Optimal. Leaf size=12 \[ \log \left (1+\sqrt {1+x^3}\right ) \]
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Rubi [A]
time = 0.04, 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, 2186, 31}
\begin {gather*} \log \left (\sqrt {x^3+1}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 2186
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} \text {Subst}\left (\int \frac {1}{1+x+\sqrt {1+x}} \, dx,x,x^3\right )\\ &=\text {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 = 0.01, size = 12, normalized size = 1.00 \begin {gather*} \log \left (1+\sqrt {1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs.
\(2(10)=20\).
time = 0.11, size = 39, normalized size = 3.25
method | result | size |
trager | \(\frac {\ln \left (-x^{3}-2 \sqrt {x^{3}+1}-2\right )}{2}\) | \(20\) |
default | \(-\frac {\ln \left (1+x \right )}{2}+\frac {3 \ln \left (x \right )}{2}-\frac {\ln \left (x^{2}-x +1\right )}{2}+\frac {\ln \left (x^{3}+1\right )}{2}+\arctanh \left (\sqrt {x^{3}+1}\right )\) | \(39\) |
elliptic | \(\frac {\left (1+\sqrt {x^{3}+1}\right ) \sqrt {x^{3}+1}\, \left (\frac {3 \ln \left (x \right )}{2}+\arctanh \left (\sqrt {x^{3}+1}\right )\right )}{1+x^{3}+\sqrt {x^{3}+1}}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs.
\(2 (10) = 20\).
time = 1.73, size = 40, normalized size = 3.33 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (10) = 20\).
time = 0.68, size = 29, normalized size = 2.42 \begin {gather*} \frac {3}{2} \, \log \left (x\right ) + \frac {1}{2} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (10) = 20\).
time = 47.96, size = 48, normalized size = 4.00 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 10, normalized size = 0.83 \begin {gather*} \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 = 0.06, size = 169, normalized size = 14.08 \begin {gather*} \frac {3\,\ln \left (x\right )}{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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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