Optimal. Leaf size=34 \[ 2 b \sqrt {c} F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )-\frac {a+b \sin ^{-1}\left (c x^2\right )}{x} \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 12
Rule 221
Rule 4842
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*}
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Mathematica [C] time = 0.06, size = 44, normalized size = 1.29 \[ -\frac {a+b \sin ^{-1}\left (c x^2\right )-2 i b \sqrt {-c} x F\left (\left .i \sinh ^{-1}\left (\sqrt {-c} x\right )\right |-1\right )}{x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arcsin \left (c x^{2}\right ) + a}{x^{2}}, x\right ) \]
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 \[ \int \frac {b \arcsin \left (c x^{2}\right ) + a}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 66, normalized size = 1.94 \[ -\frac {a}{x}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{x}+\frac {2 \sqrt {c}\, \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \EllipticF \left (x \sqrt {c}, i\right )}{\sqrt {-c^{2} x^{4}+1}}\right ) \]
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 \[ -\frac {{\left (2 \, c x \int \frac {\sqrt {-c x^{2} + 1}}{\sqrt {c x^{2} + 1} {\left (c x^{2} - 1\right )}}\,{d x} + \arctan \left (c x^{2}, \sqrt {c x^{2} + 1} \sqrt {-c x^{2} + 1}\right )\right )} b}{x} - \frac {a}{x} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x^2\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.32, size = 49, normalized size = 1.44 \[ - \frac {a}{x} + \frac {b c x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {c^{2} x^{4} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} - \frac {b \operatorname {asin}{\left (c x^{2} \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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