Optimal. Leaf size=34 \[ -\frac {a+b \text {ArcSin}\left (c x^2\right )}{x}+2 b \sqrt {c} F\left (\left .\text {ArcSin}\left (\sqrt {c} x\right )\right |-1\right ) \]
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Rubi [A]
time = 0.01, 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 = {4926, 12, 227}
\begin {gather*} 2 b \sqrt {c} F\left (\left .\text {ArcSin}\left (\sqrt {c} x\right )\right |-1\right )-\frac {a+b \text {ArcSin}\left (c x^2\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 227
Rule 4926
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] Result contains complex when optimal does not.
time = 0.05, size = 44, normalized size = 1.29 \begin {gather*} -\frac {a+b \text {ArcSin}\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} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] 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 = 0.01, size = 66, normalized size = 1.94
method | result | size |
default | \(-\frac {a}{x}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{x}+\frac {2 \sqrt {c}\, \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \EllipticF \left (x \sqrt {c}, i\right )}{\sqrt {-c^{2} x^{4}+1}}\right )\) | \(66\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.71, size = 45, normalized size = 1.32 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.81, size = 49, normalized size = 1.44 \begin {gather*} - \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (c\,x^2\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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