Optimal. Leaf size=28 \[ \frac {2 \sqrt {x^3+1}}{3}-\frac {2}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 50, 63, 207} \begin {gather*} \frac {2 \sqrt {x^3+1}}{3}-\frac {2}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^3}}{x} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,x^3\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}-\frac {2}{3} \tanh ^{-1}\left (\sqrt {1+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x^3+1}}{3}-\frac {2}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 28, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {1+x^3}}{3}-\frac {2}{3} \tanh ^{-1}\left (\sqrt {1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 34, normalized size = 1.21 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} + 1} - \frac {1}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 35, normalized size = 1.25 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} + 1} - \frac {1}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {1}{3} \, \log \left ({\left | \sqrt {x^{3} + 1} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 21, normalized size = 0.75
method | result | size |
default | \(\frac {2 \sqrt {x^{3}+1}}{3}-\frac {2 \arctanh \left (\sqrt {x^{3}+1}\right )}{3}\) | \(21\) |
elliptic | \(\frac {2 \sqrt {x^{3}+1}}{3}-\frac {2 \arctanh \left (\sqrt {x^{3}+1}\right )}{3}\) | \(21\) |
trager | \(\frac {2 \sqrt {x^{3}+1}}{3}+\frac {\ln \left (-\frac {-x^{3}+2 \sqrt {x^{3}+1}-2}{x^{3}}\right )}{3}\) | \(35\) |
meijerg | \(-\frac {-2 \left (2-2 \ln \relax (2)+3 \ln \relax (x )\right ) \sqrt {\pi }+4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {x^{3}+1}+4 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{6 \sqrt {\pi }}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 34, normalized size = 1.21 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} + 1} - \frac {1}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 174, normalized size = 6.21 \begin {gather*} \frac {2\,\sqrt {x^3+1}}{3}-\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.95, size = 48, normalized size = 1.71 \begin {gather*} \frac {2 x^{\frac {3}{2}}}{3 \sqrt {1 + \frac {1}{x^{3}}}} - \frac {2 \operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} + \frac {2}{3 x^{\frac {3}{2}} \sqrt {1 + \frac {1}{x^{3}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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