Optimal. Leaf size=67 \[ 6 \sqrt [3]{1+\sqrt {x}}-2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+\sqrt {x}}}{\sqrt {3}}\right )+3 \log \left (1-\sqrt [3]{1+\sqrt {x}}\right )-\frac {\log (x)}{2} \]
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Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 52, 59,
632, 210, 31} \begin {gather*} -2 \sqrt {3} \text {ArcTan}\left (\frac {2 \sqrt [3]{\sqrt {x}+1}+1}{\sqrt {3}}\right )+6 \sqrt [3]{\sqrt {x}+1}+3 \log \left (1-\sqrt [3]{\sqrt {x}+1}\right )-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 52
Rule 59
Rule 210
Rule 272
Rule 632
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{1+\sqrt {x}}}{x} \, dx &=2 \text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x} \, dx,x,\sqrt {x}\right )\\ &=6 \sqrt [3]{1+\sqrt {x}}+2 \text {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,\sqrt {x}\right )\\ &=6 \sqrt [3]{1+\sqrt {x}}-\frac {\log (x)}{2}-3 \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+\sqrt {x}}\right )-3 \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+\sqrt {x}}\right )\\ &=6 \sqrt [3]{1+\sqrt {x}}+3 \log \left (1-\sqrt [3]{1+\sqrt {x}}\right )-\frac {\log (x)}{2}+6 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+\sqrt {x}}\right )\\ &=6 \sqrt [3]{1+\sqrt {x}}-2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+\sqrt {x}}}{\sqrt {3}}\right )+3 \log \left (1-\sqrt [3]{1+\sqrt {x}}\right )-\frac {\log (x)}{2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 86, normalized size = 1.28 \begin {gather*} 6 \sqrt [3]{1+\sqrt {x}}-2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+\sqrt {x}}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1+\sqrt {x}}\right )-\log \left (1+\sqrt [3]{1+\sqrt {x}}+\left (1+\sqrt {x}\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 64, normalized size = 0.96
method | result | size |
meijerg | \(-\frac {2 \left (-\Gamma \left (\frac {2}{3}\right ) \sqrt {x}\, \hypergeom \left (\left [\frac {2}{3}, 1, 1\right ], \left [2, 2\right ], -\sqrt {x}\right )-3 \left (3+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+\frac {\ln \left (x \right )}{2}\right ) \Gamma \left (\frac {2}{3}\right )\right )}{3 \Gamma \left (\frac {2}{3}\right )}\) | \(48\) |
derivativedivides | \(6 \left (1+\sqrt {x}\right )^{\frac {1}{3}}+2 \ln \left (\left (1+\sqrt {x}\right )^{\frac {1}{3}}-1\right )-\ln \left (\left (1+\sqrt {x}\right )^{\frac {2}{3}}+\left (1+\sqrt {x}\right )^{\frac {1}{3}}+1\right )-2 \arctan \left (\frac {\left (1+2 \left (1+\sqrt {x}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}\) | \(64\) |
default | \(6 \left (1+\sqrt {x}\right )^{\frac {1}{3}}+2 \ln \left (\left (1+\sqrt {x}\right )^{\frac {1}{3}}-1\right )-\ln \left (\left (1+\sqrt {x}\right )^{\frac {2}{3}}+\left (1+\sqrt {x}\right )^{\frac {1}{3}}+1\right )-2 \arctan \left (\frac {\left (1+2 \left (1+\sqrt {x}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 63, normalized size = 0.94 \begin {gather*} -2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + 6 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {2}{3}} + {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 65, normalized size = 0.97 \begin {gather*} -2 \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 6 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {2}{3}} + {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.60, size = 39, normalized size = 0.58 \begin {gather*} - \frac {2 \sqrt [6]{x} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{\sqrt {x}}} \right )}}{\Gamma \left (\frac {2}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.74, size = 64, normalized size = 0.96 \begin {gather*} -2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + 6 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {2}{3}} + {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, \log \left ({\left | {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.83, size = 73, normalized size = 1.09 \begin {gather*} 2\,\ln \left ({\left (\sqrt {x}+1\right )}^{1/3}-1\right )+6\,{\left (\sqrt {x}+1\right )}^{1/3}+\ln \left ({\left (\sqrt {x}+1\right )}^{1/3}+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )-\ln \left ({\left (\sqrt {x}+1\right )}^{1/3}+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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