Optimal. Leaf size=67 \[ 6 \sqrt [3]{\sqrt {x}+1}+3 \log \left (1-\sqrt [3]{\sqrt {x}+1}\right )-\frac {\log (x)}{2}-2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{\sqrt {x}+1}+1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.03, 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 = {266, 50, 57, 618, 204, 31} \[ 6 \sqrt [3]{\sqrt {x}+1}+3 \log \left (1-\sqrt [3]{\sqrt {x}+1}\right )-\frac {\log (x)}{2}-2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{\sqrt {x}+1}+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 57
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{1+\sqrt {x}}}{x} \, dx &=2 \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x} \, dx,x,\sqrt {x}\right )\\ &=6 \sqrt [3]{1+\sqrt {x}}+2 \operatorname {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 \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+\sqrt {x}}\right )-3 \operatorname {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 \operatorname {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 = 0.02, size = 88, normalized size = 1.31 \[ 6 \sqrt [3]{\sqrt {x}+1}+2 \log \left (1-\sqrt [3]{\sqrt {x}+1}\right )-\log \left (\left (\sqrt {x}+1\right )^{2/3}+\sqrt [3]{\sqrt {x}+1}+1\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{\sqrt {x}+1}+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 65, normalized size = 0.97 \[ -2 \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 6 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {2}{3}} + {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 64, normalized size = 0.96 \[ -2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + 6 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {2}{3}} + {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, \log \left ({\left | {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 64, normalized size = 0.96 \[ -2 \sqrt {3}\, \arctan \left (\frac {\left (1+2 \left (\sqrt {x}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right )+2 \ln \left (\left (\sqrt {x}+1\right )^{\frac {1}{3}}-1\right )-\ln \left (\left (\sqrt {x}+1\right )^{\frac {2}{3}}+\left (\sqrt {x}+1\right )^{\frac {1}{3}}+1\right )+6 \left (\sqrt {x}+1\right )^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.72, size = 63, normalized size = 0.94 \[ -2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + 6 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {2}{3}} + {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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 \[ 2\,\ln \left ({\left (\sqrt {x}+1\right )}^{1/3}-1\right )+6\,{\left (\sqrt {x}+1\right )}^{1/3}+\ln \left ({\left (\sqrt {x}+1\right )}^{1/3}+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )-\ln \left ({\left (\sqrt {x}+1\right )}^{1/3}+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.05, size = 39, normalized size = 0.58 \[ - \frac {2 \sqrt [6]{x} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{\sqrt {x}}} \right )}}{\Gamma \left (\frac {2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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