Integrand size = 18, antiderivative size = 28 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {\sqrt {1+x^6}}{3}+\frac {1}{3} \text {arctanh}\left (\sqrt {1+x^6}\right ) \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {457, 81, 65, 213} \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {1}{3} \text {arctanh}\left (\sqrt {x^6+1}\right )+\frac {\sqrt {x^6+1}}{3} \]
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Rule 65
Rule 81
Rule 213
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {-1+x}{x \sqrt {1+x}} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {1+x^6}}{3}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {1+x^6}}{3}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^6}\right ) \\ & = \frac {\sqrt {1+x^6}}{3}+\frac {1}{3} \text {arctanh}\left (\sqrt {1+x^6}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {1}{3} \left (\sqrt {1+x^6}+\text {arctanh}\left (\sqrt {1+x^6}\right )\right ) \]
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Time = 0.85 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75
method | result | size |
pseudoelliptic | \(\frac {\sqrt {x^{6}+1}}{3}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{6}+1}}\right )}{3}\) | \(21\) |
trager | \(\frac {\sqrt {x^{6}+1}}{3}+\frac {\ln \left (\frac {\sqrt {x^{6}+1}+1}{x^{3}}\right )}{3}\) | \(27\) |
meijerg | \(\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{6}+1}}{6 \sqrt {\pi }}-\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{6 \sqrt {\pi }}\) | \(61\) |
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Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {1}{3} \, \sqrt {x^{6} + 1} + \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \]
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Time = 4.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^{6} + 1}}{3} - \frac {\log {\left (\sqrt {x^{6} + 1} - 1 \right )}}{6} + \frac {\log {\left (\sqrt {x^{6} + 1} + 1 \right )}}{6} \]
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Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {1}{3} \, \sqrt {x^{6} + 1} + \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {1}{3} \, \sqrt {x^{6} + 1} + \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \]
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Time = 5.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{3}+\frac {\sqrt {x^6+1}}{3} \]
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