Optimal. Leaf size=23 \[ \frac{2 \text{EllipticF}\left (\sin ^{-1}(c x),-1\right )}{c}-\frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]
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Rubi [A] time = 0.0334165, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 1199
Rule 423
Rule 424
Rule 248
Rule 221
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*}
Mathematica [C] time = 0.0117478, 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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Maple [B] time = 0.048, size = 117, normalized size = 5.1 \begin{align*}{\sqrt{-{c}^{2}{x}^{2}+1}\sqrt{{c}^{2}{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{{c}^{2}},i \right ){\frac{1}{\sqrt{{c}^{2}}}}{\frac{1}{\sqrt{-{c}^{4}{x}^{4}+1}}}}+{\sqrt{-{c}^{2}{x}^{2}+1}\sqrt{{c}^{2}{x}^{2}+1} \left ({\it EllipticF} \left ( x\sqrt{{c}^{2}},i \right ) -{\it EllipticE} \left ( x\sqrt{{c}^{2}},i \right ) \right ){\frac{1}{\sqrt{{c}^{2}}}}{\frac{1}{\sqrt{-{c}^{4}{x}^{4}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{c^{2} x^{2} - 1}{\sqrt{-c^{4} x^{4} + 1}}\,{d x} \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{\sqrt{-c^{4} x^{4} + 1}}{c^{2} x^{2} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.93362, size = 71, normalized size = 3.09 \begin{align*} - \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 )} \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{c^{2} x^{2} - 1}{\sqrt{-c^{4} x^{4} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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