Optimal. Leaf size=23 \[ \frac {2 F\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}-\frac {E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]
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Rubi [A] time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1199, 423, 424, 248, 221} \[ \frac {2 F\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}-\frac {E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]
Antiderivative was successfully verified.
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Rule 221
Rule 248
Rule 423
Rule 424
Rule 1199
Rubi steps
\begin {align*} \int \frac {1-c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx &=\int \frac {\sqrt {1-c^2 x^2}}{\sqrt {1+c^2 x^2}} \, dx\\ &=2 \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}} \, dx-\int \frac {\sqrt {1+c^2 x^2}}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}+2 \int \frac {1}{\sqrt {1-c^4 x^4}} \, dx\\ &=-\frac {E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}+\frac {2 F\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 47, normalized size = 2.04 \[ x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^4 x^4\right )-\frac {1}{3} c^2 x^3 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};c^4 x^4\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{4} x^{4} + 1}}{c^{2} x^{2} + 1}, x\right ) \]
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 \[ \int -\frac {c^{2} x^{2} - 1}{\sqrt {-c^{4} x^{4} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 117, normalized size = 5.09 \[ \frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, \EllipticF \left (\sqrt {c^{2}}\, x , i\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, \left (-\EllipticE \left (\sqrt {c^{2}}\, x , i\right )+\EllipticF \left (\sqrt {c^{2}}\, x , i\right )\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}} \]
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 \[ -\int \frac {c^{2} x^{2} - 1}{\sqrt {-c^{4} x^{4} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ -\int \frac {c^2\,x^2-1}{\sqrt {1-c^4\,x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.03, size = 71, normalized size = 3.09 \[ - \frac {c^{2} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {c^{4} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {c^{4} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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