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 = 0.09, 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 = {6874, 679,
222, 747, 858, 739, 212, 749} \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 212
Rule 222
Rule 679
Rule 739
Rule 747
Rule 749
Rule 858
Rule 6874
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 = 0.28, 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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs.
\(2(27)=54\).
time = 0.04, size = 81, normalized size = 2.61
method | result | size |
trager | \(-\frac {3 x}{4 \left (5 x +4\right )}+\frac {\sqrt {-x^{2}+1}}{5 x +4}\) | \(29\) |
default | \(\frac {3}{5 \left (5 x +4\right )}+\frac {5 \left (-\left (x +\frac {4}{5}\right )^{2}+\frac {8 x}{5}+\frac {41}{25}\right )^{\frac {3}{2}}}{9 \left (x +\frac {4}{5}\right )}+\frac {5 x \sqrt {-\left (x +\frac {4}{5}\right )^{2}+\frac {8 x}{5}+\frac {41}{25}}}{9}-\frac {\sqrt {-\left (1+x \right )^{2}+2+2 x}}{2}+\frac {\sqrt {-\left (-1+x \right )^{2}+2-2 x}}{18}\) | \(81\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{3 x^{2} + 4 x \sqrt {1 - x^{2}} + 5 \sqrt {1 - x^{2}} - 3}\, dx \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 = 3.04, 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 = 0.05, 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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