Optimal. Leaf size=127 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {1-x^3}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {1+\sqrt [3]{2} x}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt {1-x^3}\right )}{9\ 2^{2/3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {497}
\begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {1-x^3}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x+1}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt {1-x^3}\right )}{9\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 497
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1-x^3} \left (4-x^3\right )} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {1-x^3}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {1+\sqrt [3]{2} x}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt {1-x^3}\right )}{9\ 2^{2/3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 10.03, size = 28, normalized size = 0.22 \begin {gather*} \frac {1}{8} x^2 F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};x^3,\frac {x^3}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 5.86, size = 164, normalized size = 1.29
method | result | size |
default | \(\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-4\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {2}\, \sqrt {i \left (-i \sqrt {3}+2 x +1\right )}\, \sqrt {\frac {-1+x}{i \sqrt {3}-3}}\, \sqrt {-\frac {i \left (i \sqrt {3}+2 x +1\right )}{2}}\, \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1+i \sqrt {3}\, \left (1-\underline {\hspace {1.25 ex}}\alpha \right )\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {\underline {\hspace {1.25 ex}}\alpha }{2}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}-\frac {1}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-x^{3}+1}}\right )}{36}\) | \(164\) |
elliptic | \(\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-4\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {2}\, \sqrt {i \left (-i \sqrt {3}+2 x +1\right )}\, \sqrt {\frac {-1+x}{i \sqrt {3}-3}}\, \sqrt {-\frac {i \left (i \sqrt {3}+2 x +1\right )}{2}}\, \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1+i \sqrt {3}\, \left (1-\underline {\hspace {1.25 ex}}\alpha \right )\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {\underline {\hspace {1.25 ex}}\alpha }{2}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}-\frac {1}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-x^{3}+1}}\right )}{36}\) | \(164\) |
trager | \(\text {Expression too large to display}\) | \(815\) |
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. 1191 vs.
\(2 (92) = 184\).
time = 0.45, size = 1191, normalized size = 9.38
result too large to display
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^{3} \sqrt {1 - x^{3}} - 4 \sqrt {1 - x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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 [B]
time = 0.45, size = 653, normalized size = 5.14
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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