Integrand size = 20, antiderivative size = 55 \[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \left (1+x^3\right ) \sqrt {-1+x^6}-\frac {2}{3} \arctan \left (x^3+\sqrt {-1+x^6}\right )-\frac {1}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1489, 829, 858, 223, 212, 272, 65, 209} \[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x} \, dx=-\frac {1}{3} \arctan \left (\sqrt {x^6-1}\right )-\frac {1}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )+\frac {1}{3} \sqrt {x^6-1} \left (x^3+1\right ) \]
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Rule 65
Rule 209
Rule 212
Rule 223
Rule 272
Rule 829
Rule 858
Rule 1489
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(1+2 x) \sqrt {-1+x^2}}{x} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \left (1+x^3\right ) \sqrt {-1+x^6}+\frac {1}{6} \text {Subst}\left (\int \frac {-2-2 x}{x \sqrt {-1+x^2}} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \left (1+x^3\right ) \sqrt {-1+x^6}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {-1+x^2}} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \left (1+x^3\right ) \sqrt {-1+x^6}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ & = \frac {1}{3} \left (1+x^3\right ) \sqrt {-1+x^6}-\frac {1}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right ) \\ & = \frac {1}{3} \left (1+x^3\right ) \sqrt {-1+x^6}-\frac {1}{3} \arctan \left (\sqrt {-1+x^6}\right )-\frac {1}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09 \[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \left (\left (1+x^3\right ) \sqrt {-1+x^6}+2 \arctan \left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right )-2 \text {arctanh}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right )\right ) \]
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Time = 0.94 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85
method | result | size |
pseudoelliptic | \(\frac {x^{3} \sqrt {x^{6}-1}}{3}-\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}+\frac {\sqrt {x^{6}-1}}{3}+\frac {\arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right )}{3}\) | \(47\) |
trager | \(\left (\frac {x^{3}}{3}+\frac {1}{3}\right ) \sqrt {x^{6}-1}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}+\frac {\ln \left (-x^{3}+\sqrt {x^{6}-1}\right )}{3}\) | \(62\) |
meijerg | \(-\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-x^{6}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )-2 \left (2-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}+\frac {i \sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 i \sqrt {\pi }\, x^{3} \sqrt {-x^{6}+1}-2 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}\) | \(136\) |
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Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \, \sqrt {x^{6} - 1} {\left (x^{3} + 1\right )} - \frac {2}{3} \, \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + \frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \]
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Time = 4.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02 \[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x} \, dx=\frac {x^{3} \sqrt {x^{6} - 1}}{3} + \frac {\begin {cases} \sqrt {x^{6} - 1} - \operatorname {acos}{\left (\frac {1}{x^{3}} \right )} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} - \frac {\log {\left (2 x^{3} + 2 \sqrt {x^{6} - 1} \right )}}{3} \]
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Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.40 \[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {\sqrt {x^{6} - 1}}{3 \, x^{3} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) + \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \]
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\[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x} \, dx=\int { \frac {\sqrt {x^{6} - 1} {\left (2 \, x^{3} + 1\right )}}{x} \,d x } \]
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Time = 6.38 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.98 \[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x} \, dx=\frac {\sqrt {x^6-1}}{3}-\frac {\ln \left (\sqrt {x^6-1}+x^3\right )}{3}+\frac {x^3\,\sqrt {x^6-1}}{3}-\frac {\ln \left (\frac {\sqrt {x^6-1}+1{}\mathrm {i}}{x^3}\right )\,1{}\mathrm {i}}{3} \]
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