Integrand size = 14, antiderivative size = 34 \[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^2} \, dx=-\frac {a+b \arcsin \left (c x^2\right )}{x}+2 b \sqrt {c} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {c} x\right ),-1\right ) \]
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Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4926, 12, 227} \[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^2} \, dx=2 b \sqrt {c} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {c} x\right ),-1\right )-\frac {a+b \arcsin \left (c x^2\right )}{x} \]
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Rule 12
Rule 227
Rule 4926
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arcsin \left (c x^2\right )}{x}+b \int \frac {2 c}{\sqrt {1-c^2 x^4}} \, dx \\ & = -\frac {a+b \arcsin \left (c x^2\right )}{x}+(2 b c) \int \frac {1}{\sqrt {1-c^2 x^4}} \, dx \\ & = -\frac {a+b \arcsin \left (c x^2\right )}{x}+2 b \sqrt {c} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {c} x\right ),-1\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29 \[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^2} \, dx=-\frac {a+b \arcsin \left (c x^2\right )-2 i b \sqrt {-c} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c} x\right ),-1\right )}{x} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (32 ) = 64\).
Time = 0.39 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.94
method | result | size |
default | \(-\frac {a}{x}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{x}+\frac {2 \sqrt {c}\, \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {c}, i\right )}{\sqrt {-c^{2} x^{4}+1}}\right )\) | \(66\) |
parts | \(-\frac {a}{x}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{x}+\frac {2 \sqrt {c}\, \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {c}, i\right )}{\sqrt {-c^{2} x^{4}+1}}\right )\) | \(66\) |
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none
Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.32 \[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^2} \, dx=\frac {b x \arctan \left (\frac {\sqrt {-c^{2} x^{4} + 1}}{c x^{2}}\right ) + {\left (b x - b\right )} \arcsin \left (c x^{2}\right ) - a}{x} \]
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Time = 0.76 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^2} \, dx=- \frac {a}{x} + \frac {b c x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {c^{2} x^{4} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} - \frac {b \operatorname {asin}{\left (c x^{2} \right )}}{x} \]
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\[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^2} \, dx=\int { \frac {b \arcsin \left (c x^{2}\right ) + a}{x^{2}} \,d x } \]
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\[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^2} \, dx=\int { \frac {b \arcsin \left (c x^{2}\right ) + a}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^2} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x^2\right )}{x^2} \,d x \]
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