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 = 0.13, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 206
Rule 216
Rule 665
Rule 725
Rule 733
Rule 735
Rule 844
Rule 6742
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*}
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Mathematica [A] time = 0.06, size = 23, normalized size = 0.74 \[ \frac {5 \sqrt {1-x^2}+3}{25 x+20} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 25, normalized size = 0.81 \[ \frac {25 \, x + 20 \, \sqrt {-x^{2} + 1} + 32}{20 \, {\left (5 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.37, size = 68, normalized size = 2.19 \[ \frac {\frac {5 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} - 4}{4 \, {\left (\frac {5 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} - \frac {2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 2\right )}} + \frac {3}{5 \, {\left (5 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 81, normalized size = 2.61 \[ \frac {5 \sqrt {\frac {8 x}{5}-\left (x +\frac {4}{5}\right )^{2}+\frac {41}{25}}\, x}{9}+\frac {3}{5 \left (5 x +4\right )}+\frac {\sqrt {-2 x -\left (x -1\right )^{2}+2}}{18}-\frac {\sqrt {2 x -\left (x +1\right )^{2}+2}}{2}+\frac {5 \left (\frac {8 x}{5}-\left (x +\frac {4}{5}\right )^{2}+\frac {41}{25}\right )^{\frac {3}{2}}}{9 \left (x +\frac {4}{5}\right )} \]
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 \[ -\int \frac {1}{3 \, x^{2} + 4 \, \sqrt {-x^{2} + 1} x + 5 \, \sqrt {-x^{2} + 1} - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 19, normalized size = 0.61 \[ \frac {\sqrt {1-x^2}+\frac {3}{5}}{5\,x+4} \]
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 \[ - \int \frac {1}{3 x^{2} + 4 x \sqrt {1 - x^{2}} + 5 \sqrt {1 - x^{2}} - 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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