Optimal. Leaf size=29 \[ -2 \sqrt{a} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{a} x\right ),-1\right )-\frac{\cos ^{-1}\left (a x^2\right )}{x} \]
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Rubi [A] time = 0.014442, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 4843
Rule 12
Rule 221
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*}
Mathematica [C] time = 0.0385381, size = 40, normalized size = 1.38 \[ -\frac{\cos ^{-1}\left (a x^2\right )+2 i \sqrt{-a} x \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-a} x\right ),-1\right )}{x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 57, normalized size = 2. \begin{align*} -{\frac{\arccos \left ( a{x}^{2} \right ) }{x}}-2\,{\frac{\sqrt{a}\sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{a},i \right ) }{\sqrt{-{a}^{2}{x}^{4}+1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\arccos \left (a x^{2}\right )}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.21139, size = 44, normalized size = 1.52 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arccos \left (a x^{2}\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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