Optimal. Leaf size=23 \[ \frac {2 \operatorname {EllipticF}\left (\sin ^{-1}(c x),-1\right )}{c}-\frac {E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]
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Rubi [A] time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {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
Rubi steps
\begin {align*} \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 [A] time = 0.01, size = 24, normalized size = 1.04 \[ \frac {E\left (\left .\sin ^{-1}\left (\sqrt {-c^2} x\right )\right |-1\right )}{\sqrt {-c^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1}}{\sqrt {c^{2} x^{2} + 1}}, x\right ) \]
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 \[ \int \frac {\sqrt {-c^{2} x^{2} + 1}}{\sqrt {c^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 28, normalized size = 1.22 \[ \frac {\left (-\EllipticE \left (c x \,\mathrm {csgn}\relax (c ), i\right )+2 \EllipticF \left (c x \,\mathrm {csgn}\relax (c ), i\right )\right ) \mathrm {csgn}\relax (c )}{c} \]
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 \[ \int \frac {\sqrt {-c^{2} x^{2} + 1}}{\sqrt {c^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {\sqrt {1-c^2\,x^2}}{\sqrt {c^2\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{\sqrt {c^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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