Optimal. Leaf size=31 \[ \frac {3}{5 (4+5 x)}+\frac {\sqrt {1-x^2}}{4+5 x} \]
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Rubi [A]
time = 0.45, 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 = {6874, 283,
222, 272, 52, 65, 212, 747, 858, 739, 749, 270, 745} \begin {gather*} \frac {\sqrt {1-x^2}}{5 x+4}+\frac {3}{5 (5 x+4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 222
Rule 270
Rule 272
Rule 283
Rule 739
Rule 745
Rule 747
Rule 749
Rule 858
Rule 6874
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} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )-\frac {1}{2} \text {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 \text {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} \text {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 \text {Subst}\left (\int \frac {1}{9-x^2} \, dx,x,\frac {5+4 x}{\sqrt {1-x^2}}\right )-\text {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} \text {Subst}\left (\int \frac {1}{9-x^2} \, dx,x,\frac {5+4 x}{\sqrt {1-x^2}}\right )-\frac {9}{5} \text {Subst}\left (\int \frac {1}{9-x^2} \, dx,x,\frac {5+4 x}{\sqrt {1-x^2}}\right )+\text {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 = 0.28, size = 23, normalized size = 0.74 \begin {gather*} \frac {3+5 \sqrt {1-x^2}}{20+25 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 32, normalized size = 1.03
method | result | size |
trager | \(-\frac {3 x}{4 \left (5 x +4\right )}+\frac {\sqrt {-x^{2}+1}}{5 x +4}\) | \(29\) |
default | \(\frac {\sqrt {-\left (x +\frac {4}{5}\right )^{2}+\frac {8 x}{5}+\frac {41}{25}}}{5 x +4}+\frac {3}{5 \left (5 x +4\right )}\) | \(32\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 25, normalized size = 0.81 \begin {gather*} \frac {25 \, x + 20 \, \sqrt {-x^{2} + 1} + 32}{20 \, {\left (5 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (27) = 54\).
time = 5.72, size = 68, normalized size = 2.19 \begin {gather*} \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 {gather*}
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 \begin {gather*} \frac {\sqrt {1-x^2}+\frac {3}{5}}{5\,x+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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