Integrand size = 15, antiderivative size = 59 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {3-2 \sqrt [3]{-27+2 x^7}}{3 \sqrt {3}}\right )}{21 \sqrt {3}}-\frac {\log (x)}{18}+\frac {1}{42} \log \left (3+\sqrt [3]{-27+2 x^7}\right ) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 60, 632, 210, 31} \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {3-2 \sqrt [3]{2 x^7-27}}{3 \sqrt {3}}\right )}{21 \sqrt {3}}+\frac {1}{42} \log \left (\sqrt [3]{2 x^7-27}+3\right )-\frac {\log (x)}{18} \]
[In]
[Out]
Rule 31
Rule 60
Rule 210
Rule 272
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} \text {Subst}\left (\int \frac {1}{x (-27+2 x)^{2/3}} \, dx,x,x^7\right ) \\ & = -\frac {\log (x)}{18}+\frac {1}{42} \text {Subst}\left (\int \frac {1}{3+x} \, dx,x,\sqrt [3]{-27+2 x^7}\right )+\frac {1}{14} \text {Subst}\left (\int \frac {1}{9-3 x+x^2} \, dx,x,\sqrt [3]{-27+2 x^7}\right ) \\ & = -\frac {\log (x)}{18}+\frac {1}{42} \log \left (3+\sqrt [3]{-27+2 x^7}\right )-\frac {1}{7} \text {Subst}\left (\int \frac {1}{-27-x^2} \, dx,x,-3+2 \sqrt [3]{-27+2 x^7}\right ) \\ & = -\frac {\arctan \left (\frac {3-2 \sqrt [3]{-27+2 x^7}}{3 \sqrt {3}}\right )}{21 \sqrt {3}}-\frac {\log (x)}{18}+\frac {1}{42} \log \left (3+\sqrt [3]{-27+2 x^7}\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {1}{126} \left (-2 \sqrt {3} \arctan \left (\frac {3-2 \sqrt [3]{-27+2 x^7}}{3 \sqrt {3}}\right )+2 \log \left (3+\sqrt [3]{-27+2 x^7}\right )-\log \left (9-3 \sqrt [3]{-27+2 x^7}+\left (-27+2 x^7\right )^{2/3}\right )\right ) \]
[In]
[Out]
Time = 7.37 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.14
method | result | size |
pseudoelliptic | \(\frac {\ln \left (3+\left (2 x^{7}-27\right )^{\frac {1}{3}}\right )}{63}-\frac {\ln \left (\left (2 x^{7}-27\right )^{\frac {2}{3}}-3 \left (2 x^{7}-27\right )^{\frac {1}{3}}+9\right )}{126}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (2 x^{7}-27\right )^{\frac {1}{3}}}{9}-\frac {\sqrt {3}}{3}\right )}{63}\) | \(67\) |
meijerg | \(\frac {{\left (-\operatorname {signum}\left (-1+\frac {2 x^{7}}{27}\right )\right )}^{\frac {2}{3}} \left (\left (\frac {\pi \sqrt {3}}{6}-\frac {9 \ln \left (3\right )}{2}+7 \ln \left (x \right )+\ln \left (2\right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+\frac {4 \Gamma \left (\frac {2}{3}\right ) x^{7} {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (1,1,\frac {5}{3};2,2;\frac {2 x^{7}}{27}\right )}{81}\right )}{63 \operatorname {signum}\left (-1+\frac {2 x^{7}}{27}\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )}\) | \(74\) |
trager | \(-\frac {\ln \left (-\frac {757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2} x^{7}+119351332086100723341414 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) x^{7}-20295123418338509861200 x^{7}+347409114848503477844697 \left (2 x^{7}-27\right )^{\frac {2}{3}} \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2}-1042227344545510433534091 \left (2 x^{7}-27\right )^{\frac {1}{3}} \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )+94802367093259243458870 \left (2 x^{7}-27\right )^{\frac {2}{3}}+3042531384693169740692067 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-284407101279777730376610 \left (2 x^{7}-27\right )^{\frac {1}{3}}+843871231734515240028696}{x^{7}}\right )}{63}-\frac {\ln \left (-\frac {757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2} x^{7}+119351332086100723341414 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) x^{7}-20295123418338509861200 x^{7}+347409114848503477844697 \left (2 x^{7}-27\right )^{\frac {2}{3}} \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2}-1042227344545510433534091 \left (2 x^{7}-27\right )^{\frac {1}{3}} \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )+94802367093259243458870 \left (2 x^{7}-27\right )^{\frac {2}{3}}+3042531384693169740692067 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-284407101279777730376610 \left (2 x^{7}-27\right )^{\frac {1}{3}}+843871231734515240028696}{x^{7}}\right ) \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )}{7}+\frac {\operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) \ln \left (-\frac {757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2} x^{7}+48949965800622396478998 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) x^{7}-24206310434198416909112 x^{7}-347409114848503477844697 \left (2 x^{7}-27\right )^{\frac {2}{3}} \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2}+1042227344545510433534091 \left (2 x^{7}-27\right )^{\frac {1}{3}} \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )+56201354332314412587237 \left (2 x^{7}-27\right )^{\frac {2}{3}}-3210832682579892860512479 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-168604062996943237761711 \left (2 x^{7}-27\right )^{\frac {1}{3}}+496462116886011762183999}{x^{7}}\right )}{7}\) | \(453\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {1}{63} \, \sqrt {3} \arctan \left (\frac {2}{9} \, \sqrt {3} {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{126} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {2}{3}} - 3 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 9\right ) + \frac {1}{63} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 3\right ) \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=- \frac {\sqrt [3]{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {27 e^{2 i \pi }}{2 x^{7}}} \right )}}{14 x^{\frac {14}{3}} \Gamma \left (\frac {5}{3}\right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {1}{63} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (2 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} - 3\right )}\right ) - \frac {1}{126} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {2}{3}} - 3 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 9\right ) + \frac {1}{63} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 3\right ) \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {1}{63} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (2 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} - 3\right )}\right ) - \frac {1}{126} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {2}{3}} - 3 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 9\right ) + \frac {1}{63} \, \log \left ({\left | {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 3 \right |}\right ) \]
[In]
[Out]
Time = 0.48 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {\ln \left (\frac {{\left (2\,x^7-27\right )}^{1/3}}{49}+\frac {3}{49}\right )}{63}-\ln \left (\frac {27}{14}-\frac {9\,{\left (2\,x^7-27\right )}^{1/3}}{7}+\frac {\sqrt {3}\,27{}\mathrm {i}}{14}\right )\,\left (\frac {1}{126}+\frac {\sqrt {3}\,1{}\mathrm {i}}{126}\right )+\ln \left (\frac {9\,{\left (2\,x^7-27\right )}^{1/3}}{7}-\frac {27}{14}+\frac {\sqrt {3}\,27{}\mathrm {i}}{14}\right )\,\left (-\frac {1}{126}+\frac {\sqrt {3}\,1{}\mathrm {i}}{126}\right ) \]
[In]
[Out]