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 = 0.65, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 13, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.302, Rules used = {6742, 277, 216, 266, 50, 63, 206, 733, 844, 725, 735, 264, 731} \[ \frac {\sqrt {1-x^2}}{5 x+4}+\frac {3}{5 (5 x+4)} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 216
Rule 264
Rule 266
Rule 277
Rule 725
Rule 731
Rule 733
Rule 735
Rule 844
Rule 6742
Rubi steps
\begin {align*} \int \frac {-1+\sqrt {1-x^2}}{\sqrt {1-x^2} \left (2+x-2 \sqrt {1-x^2}\right )^2} \, dx &=\int \left (\frac {1}{\left (-2-x+2 \sqrt {1-x^2}\right )^2}-\frac {1}{\sqrt {1-x^2} \left (-2-x+2 \sqrt {1-x^2}\right )^2}\right ) \, dx\\ &=\int \frac {1}{\left (-2-x+2 \sqrt {1-x^2}\right )^2} \, dx-\int \frac {1}{\sqrt {1-x^2} \left (-2-x+2 \sqrt {1-x^2}\right )^2} \, dx\\ &=-\int \left (\frac {1}{2 x^2}-\frac {1}{x}+\frac {15}{2 (4+5 x)^2}+\frac {5}{4+5 x}+\frac {1}{2 x^2 \sqrt {1-x^2}}-\frac {1}{x \sqrt {1-x^2}}+\frac {9}{2 (4+5 x)^2 \sqrt {1-x^2}}+\frac {5}{(4+5 x) \sqrt {1-x^2}}\right ) \, dx+\int \left (\frac {1}{2 x^2}-\frac {1}{x}+\frac {9}{2 (4+5 x)^2}+\frac {5}{4+5 x}+\frac {\sqrt {1-x^2}}{2 x^2}-\frac {\sqrt {1-x^2}}{x}+\frac {15 \sqrt {1-x^2}}{2 (4+5 x)^2}+\frac {5 \sqrt {1-x^2}}{4+5 x}\right ) \, dx\\ &=\frac {3}{5 (4+5 x)}-\frac {1}{2} \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx+\frac {1}{2} \int \frac {\sqrt {1-x^2}}{x^2} \, dx-\frac {9}{2} \int \frac {1}{(4+5 x)^2 \sqrt {1-x^2}} \, dx-5 \int \frac {1}{(4+5 x) \sqrt {1-x^2}} \, dx+5 \int \frac {\sqrt {1-x^2}}{4+5 x} \, dx+\frac {15}{2} \int \frac {\sqrt {1-x^2}}{(4+5 x)^2} \, dx+\int \frac {1}{x \sqrt {1-x^2}} \, dx-\int \frac {\sqrt {1-x^2}}{x} \, dx\\ &=\frac {3}{5 (4+5 x)}+\sqrt {1-x^2}+\frac {\sqrt {1-x^2}}{4+5 x}-\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {1-x}}{x} \, dx,x,x^2\right )-\frac {3}{2} \int \frac {x}{(4+5 x) \sqrt {1-x^2}} \, dx+2 \int \frac {1}{(4+5 x) \sqrt {1-x^2}} \, dx+5 \operatorname {Subst}\left (\int \frac {1}{9-x^2} \, dx,x,\frac {5+4 x}{\sqrt {1-x^2}}\right )+\int \frac {5+4 x}{(4+5 x) \sqrt {1-x^2}} \, dx\\ &=\frac {3}{5 (4+5 x)}+\frac {\sqrt {1-x^2}}{4+5 x}-\frac {1}{2} \sin ^{-1}(x)+\frac {5}{3} \tanh ^{-1}\left (\frac {5+4 x}{3 \sqrt {1-x^2}}\right )-\frac {3}{10} \int \frac {1}{\sqrt {1-x^2}} \, dx-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )+\frac {4}{5} \int \frac {1}{\sqrt {1-x^2}} \, dx+\frac {6}{5} \int \frac {1}{(4+5 x) \sqrt {1-x^2}} \, dx+\frac {9}{5} \int \frac {1}{(4+5 x) \sqrt {1-x^2}} \, dx-2 \operatorname {Subst}\left (\int \frac {1}{9-x^2} \, dx,x,\frac {5+4 x}{\sqrt {1-x^2}}\right )-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=\frac {3}{5 (4+5 x)}+\frac {\sqrt {1-x^2}}{4+5 x}+\tanh ^{-1}\left (\frac {5+4 x}{3 \sqrt {1-x^2}}\right )-\tanh ^{-1}\left (\sqrt {1-x^2}\right )-\frac {6}{5} \operatorname {Subst}\left (\int \frac {1}{9-x^2} \, dx,x,\frac {5+4 x}{\sqrt {1-x^2}}\right )-\frac {9}{5} \operatorname {Subst}\left (\int \frac {1}{9-x^2} \, dx,x,\frac {5+4 x}{\sqrt {1-x^2}}\right )+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=\frac {3}{5 (4+5 x)}+\frac {\sqrt {1-x^2}}{4+5 x}\\ \end {align*}
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Mathematica [A] time = 0.18, 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 = 0.41, 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 = 0.62, 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 [A] time = 0.01, size = 32, normalized size = 1.03 \[ \frac {\sqrt {\frac {8 x}{5}-\left (x +\frac {4}{5}\right )^{2}+\frac {41}{25}}}{5 x +4}+\frac {3}{5 \left (5 x +4\right )} \]
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 \[ -\frac {1}{56} \, \sqrt {7} \log \left (\frac {3 \, x - 2 \, \sqrt {7} - 2}{3 \, x + 2 \, \sqrt {7} - 2}\right ) - \int -\frac {100 \, x^{7} + 285 \, x^{6} + 264 \, x^{5} + 80 \, x^{4}}{8 \, {\left (21 \, x^{9} + 278 \, x^{8} + 283 \, x^{7} - 2022 \, x^{6} - 3632 \, x^{5} + 2256 \, x^{4} + 7424 \, x^{3} + 1536 \, x^{2} - 8 \, {\left (9 \, x^{8} + 12 \, x^{7} - 101 \, x^{6} - 172 \, x^{5} + 284 \, x^{4} + 672 \, x^{3} + 64 \, x^{2} - 512 \, x - 256\right )} \sqrt {x + 1} \sqrt {-x + 1} - 4096 \, x - 2048\right )}}\,{d x} - \frac {1}{24} \, \log \left (x + 2\right ) + \frac {1}{16} \, \log \left (x + 1\right ) - \frac {1}{48} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.32, 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 \[ \int \frac {\sqrt {1 - x^{2}} - 1}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \left (x - 2 \sqrt {1 - x^{2}} + 2\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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