Optimal. Leaf size=154 \[ -\frac {1}{3} \tan ^{-1}\left (\frac {\sqrt [6]{\frac {x^3+6}{x^3-6}}}{x-1}\right )-\frac {1}{6} \tan ^{-1}\left (\frac {\sqrt [3]{\frac {x^3+6}{x^3-6}}-x^2+2 x-1}{(x-1) \sqrt [6]{\frac {x^3+6}{x^3-6}}}\right )+\frac {\tanh ^{-1}\left (\frac {\left (\sqrt {3} x-\sqrt {3}\right ) \sqrt [6]{\frac {x^3+6}{x^3-6}}}{\sqrt [3]{\frac {x^3+6}{x^3-6}}+x^2-2 x+1}\right )}{2 \sqrt {3}} \]
________________________________________________________________________________________
Rubi [F] time = 10.60, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx &=\frac {\sqrt [6]{6+x^3} \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=\frac {\sqrt [6]{6+x^3} \int \left (-\frac {1}{x \left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3}}+\frac {-90+116 x-90 x^2+51 x^3-26 x^4+16 x^5-6 x^6+x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}\right ) \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=-\frac {\sqrt [6]{6+x^3} \int \frac {1}{x \left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3}} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \int \frac {-90+116 x-90 x^2+51 x^3-26 x^4+16 x^5-6 x^6+x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=-\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {1}{(-6+x)^{5/6} x \sqrt [6]{6+x}} \, dx,x,x^3\right )}{3 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \int \left (-\frac {90}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}+\frac {116 x}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}-\frac {90 x^2}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}+\frac {51 x^3}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}-\frac {26 x^4}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}+\frac {16 x^5}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}-\frac {6 x^6}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}+\frac {x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )}\right ) \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=\frac {\sqrt [6]{6+x^3} \int \frac {x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (2 \sqrt [6]{6+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{-6-6 x^6} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (6 \sqrt [6]{6+x^3}\right ) \int \frac {x^6}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (16 \sqrt [6]{6+x^3}\right ) \int \frac {x^5}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (26 \sqrt [6]{6+x^3}\right ) \int \frac {x^4}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (51 \sqrt [6]{6+x^3}\right ) \int \frac {x^3}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {1}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {x^2}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (116 \sqrt [6]{6+x^3}\right ) \int \frac {x}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{9 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {-\frac {1}{2}+\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{9 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {-\frac {1}{2}-\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{9 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \int \frac {x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (6 \sqrt [6]{6+x^3}\right ) \int \frac {x^6}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (16 \sqrt [6]{6+x^3}\right ) \int \frac {x^5}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (26 \sqrt [6]{6+x^3}\right ) \int \frac {x^4}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (51 \sqrt [6]{6+x^3}\right ) \int \frac {x^3}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {1}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {x^2}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (116 \sqrt [6]{6+x^3}\right ) \int \frac {x}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=\frac {\sqrt [6]{6+x^3} \tan ^{-1}\left (\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{9 \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}+\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{36 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{36 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \int \frac {x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (6 \sqrt [6]{6+x^3}\right ) \int \frac {x^6}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (16 \sqrt [6]{6+x^3}\right ) \int \frac {x^5}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (26 \sqrt [6]{6+x^3}\right ) \int \frac {x^4}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (51 \sqrt [6]{6+x^3}\right ) \int \frac {x^3}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {1}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {x^2}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (116 \sqrt [6]{6+x^3}\right ) \int \frac {x}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{12 \sqrt {3} \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{12 \sqrt {3} \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=\frac {\sqrt [6]{6+x^3} \tan ^{-1}\left (\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{9 \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}+\frac {\sqrt [6]{6+x^3} \log \left (1-\frac {\sqrt {3} \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}+\frac {\sqrt [3]{6+x^3}}{\sqrt [3]{-6+x^3}}\right )}{12 \sqrt {3} \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}-\frac {\sqrt [6]{6+x^3} \log \left (1+\frac {\sqrt {3} \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}+\frac {\sqrt [3]{6+x^3}}{\sqrt [3]{-6+x^3}}\right )}{12 \sqrt {3} \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}-\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+\frac {2 \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{18 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\sqrt [6]{6+x^3} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+\frac {2 \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{18 \sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\sqrt [6]{6+x^3} \int \frac {x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (6 \sqrt [6]{6+x^3}\right ) \int \frac {x^6}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (16 \sqrt [6]{6+x^3}\right ) \int \frac {x^5}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (26 \sqrt [6]{6+x^3}\right ) \int \frac {x^4}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (51 \sqrt [6]{6+x^3}\right ) \int \frac {x^3}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {1}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {x^2}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (116 \sqrt [6]{6+x^3}\right ) \int \frac {x}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ &=\frac {\sqrt [6]{6+x^3} \tan ^{-1}\left (\frac {\sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{9 \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}-\frac {\sqrt [6]{6+x^3} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{18 \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}+\frac {\sqrt [6]{6+x^3} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}\right )}{18 \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}+\frac {\sqrt [6]{6+x^3} \log \left (1-\frac {\sqrt {3} \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}+\frac {\sqrt [3]{6+x^3}}{\sqrt [3]{-6+x^3}}\right )}{12 \sqrt {3} \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}-\frac {\sqrt [6]{6+x^3} \log \left (1+\frac {\sqrt {3} \sqrt [6]{6+x^3}}{\sqrt [6]{-6+x^3}}+\frac {\sqrt [3]{6+x^3}}{\sqrt [3]{-6+x^3}}\right )}{12 \sqrt {3} \sqrt [6]{-6+x^3} \sqrt [6]{-\frac {6+x^3}{6-x^3}}}+\frac {\sqrt [6]{6+x^3} \int \frac {x^7}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (6 \sqrt [6]{6+x^3}\right ) \int \frac {x^6}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (16 \sqrt [6]{6+x^3}\right ) \int \frac {x^5}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (26 \sqrt [6]{6+x^3}\right ) \int \frac {x^4}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (51 \sqrt [6]{6+x^3}\right ) \int \frac {x^3}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {1}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}-\frac {\left (90 \sqrt [6]{6+x^3}\right ) \int \frac {x^2}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}+\frac {\left (116 \sqrt [6]{6+x^3}\right ) \int \frac {x}{\left (-6+x^3\right )^{5/6} \sqrt [6]{6+x^3} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx}{\sqrt [6]{-6+x^3} \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 1.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 3.02, size = 154, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \tan ^{-1}\left (\frac {\sqrt [6]{\frac {x^3+6}{x^3-6}}}{x-1}\right )-\frac {1}{6} \tan ^{-1}\left (\frac {\sqrt [3]{\frac {x^3+6}{x^3-6}}-x^2+2 x-1}{(x-1) \sqrt [6]{\frac {x^3+6}{x^3-6}}}\right )+\frac {\tanh ^{-1}\left (\frac {\left (\sqrt {3} x-\sqrt {3}\right ) \sqrt [6]{\frac {x^3+6}{x^3-6}}}{\sqrt [3]{\frac {x^3+6}{x^3-6}}+x^2-2 x+1}\right )}{2 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + 6 \, x^{3} - 6 \, x^{2} - 36}{{\left (x^{8} - 6 \, x^{7} + 15 \, x^{6} - 26 \, x^{5} + 51 \, x^{4} - 96 \, x^{3} + 122 \, x^{2} - 90 \, x + 36\right )} {\left (x^{3} - 6\right )} x \left (\frac {x^{3} + 6}{x^{3} - 6}\right )^{\frac {1}{6}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}+6 x^{3}-6 x^{2}-36}{x \left (x^{3}-6\right ) \left (\frac {x^{3}+6}{x^{3}-6}\right )^{\frac {1}{6}} \left (x^{8}-6 x^{7}+15 x^{6}-26 x^{5}+51 x^{4}-96 x^{3}+122 x^{2}-90 x +36\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + 6 \, x^{3} - 6 \, x^{2} - 36}{{\left (x^{8} - 6 \, x^{7} + 15 \, x^{6} - 26 \, x^{5} + 51 \, x^{4} - 96 \, x^{3} + 122 \, x^{2} - 90 \, x + 36\right )} {\left (x^{3} - 6\right )} x \left (\frac {x^{3} + 6}{x^{3} - 6}\right )^{\frac {1}{6}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {-x^6-6\,x^3+6\,x^2+36}{x\,{\left (\frac {x^3+6}{x^3-6}\right )}^{1/6}\,\left (x^3-6\right )\,\left (x^8-6\,x^7+15\,x^6-26\,x^5+51\,x^4-96\,x^3+122\,x^2-90\,x+36\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________