Integrand size = 13, antiderivative size = 28 \[ \int \frac {\sqrt {1+x^3}}{x} \, dx=\frac {2 \sqrt {1+x^3}}{3}-\frac {2}{3} \text {arctanh}\left (\sqrt {1+x^3}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 52, 65, 213} \[ \int \frac {\sqrt {1+x^3}}{x} \, dx=\frac {2 \sqrt {x^3+1}}{3}-\frac {2}{3} \text {arctanh}\left (\sqrt {x^3+1}\right ) \]
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Rule 52
Rule 65
Rule 213
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,x^3\right ) \\ & = \frac {2 \sqrt {1+x^3}}{3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right ) \\ & = \frac {2 \sqrt {1+x^3}}{3}+\frac {2}{3} \text {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} \text {arctanh}\left (\sqrt {1+x^3}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x^3}}{x} \, dx=\frac {2 \sqrt {1+x^3}}{3}-\frac {2}{3} \text {arctanh}\left (\sqrt {1+x^3}\right ) \]
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Time = 1.84 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {2 \sqrt {x^{3}+1}}{3}-\frac {2 \,\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{3}\) | \(21\) |
elliptic | \(\frac {2 \sqrt {x^{3}+1}}{3}-\frac {2 \,\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{3}\) | \(21\) |
trager | \(\frac {2 \sqrt {x^{3}+1}}{3}+\frac {\ln \left (-\frac {-x^{3}+2 \sqrt {x^{3}+1}-2}{x^{3}}\right )}{3}\) | \(35\) |
pseudoelliptic | \(\frac {2 \sqrt {x^{3}+1}}{3}+\frac {\ln \left (\sqrt {x^{3}+1}-1\right )}{3}-\frac {\ln \left (\sqrt {x^{3}+1}+1\right )}{3}\) | \(35\) |
meijerg | \(-\frac {4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {x^{3}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right )-2 \left (2-2 \ln \left (2\right )+3 \ln \left (x \right )\right ) \sqrt {\pi }}{6 \sqrt {\pi }}\) | \(56\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {1+x^3}}{x} \, dx=\frac {2}{3} \, \sqrt {x^{3} + 1} - \frac {1}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \]
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Time = 0.65 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {1+x^3}}{x} \, dx=\frac {2 x^{\frac {3}{2}}}{3 \sqrt {1 + \frac {1}{x^{3}}}} - \frac {2 \operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} + \frac {2}{3 x^{\frac {3}{2}} \sqrt {1 + \frac {1}{x^{3}}}} \]
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Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {1+x^3}}{x} \, dx=\frac {2}{3} \, \sqrt {x^{3} + 1} - \frac {1}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {1+x^3}}{x} \, dx=\frac {2}{3} \, \sqrt {x^{3} + 1} - \frac {1}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {1}{3} \, \log \left ({\left | \sqrt {x^{3} + 1} - 1 \right |}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 174, normalized size of antiderivative = 6.21 \[ \int \frac {\sqrt {1+x^3}}{x} \, dx=\frac {2\,\sqrt {x^3+1}}{3}-\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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