Optimal. Leaf size=14 \[ -\frac {2}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {266, 63, 207} \begin {gather*} -\frac {2}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {1+x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )\\ &=\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=-\frac {2}{3} \tanh ^{-1}\left (\sqrt {1+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {2}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {2}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 25, normalized size = 1.79 \begin {gather*} -\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 [B] time = 0.34, size = 26, normalized size = 1.86 \begin {gather*} -\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 = 0.29, size = 11, normalized size = 0.79 \begin {gather*} -\frac {2 \arctanh \left (\sqrt {x^{3}+1}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 25, normalized size = 1.79 \begin {gather*} -\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 = 0.25, size = 164, normalized size = 11.71 \begin {gather*} -\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\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 = 0.78, size = 12, normalized size = 0.86 \begin {gather*} - \frac {2 \operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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