Optimal. Leaf size=90 \[ \frac {x \sqrt {-b^2 x^4-1}}{b x^2+1}+\frac {\left (b x^2+1\right ) \sqrt {\frac {b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt {b} x\right )|\frac {1}{2}\right )}{\sqrt {b} \sqrt {-b^2 x^4-1}} \]
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Rubi [A] time = 0.01, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1196} \[ \frac {x \sqrt {-b^2 x^4-1}}{b x^2+1}+\frac {\left (b x^2+1\right ) \sqrt {\frac {b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt {b} x\right )|\frac {1}{2}\right )}{\sqrt {b} \sqrt {-b^2 x^4-1}} \]
Antiderivative was successfully verified.
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Rule 1196
Rubi steps
\begin {align*} \int \frac {1-b x^2}{\sqrt {-1-b^2 x^4}} \, dx &=\frac {x \sqrt {-1-b^2 x^4}}{1+b x^2}+\frac {\left (1+b x^2\right ) \sqrt {\frac {1+b^2 x^4}{\left (1+b x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt {b} x\right )|\frac {1}{2}\right )}{\sqrt {b} \sqrt {-1-b^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 76, normalized size = 0.84 \[ -\frac {\sqrt {b^2 x^4+1} \left (b x^3 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-b^2 x^4\right )-3 x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-b^2 x^4\right )\right )}{3 \sqrt {-b^2 x^4-1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ \frac {b x {\rm integral}\left (-\frac {\sqrt {-b^{2} x^{4} - 1} {\left (b x^{2} - 1\right )}}{b^{3} x^{6} + b x^{2}}, x\right ) + \sqrt {-b^{2} x^{4} - 1}}{b x} \]
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 \[ \int -\frac {b x^{2} - 1}{\sqrt {-b^{2} x^{4} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 122, normalized size = 1.36 \[ \frac {\sqrt {i b \,x^{2}+1}\, \sqrt {-i b \,x^{2}+1}\, \EllipticF \left (\sqrt {-i b}\, x , i\right )}{\sqrt {-i b}\, \sqrt {-b^{2} x^{4}-1}}+\frac {i \sqrt {i b \,x^{2}+1}\, \sqrt {-i b \,x^{2}+1}\, \left (-\EllipticE \left (\sqrt {-i b}\, x , i\right )+\EllipticF \left (\sqrt {-i b}\, x , i\right )\right )}{\sqrt {-i b}\, \sqrt {-b^{2} x^{4}-1}} \]
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 \[ -\int \frac {b x^{2} - 1}{\sqrt {-b^{2} x^{4} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {b\,x^2-1}{\sqrt {-b^2\,x^4-1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.10, size = 70, normalized size = 0.78 \[ \frac {i b x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} - \frac {i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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