Optimal. Leaf size=45 \[ \frac {1}{8} \sqrt {4 x^4-3 x^2}+\frac {3}{16} \tanh ^{-1}\left (\frac {2 x^2}{\sqrt {4 x^4-3 x^2}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2018, 640, 620, 206} \begin {gather*} \frac {1}{8} \sqrt {4 x^4-3 x^2}+\frac {3}{16} \tanh ^{-1}\left (\frac {2 x^2}{\sqrt {4 x^4-3 x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 2018
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {-3 x^2+4 x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {-3 x+4 x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{8} \sqrt {-3 x^2+4 x^4}+\frac {3}{16} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-3 x+4 x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{8} \sqrt {-3 x^2+4 x^4}+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\frac {x^2}{\sqrt {-3 x^2+4 x^4}}\right )\\ &=\frac {1}{8} \sqrt {-3 x^2+4 x^4}+\frac {3}{16} \tanh ^{-1}\left (\frac {2 x^2}{\sqrt {-3 x^2+4 x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 57, normalized size = 1.27 \begin {gather*} \frac {x \left (8 x^3+3 \sqrt {4 x^2-3} \tanh ^{-1}\left (\frac {2 x}{\sqrt {4 x^2-3}}\right )-6 x\right )}{16 \sqrt {x^2 \left (4 x^2-3\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 51, normalized size = 1.13 \begin {gather*} \frac {1}{8} \sqrt {4 x^4-3 x^2}+\frac {3}{16} \tanh ^{-1}\left (\frac {2 \sqrt {4 x^4-3 x^2}}{4 x^2-3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 45, normalized size = 1.00 \begin {gather*} \frac {1}{8} \, \sqrt {4 \, x^{4} - 3 \, x^{2}} - \frac {3}{16} \, \log \left (-\frac {2 \, x^{2} - \sqrt {4 \, x^{4} - 3 \, x^{2}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 42, normalized size = 0.93 \begin {gather*} \frac {1}{8} \, \sqrt {4 \, x^{4} - 3 \, x^{2}} - \frac {3}{32} \, \log \left ({\left | -8 \, x^{2} + 4 \, \sqrt {4 \, x^{4} - 3 \, x^{2}} + 3 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 1.33 \begin {gather*} \frac {\sqrt {4 x^{2}-3}\, \left (4 \sqrt {4 x^{2}-3}\, x +3 \sqrt {4}\, \ln \left (\sqrt {4}\, x +\sqrt {4 x^{2}-3}\right )\right ) x}{32 \sqrt {4 x^{4}-3 x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 41, normalized size = 0.91 \begin {gather*} \frac {1}{8} \, \sqrt {4 \, x^{4} - 3 \, x^{2}} + \frac {3}{32} \, \log \left (8 \, x^{2} + 4 \, \sqrt {4 \, x^{4} - 3 \, x^{2}} - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.46, size = 40, normalized size = 0.89 \begin {gather*} \frac {3\,\ln \left (\frac {\sqrt {4\,x^2-3}\,\sqrt {x^2}}{2}+x^2-\frac {3}{8}\right )}{32}+\frac {\sqrt {4\,x^4-3\,x^2}}{8} \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^{3}}{\sqrt {x^{2} \left (4 x^{2} - 3\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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