Optimal. Leaf size=138 \[ -\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1}-2 x\right )}{3 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-1}+x}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\left (x^3-1\right )^{2/3} \left (3 x^3+2\right )}{10 x^5}+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1} x+\sqrt [3]{2} \left (x^3-1\right )^{2/3}+2 x^2\right )}{6 \sqrt [3]{2}} \]
________________________________________________________________________________________
Rubi [C] time = 0.78, antiderivative size = 402, normalized size of antiderivative = 2.91, number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1586, 6725, 271, 264, 2148, 6728} \begin {gather*} -\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1}-x+1\right )}{4 \sqrt [3]{2}}-\frac {\log \left (2\ 2^{2/3} \sqrt [3]{x^3-1}-2 x-i \sqrt {3}-1\right )}{4 \sqrt [3]{2}}-\frac {\log \left (2\ 2^{2/3} \sqrt [3]{x^3-1}-2 x+i \sqrt {3}-1\right )}{4 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (2 x-i \sqrt {3}+1\right )}{\sqrt [3]{x^3-1}}}{2 \sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (2 x+i \sqrt {3}+1\right )}{\sqrt [3]{x^3-1}}}{2 \sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\left (x^3-1\right )^{2/3}}{5 x^5}+\frac {3 \left (x^3-1\right )^{2/3}}{10 x^2}+\frac {\log \left ((1-x) (x+1)^2\right )}{12 \sqrt [3]{2}}+\frac {\log \left (-\left (-2 x-i \sqrt {3}+1\right )^2 \left (2 x-i \sqrt {3}+1\right )\right )}{12 \sqrt [3]{2}}+\frac {\log \left (-\left (-2 x+i \sqrt {3}+1\right )^2 \left (2 x+i \sqrt {3}+1\right )\right )}{12 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 271
Rule 1586
Rule 2148
Rule 6725
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3+x^6\right )}{x^6 \left (-1+x^6\right )} \, dx &=\int \frac {1+x^3+x^6}{x^6 \sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx\\ &=\int \left (\frac {1}{x^6 \sqrt [3]{-1+x^3}}+\frac {1}{3 (1+x) \sqrt [3]{-1+x^3}}+\frac {2-x}{3 \left (1-x+x^2\right ) \sqrt [3]{-1+x^3}}\right ) \, dx\\ &=\frac {1}{3} \int \frac {1}{(1+x) \sqrt [3]{-1+x^3}} \, dx+\frac {1}{3} \int \frac {2-x}{\left (1-x+x^2\right ) \sqrt [3]{-1+x^3}} \, dx+\int \frac {1}{x^6 \sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {\tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left ((1-x) (1+x)^2\right )}{12 \sqrt [3]{2}}-\frac {\log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{4 \sqrt [3]{2}}+\frac {1}{3} \int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^3}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^3}}\right ) \, dx+\frac {3}{5} \int \frac {1}{x^3 \sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {3 \left (-1+x^3\right )^{2/3}}{10 x^2}+\frac {\tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left ((1-x) (1+x)^2\right )}{12 \sqrt [3]{2}}-\frac {\log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{4 \sqrt [3]{2}}+\frac {1}{3} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^3}} \, dx+\frac {1}{3} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {3 \left (-1+x^3\right )^{2/3}}{10 x^2}+\frac {\tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1-i \sqrt {3}+2 x\right )}{\sqrt [3]{-1+x^3}}}{2 \sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1+i \sqrt {3}+2 x\right )}{\sqrt [3]{-1+x^3}}}{2 \sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left ((1-x) (1+x)^2\right )}{12 \sqrt [3]{2}}+\frac {\log \left (-\left (1-i \sqrt {3}-2 x\right )^2 \left (1-i \sqrt {3}+2 x\right )\right )}{12 \sqrt [3]{2}}+\frac {\log \left (-\left (1+i \sqrt {3}-2 x\right )^2 \left (1+i \sqrt {3}+2 x\right )\right )}{12 \sqrt [3]{2}}-\frac {\log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{4 \sqrt [3]{2}}-\frac {\log \left (-1-i \sqrt {3}-2 x+2\ 2^{2/3} \sqrt [3]{-1+x^3}\right )}{4 \sqrt [3]{2}}-\frac {\log \left (-1+i \sqrt {3}-2 x+2\ 2^{2/3} \sqrt [3]{-1+x^3}\right )}{4 \sqrt [3]{2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3+x^6\right )}{x^6 \left (-1+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.37, size = 138, normalized size = 1.00 \begin {gather*} -\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1}-2 x\right )}{3 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-1}+x}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\left (x^3-1\right )^{2/3} \left (3 x^3+2\right )}{10 x^5}+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1} x+\sqrt [3]{2} \left (x^3-1\right )^{2/3}+2 x^2\right )}{6 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 2.45, size = 301, normalized size = 2.18 \begin {gather*} -\frac {10 \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 12 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{6 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 10 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 5 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 12 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 18 \, {\left (3 \, x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{180 \, x^{5}} \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 {{\left (x^{6} + x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 2.68, size = 929, normalized size = 6.73
result too large to display
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 {{\left (x^{6} + x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 1\right )} x^{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 {{\left (x^3-1\right )}^{2/3}\,\left (x^6+x^3+1\right )}{x^6\,\left (x^6-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + x^{3} + 1\right )}{x^{6} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________