Optimal. Leaf size=78 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [3]{5}-2 x}{\sqrt {3} \sqrt [3]{5}}\right )}{\sqrt {3} 5^{2/3}}+\frac {\log \left (\sqrt [3]{5}+x\right )}{3\ 5^{2/3}}-\frac {\log \left (5^{2/3}-\sqrt [3]{5} x+x^2\right )}{6\ 5^{2/3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {206, 31, 648,
631, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{5}-2 x}{\sqrt {3} \sqrt [3]{5}}\right )}{\sqrt {3} 5^{2/3}}-\frac {\log \left (x^2-\sqrt [3]{5} x+5^{2/3}\right )}{6\ 5^{2/3}}+\frac {\log \left (x+\sqrt [3]{5}\right )}{3\ 5^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{5+x^3} \, dx &=\frac {\int \frac {1}{\sqrt [3]{5}+x} \, dx}{3\ 5^{2/3}}+\frac {\int \frac {2 \sqrt [3]{5}-x}{5^{2/3}-\sqrt [3]{5} x+x^2} \, dx}{3\ 5^{2/3}}\\ &=\frac {\log \left (\sqrt [3]{5}+x\right )}{3\ 5^{2/3}}-\frac {\int \frac {-\sqrt [3]{5}+2 x}{5^{2/3}-\sqrt [3]{5} x+x^2} \, dx}{6\ 5^{2/3}}+\frac {\int \frac {1}{5^{2/3}-\sqrt [3]{5} x+x^2} \, dx}{2 \sqrt [3]{5}}\\ &=\frac {\log \left (\sqrt [3]{5}+x\right )}{3\ 5^{2/3}}-\frac {\log \left (5^{2/3}-\sqrt [3]{5} x+x^2\right )}{6\ 5^{2/3}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{5}}\right )}{5^{2/3}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{5}-2 x}{\sqrt {3} \sqrt [3]{5}}\right )}{\sqrt {3} 5^{2/3}}+\frac {\log \left (\sqrt [3]{5}+x\right )}{3\ 5^{2/3}}-\frac {\log \left (5^{2/3}-\sqrt [3]{5} x+x^2\right )}{6\ 5^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 71, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {3} \tan ^{-1}\left (\frac {-5+2\ 5^{2/3} x}{5 \sqrt {3}}\right )+2 \log \left (5+5^{2/3} x\right )-\log \left (5-5^{2/3} x+\sqrt [3]{5} x^2\right )}{6\ 5^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 54, normalized size = 0.69
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+5\right )}{\sum }\frac {\ln \left (-\textit {\_R} +x \right )}{\textit {\_R}^{2}}\right )}{3}\) | \(22\) |
default | \(\frac {\ln \left (5^{\frac {1}{3}}+x \right ) 5^{\frac {1}{3}}}{15}-\frac {\ln \left (5^{\frac {2}{3}}-5^{\frac {1}{3}} x +x^{2}\right ) 5^{\frac {1}{3}}}{30}+\frac {5^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,5^{\frac {2}{3}} x}{5}-1\right )}{3}\right )}{15}\) | \(54\) |
meijerg | \(\frac {5^{\frac {1}{3}} \left (\frac {x \ln \left (1+\frac {5^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{5}\right )}{\left (x^{3}\right )^{\frac {1}{3}}}-\frac {x \ln \left (1-\frac {5^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{5}+\frac {5^{\frac {1}{3}} \left (x^{3}\right )^{\frac {2}{3}}}{5}\right )}{2 \left (x^{3}\right )^{\frac {1}{3}}}+\frac {x \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, 5^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{10-5^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}\right )}{\left (x^{3}\right )^{\frac {1}{3}}}\right )}{15}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.61, size = 57, normalized size = 0.73 \begin {gather*} \frac {1}{15} \cdot 5^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {1}{15} \cdot 5^{\frac {2}{3}} \sqrt {3} {\left (2 \, x - 5^{\frac {1}{3}}\right )}\right ) - \frac {1}{30} \cdot 5^{\frac {1}{3}} \log \left (x^{2} - 5^{\frac {1}{3}} x + 5^{\frac {2}{3}}\right ) + \frac {1}{15} \cdot 5^{\frac {1}{3}} \log \left (x + 5^{\frac {1}{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.81, size = 69, normalized size = 0.88 \begin {gather*} \frac {1}{15} \cdot 25^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {1}{75} \cdot 25^{\frac {1}{6}} {\left (2 \cdot 25^{\frac {2}{3}} \sqrt {3} x - 5 \cdot 25^{\frac {1}{3}} \sqrt {3}\right )}\right ) - \frac {1}{150} \cdot 25^{\frac {2}{3}} \log \left (5 \, x^{2} - 25^{\frac {2}{3}} x + 5 \cdot 25^{\frac {1}{3}}\right ) + \frac {1}{75} \cdot 25^{\frac {2}{3}} \log \left (5 \, x + 25^{\frac {2}{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 73, normalized size = 0.94 \begin {gather*} \frac {\sqrt [3]{5} \log {\left (x + \sqrt [3]{5} \right )}}{15} - \frac {\sqrt [3]{5} \log {\left (x^{2} - \sqrt [3]{5} x + 5^{\frac {2}{3}} \right )}}{30} + \frac {\sqrt {3} \cdot \sqrt [3]{5} \operatorname {atan}{\left (\frac {2 \sqrt {3} \cdot 5^{\frac {2}{3}} x}{15} - \frac {\sqrt {3}}{3} \right )}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 58, normalized size = 0.74 \begin {gather*} \frac {1}{15} \cdot 5^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {1}{15} \cdot 5^{\frac {2}{3}} \sqrt {3} {\left (2 \, x - 5^{\frac {1}{3}}\right )}\right ) - \frac {1}{30} \cdot 5^{\frac {1}{3}} \log \left (x^{2} - 5^{\frac {1}{3}} x + 5^{\frac {2}{3}}\right ) + \frac {1}{15} \cdot 5^{\frac {1}{3}} \log \left ({\left | x + 5^{\frac {1}{3}} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 70, normalized size = 0.90 \begin {gather*} \frac {5^{1/3}\,\ln \left (x+5^{1/3}\right )}{15}+\frac {5^{1/3}\,\ln \left (x+\frac {5^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{30}-\frac {5^{1/3}\,\ln \left (x-\frac {5^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{30} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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