Optimal. Leaf size=31 \[ \frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)} \]
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Rubi [A] time = 0.125123, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6742, 665, 216, 733, 844, 725, 206, 735} \[ \frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 665
Rule 216
Rule 733
Rule 844
Rule 725
Rule 206
Rule 735
Rubi steps
\begin{align*} \int \frac{1}{3-3 x^2-5 \sqrt{1-x^2}-4 x \sqrt{1-x^2}} \, dx &=\int \left (-\frac{3}{(4+5 x)^2}+\frac{\sqrt{1-x^2}}{18 (-1+x)}-\frac{\sqrt{1-x^2}}{2 (1+x)}-\frac{5 \sqrt{1-x^2}}{(4+5 x)^2}+\frac{20 \sqrt{1-x^2}}{9 (4+5 x)}\right ) \, dx\\ &=\frac{3}{5 (4+5 x)}+\frac{1}{18} \int \frac{\sqrt{1-x^2}}{-1+x} \, dx-\frac{1}{2} \int \frac{\sqrt{1-x^2}}{1+x} \, dx+\frac{20}{9} \int \frac{\sqrt{1-x^2}}{4+5 x} \, dx-5 \int \frac{\sqrt{1-x^2}}{(4+5 x)^2} \, dx\\ &=\frac{3}{5 (4+5 x)}+\frac{\sqrt{1-x^2}}{4+5 x}-\frac{1}{18} \int \frac{1}{\sqrt{1-x^2}} \, dx+\frac{4}{9} \int \frac{5+4 x}{(4+5 x) \sqrt{1-x^2}} \, dx-\frac{1}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx+\int \frac{x}{(4+5 x) \sqrt{1-x^2}} \, dx\\ &=\frac{3}{5 (4+5 x)}+\frac{\sqrt{1-x^2}}{4+5 x}-\frac{5}{9} \sin ^{-1}(x)+\frac{1}{5} \int \frac{1}{\sqrt{1-x^2}} \, dx+\frac{16}{45} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{3}{5 (4+5 x)}+\frac{\sqrt{1-x^2}}{4+5 x}\\ \end{align*}
Mathematica [A] time = 0.0581363, size = 23, normalized size = 0.74 \[ \frac{5 \sqrt{1-x^2}+3}{25 x+20} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 81, normalized size = 2.6 \begin{align*}{\frac{3}{20+25\,x}}+{\frac{1}{18}\sqrt{- \left ( x-1 \right ) ^{2}-2\,x+2}}+{\frac{5}{9} \left ( - \left ( x+{\frac{4}{5}} \right ) ^{2}+{\frac{8\,x}{5}}+{\frac{41}{25}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{4}{5}} \right ) ^{-1}}+{\frac{5\,x}{9}\sqrt{- \left ( x+{\frac{4}{5}} \right ) ^{2}+{\frac{8\,x}{5}}+{\frac{41}{25}}}}-{\frac{1}{2}\sqrt{- \left ( 1+x \right ) ^{2}+2\,x+2}} \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{1}{3 \, x^{2} + 4 \, \sqrt{-x^{2} + 1} x + 5 \, \sqrt{-x^{2} + 1} - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61212, size = 65, normalized size = 2.1 \begin{align*} \frac{25 \, x + 20 \, \sqrt{-x^{2} + 1} + 32}{20 \,{\left (5 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{3 x^{2} + 4 x \sqrt{1 - x^{2}} + 5 \sqrt{1 - x^{2}} - 3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12394, size = 92, normalized size = 2.97 \begin{align*} \frac{\frac{5 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - 4}{4 \,{\left (\frac{5 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - \frac{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 2\right )}} + \frac{3}{5 \,{\left (5 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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