Optimal. Leaf size=96 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} (1+x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2^{2/3}}-\frac {\log \left (3+x^2\right )}{2\ 2^{2/3}}+\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{1-x}+(1+x)^{2/3}\right )}{2\ 2^{2/3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1022}
\begin {gather*} -\frac {\sqrt {3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} (x+1)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2^{2/3}}-\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}+\frac {3 \log \left ((x+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{1-x}\right )}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1022
Rubi steps
\begin {align*} \int \frac {3-x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} (1+x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2^{2/3}}-\frac {\log \left (3+x^2\right )}{2\ 2^{2/3}}+\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{1-x}+(1+x)^{2/3}\right )}{2\ 2^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 155, normalized size = 1.61 \begin {gather*} \frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1-x^2}}{-2^{2/3}-2^{2/3} x+\sqrt [3]{1-x^2}}\right )+2 \log \left (2^{2/3}+2^{2/3} x+2 \sqrt [3]{1-x^2}\right )-\log \left (-\sqrt [3]{2}-2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+2^{2/3} (1+x) \sqrt [3]{1-x^2}-2 \left (1-x^2\right )^{2/3}\right )}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 8.69, size = 1033, normalized size = 10.76
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1033\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 285 vs.
\(2 (70) = 140\).
time = 10.19, size = 285, normalized size = 2.97 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (x^{4} + 3 \, x^{3} + 3 \, x^{2} + 9 \, x\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{6} - 18 \, x^{5} - 117 \, x^{4} - 36 \, x^{3} + 207 \, x^{2} + 54 \, x - 27\right )} + 12 \, {\left (x^{5} + 19 \, x^{4} + 42 \, x^{3} + 6 \, x^{2} - 27 \, x - 9\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{6} + 54 \, x^{5} + 171 \, x^{4} + 108 \, x^{3} - 81 \, x^{2} - 162 \, x - 27\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{2} + 3 \, x\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{4} + 18 \, x^{3} + 24 \, x^{2} - 18 \, x - 9\right )} - 6 \, {\left (x^{3} + 7 \, x^{2} + 3 \, x - 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}}{x^{4} + 6 \, x^{2} + 9}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (x^{2} + 3\right )} + 6 \cdot 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + 12 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{x^{2} + 3}\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 {x}{x^{2} \sqrt [3]{1 - x^{2}} + 3 \sqrt [3]{1 - x^{2}}}\, dx - \int \left (- \frac {3}{x^{2} \sqrt [3]{1 - x^{2}} + 3 \sqrt [3]{1 - x^{2}}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x-3}{{\left (1-x^2\right )}^{1/3}\,\left (x^2+3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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