (a x^2\right )}{x}-2 \sqrt {a} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4843, 12, 221} \[ -\frac {\cos ^{-1}\left (a x^2\right )}{x}-2 \sqrt {a} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 221
Rule 4843
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}\left (a x^2\right )}{x^2} \, dx &=-\frac {\cos ^{-1}\left (a x^2\right )}{x}-\int \frac {2 a}{\sqrt {1-a^2 x^4}} \, dx\\ &=-\frac {\cos ^{-1}\left (a x^2\right )}{x}-(2 a) \int \frac {1}{\sqrt {1-a^2 x^4}} \, dx\\ &=-\frac {\cos ^{-1}\left (a x^2\right )}{x}-2 \sqrt {a} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.04, size = 40, normalized size = 1.38 \[ -\frac {\cos ^{-1}\left (a x^2\right )+2 i \sqrt {-a} x F\left (\left .i \sinh ^{-1}\left (\sqrt {-a} x\right )\right |-1\right )}{x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arccos \left (a x^{2}\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arccos \left (a x^{2}\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 57, normalized size = 1.97 \[ -\frac {\arccos \left (a \,x^{2}\right )}{x}-\frac {2 \sqrt {a}\, \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, \EllipticF \left (x \sqrt {a}, i\right )}{\sqrt {-a^{2} x^{4}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, a x \int \frac {\sqrt {-a x^{2} + 1}}{\sqrt {a x^{2} + 1} {\left (a x^{2} - 1\right )}}\,{d x} - \arctan \left (\sqrt {a x^{2} + 1} \sqrt {-a x^{2} + 1}, a x^{2}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {acos}\left (a\,x^2\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.07, size = 44, normalized size = 1.52 \[ - \frac {a x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {a^{2} x^{4} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} - \frac {\operatorname {acos}{\left (a x^{2} \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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