Integrand size = 18, antiderivative size = 97 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=\frac {\sqrt [3]{1+x^3} \left (1+3 x^3\right )}{3 x^3}-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \log \left (-1+\sqrt [3]{1+x^3}\right )-\frac {1}{9} \log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {457, 79, 52, 59, 632, 210, 31} \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=-\frac {2 \arctan \left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\left (x^3+1\right )^{4/3}}{3 x^3}+\frac {2}{3} \sqrt [3]{x^3+1}+\frac {1}{3} \log \left (1-\sqrt [3]{x^3+1}\right )-\frac {\log (x)}{3} \]
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Rule 31
Rule 52
Rule 59
Rule 79
Rule 210
Rule 457
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(-1+x) \sqrt [3]{1+x}}{x^2} \, dx,x,x^3\right ) \\ & = \frac {\left (1+x^3\right )^{4/3}}{3 x^3}+\frac {2}{9} \text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x} \, dx,x,x^3\right ) \\ & = \frac {2}{3} \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{3 x^3}+\frac {2}{9} \text {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^3\right ) \\ & = \frac {2}{3} \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{3 x^3}-\frac {\log (x)}{3}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^3}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^3}\right ) \\ & = \frac {2}{3} \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{3 x^3}-\frac {\log (x)}{3}+\frac {1}{3} \log \left (1-\sqrt [3]{1+x^3}\right )+\frac {2}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^3}\right ) \\ & = \frac {2}{3} \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{3 x^3}-\frac {2 \arctan \left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\log (x)}{3}+\frac {1}{3} \log \left (1-\sqrt [3]{1+x^3}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=\frac {1}{9} \left (\frac {3 \sqrt [3]{1+x^3} \left (1+3 x^3\right )}{x^3}-2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1+x^3}\right )-\log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.37 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.73
method | result | size |
risch | \(\frac {3 x^{6}+4 x^{3}+1}{3 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}}+\frac {-\frac {4 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{27}+\frac {2 \left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{9}}{\Gamma \left (\frac {2}{3}\right )}\) | \(71\) |
meijerg | \(\frac {\frac {\Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], -x^{3}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+3 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )+\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{3}}}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {-\Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [\frac {2}{3}, 1, 1\right ], \left [2, 2\right ], -x^{3}\right )-3 \left (3+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\) | \(103\) |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right ) x^{3}+9 x^{3} \left (x^{3}+1\right )^{\frac {1}{3}}+2 \ln \left (-1+\left (x^{3}+1\right )^{\frac {1}{3}}\right ) x^{3}-\ln \left (1+\left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}\right ) x^{3}+3 \left (x^{3}+1\right )^{\frac {1}{3}}}{9 \left (-1+\left (x^{3}+1\right )^{\frac {1}{3}}\right ) \left (1+\left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}\right )}\) | \(116\) |
trager | \(\frac {\left (x^{3}+1\right )^{\frac {1}{3}} \left (3 x^{3}+1\right )}{3 x^{3}}+\frac {2 \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+15 \left (x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-12 x^{3}+24 \left (x^{3}+1\right )^{\frac {2}{3}}-24 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-9 \left (x^{3}+1\right )^{\frac {1}{3}}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-16}{x^{3}}\right )}{9}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+15 \left (x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-8 x^{3}-9 \left (x^{3}+1\right )^{\frac {2}{3}}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+24 \left (x^{3}+1\right )^{\frac {1}{3}}-29 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-20}{x^{3}}\right )}{9}\) | \(248\) |
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Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=-\frac {2 \, \sqrt {3} x^{3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + x^{3} \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{3} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (3 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{9 \, x^{3}} \]
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Result contains complex when optimal does not.
Time = 51.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.67 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=- \frac {x \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 \Gamma \left (\frac {2}{3}\right )} + \frac {\Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{2} \Gamma \left (\frac {5}{3}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=-\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {1}{9} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{9} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.76 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=-\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {1}{9} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{9} \, \log \left ({\left | {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Time = 6.82 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.57 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=\frac {\ln \left ({\left (x^3+1\right )}^{1/3}-1\right )}{3}-\frac {\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{9}-\frac {1}{9}\right )}{9}+{\left (x^3+1\right )}^{1/3}-\ln \left ({\left (x^3+1\right )}^{1/3}+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )+\ln \left ({\left (x^3+1\right )}^{1/3}+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )+\frac {{\left (x^3+1\right )}^{1/3}}{3\,x^3}+\ln \left (3\,{\left (x^3+1\right )}^{1/3}+\frac {3}{2}-\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (3\,{\left (x^3+1\right )}^{1/3}+\frac {3}{2}+\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]
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