Optimal. Leaf size=61 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{2 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{2 \sqrt {6}} \]
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Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {407}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{2 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{2 \sqrt {6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 407
Rubi steps
\begin {align*} \int \frac {1}{\left (-2-3 x^2\right ) \sqrt [4]{-1-3 x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{2 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{2 \sqrt {6}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 56, normalized size = 0.92 \begin {gather*} -\frac {-\tan ^{-1}\left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{-1-3 x^2}}{x}\right )+\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{2 \sqrt {6}} \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 = 0.96, size = 137, normalized size = 2.25
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \left (-3 x^{2}-1\right )^{\frac {3}{4}}+3 \sqrt {-3 x^{2}-1}\, x -\RootOf \left (\textit {\_Z}^{2}-6\right ) \left (-3 x^{2}-1\right )^{\frac {1}{4}}-3 x}{3 x^{2}+2}\right )}{12}-\frac {\RootOf \left (\textit {\_Z}^{2}+6\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+6\right ) \left (-3 x^{2}-1\right )^{\frac {3}{4}}+3 \sqrt {-3 x^{2}-1}\, x +\RootOf \left (\textit {\_Z}^{2}+6\right ) \left (-3 x^{2}-1\right )^{\frac {1}{4}}+3 x}{3 x^{2}+2}\right )}{12}\) | \(137\) |
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 [C] Result contains complex when optimal does not.
time = 3.49, size = 243, normalized size = 3.98 \begin {gather*} -\frac {1}{24} \, \sqrt {6} \log \left (\frac {\sqrt {6} \sqrt {-3 \, x^{2} - 1} x - \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, {\left (3 \, x^{2} + 2\right )}}\right ) + \frac {1}{24} \, \sqrt {6} \log \left (-\frac {\sqrt {6} \sqrt {-3 \, x^{2} - 1} x - \sqrt {6} x - 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, {\left (3 \, x^{2} + 2\right )}}\right ) + \frac {1}{24} i \, \sqrt {6} \log \left (\frac {i \, \sqrt {6} \sqrt {-3 \, x^{2} - 1} x + i \, \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, {\left (3 \, x^{2} + 2\right )}}\right ) - \frac {1}{24} i \, \sqrt {6} \log \left (\frac {-i \, \sqrt {6} \sqrt {-3 \, x^{2} - 1} x - i \, \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, {\left (3 \, x^{2} + 2\right )}}\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} \sqrt [4]{- 3 x^{2} - 1} + 2 \sqrt [4]{- 3 x^{2} - 1}}\, 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 = -0.02 \begin {gather*} -\int \frac {1}{{\left (-3\,x^2-1\right )}^{1/4}\,\left (3\,x^2+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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