Optimal. Leaf size=34 \[ 2 b \sqrt{c} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{c} x\right ),-1\right )-\frac{a+b \sin ^{-1}\left (c x^2\right )}{x} \]
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Rubi [A] time = 0.0224676, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4842, 12, 221} \[ 2 b \sqrt{c} F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )-\frac{a+b \sin ^{-1}\left (c x^2\right )}{x} \]
Antiderivative was successfully verified.
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Rule 4842
Rule 12
Rule 221
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}\left (c x^2\right )}{x^2} \, dx &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{x}+b \int \frac{2 c}{\sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{x}+(2 b c) \int \frac{1}{\sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{x}+2 b \sqrt{c} F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.0521546, size = 44, normalized size = 1.29 \[ -\frac{-2 i b \sqrt{-c} x \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-c} x\right ),-1\right )+a+b \sin ^{-1}\left (c x^2\right )}{x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 66, normalized size = 1.9 \begin{align*} -{\frac{a}{x}}+b \left ( -{\frac{\arcsin \left ( c{x}^{2} \right ) }{x}}+2\,{\frac{\sqrt{c}\sqrt{-c{x}^{2}+1}\sqrt{c{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{c},i \right ) }{\sqrt{-{c}^{2}{x}^{4}+1}}} \right ) \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{b \arcsin \left (c x^{2}\right ) + a}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.39979, size = 49, normalized size = 1.44 \begin{align*} - \frac{a}{x} + \frac{b c 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 |{c^{2} x^{4} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac{5}{4}\right )} - \frac{b \operatorname{asin}{\left (c 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{b \arcsin \left (c x^{2}\right ) + a}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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