Optimal. Leaf size=66 \[ \frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 x}{\sqrt {3} \sqrt [3]{x \left (-q+x^2\right )}}\right )+\frac {\log (x)}{4}-\frac {3}{4} \log \left (-x+\sqrt [3]{x \left (-q+x^2\right )}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 117, normalized size of antiderivative = 1.77, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2004, 2036,
335, 281, 245} \begin {gather*} \frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{x^2-q} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-q}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{x^3-q x}}-\frac {3 \sqrt [3]{x} \sqrt [3]{x^2-q} \log \left (x^{2/3}-\sqrt [3]{x^2-q}\right )}{4 \sqrt [3]{x^3-q x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 281
Rule 335
Rule 2004
Rule 2036
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{x \left (-q+x^2\right )}} \, dx &=\int \frac {1}{\sqrt [3]{-q x+x^3}} \, dx\\ &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-q+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-q+x^2}} \, dx}{\sqrt [3]{-q x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-q+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-q+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-q x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-q+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-q+x^3}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-q x+x^3}}\\ &=\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{-q+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-q+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-q x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{-q+x^2} \log \left (x^{2/3}-\sqrt [3]{-q+x^2}\right )}{4 \sqrt [3]{-q x+x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 130, normalized size = 1.97 \begin {gather*} \frac {\sqrt [3]{x} \sqrt [3]{-q+x^2} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-q+x^2}}\right )-2 \log \left (-x^{2/3}+\sqrt [3]{-q+x^2}\right )+\log \left (x^{4/3}+x^{2/3} \sqrt [3]{-q+x^2}+\left (-q+x^2\right )^{2/3}\right )\right )}{4 \sqrt [3]{-q x+x^3}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x \left (x^{2}-q \right )\right )^{\frac {1}{3}}}\, dx\]
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. 415 vs.
\(2 (52) = 104\).
time = 1.00, size = 415, normalized size = 6.29 \begin {gather*} \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (q^{12} - 15 \, q^{10} + 90 \, q^{8} - 351 \, q^{6} + 810 \, q^{4} - 1215 \, q^{2} + 729\right )} {\left (x^{3} - q x\right )}^{\frac {1}{3}} x - 2 \, \sqrt {3} {\left (q^{12} + 6 \, q^{11} - 15 \, q^{10} - 54 \, q^{9} + 90 \, q^{8} + 270 \, q^{7} - 351 \, q^{6} - 810 \, q^{5} + 810 \, q^{4} + 1458 \, q^{3} - 1215 \, q^{2} - 1458 \, q + 729\right )} {\left (x^{3} - q x\right )}^{\frac {2}{3}} - \sqrt {3} {\left (q^{13} + 10 \, q^{12} - 15 \, q^{11} - 282 \, q^{10} + 90 \, q^{9} + 2178 \, q^{8} - 351 \, q^{7} - 6534 \, q^{6} + 810 \, q^{5} + 7614 \, q^{4} - 1215 \, q^{3} - {\left (q^{12} - 6 \, q^{11} - 15 \, q^{10} + 54 \, q^{9} + 90 \, q^{8} - 270 \, q^{7} - 351 \, q^{6} + 810 \, q^{5} + 810 \, q^{4} - 1458 \, q^{3} - 1215 \, q^{2} + 1458 \, q + 729\right )} x^{2} - 2430 \, q^{2} + 729 \, q\right )}}{q^{13} + 18 \, q^{12} + 81 \, q^{11} - 162 \, q^{10} - 1350 \, q^{9} + 810 \, q^{8} + 6561 \, q^{7} - 2430 \, q^{6} - 12150 \, q^{5} + 4374 \, q^{4} + 6561 \, q^{3} - 9 \, {\left (q^{12} + 2 \, q^{11} - 15 \, q^{10} - 18 \, q^{9} + 90 \, q^{8} + 90 \, q^{7} - 351 \, q^{6} - 270 \, q^{5} + 810 \, q^{4} + 486 \, q^{3} - 1215 \, q^{2} - 486 \, q + 729\right )} x^{2} - 4374 \, q^{2} + 729 \, q}\right ) - \frac {1}{4} \, \log \left (-3 \, {\left (x^{3} - q x\right )}^{\frac {1}{3}} x + q + 3 \, {\left (x^{3} - q x\right )}^{\frac {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 {1}{\sqrt [3]{x \left (- q + x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 97, normalized size = 1.47 \begin {gather*} -\frac {3 q \left (-\frac {\ln \left (\left (\left (-q \left (\frac 1{x}\right )^{2}+1\right )^{\frac {1}{3}}\right )^{2}+\left (-q \left (\frac 1{x}\right )^{2}+1\right )^{\frac {1}{3}}+1\right )}{6}+\frac {1}{3} \sqrt {3} \arctan \left (\frac {2 \left (\left (-q \left (\frac 1{x}\right )^{2}+1\right )^{\frac {1}{3}}+\frac 1{2}\right )}{\sqrt {3}}\right )+\frac {\ln \left |\left (-q \left (\frac 1{x}\right )^{2}+1\right )^{\frac {1}{3}}-1\right |}{3}\right )}{2 q} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 37, normalized size = 0.56 \begin {gather*} \frac {3\,x\,{\left (1-\frac {x^2}{q}\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ \frac {x^2}{q}\right )}{2\,{\left (x^3-q\,x\right )}^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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