Optimal. Leaf size=12 \[ \log \left (\sqrt {x^3+1}+1\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {12, 2155, 31} \[ \log \left (\sqrt {x^3+1}+1\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 2155
Rubi steps
\begin {align*} \int \frac {3 x^2}{2 \left (1+x^3+\sqrt {1+x^3}\right )} \, dx &=\frac {3}{2} \int \frac {x^2}{1+x^3+\sqrt {1+x^3}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x+\sqrt {1+x}} \, dx,x,x^3\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt {1+x^3}\right )\\ &=\log \left (1+\sqrt {1+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 12, normalized size = 1.00 \[ \log \left (\sqrt {x^3+1}+1\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 29, normalized size = 2.42 \[ \frac {3}{2} \, \log \relax (x) + \frac {1}{2} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 10, normalized size = 0.83 \[ \log \left (\sqrt {x^{3} + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 39, normalized size = 3.25 \[ \arctanh \left (\sqrt {x^{3}+1}\right )+\frac {3 \ln \relax (x )}{2}-\frac {\ln \left (x +1\right )}{2}+\frac {\ln \left (x^{3}+1\right )}{2}-\frac {\ln \left (x^{2}-x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.98, size = 40, normalized size = 3.33 \[ -\frac {1}{2} \, \log \left (x^{2} - x + 1\right ) + \log \left (\frac {x^{3} + \sqrt {x^{2} - x + 1} \sqrt {x + 1} + 1}{\sqrt {x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 169, normalized size = 14.08 \[ \frac {3\,\ln \relax (x)}{2}+\frac {3\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 154.69, size = 48, normalized size = 4.00 \[ - \frac {\log {\left (2 \sqrt {x^{3} + 1} \right )}}{2} + \frac {\log {\left (2 \sqrt {x^{3} + 1} + 2 \right )}}{2} + \frac {\log {\left (3 x^{3} + 3 \sqrt {x^{3} + 1} + 3 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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