Optimal. Leaf size=38 \[ \frac {\sqrt [3]{-1+x^6} \left (4-7 x^3-8 x^6+7 x^9+4 x^{12}\right )}{28 x^7} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 49, normalized size of antiderivative = 1.29, number of steps
used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1849, 1600,
1598, 460} \begin {gather*} \frac {\left (x^6-1\right )^{4/3}}{7 x}-\frac {\left (x^6-1\right )^{4/3}}{7 x^7}+\frac {\left (x^6-1\right )^{4/3}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 460
Rule 1598
Rule 1600
Rule 1849
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right ) \left (-1+x^3+x^6\right )}{x^8} \, dx &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt [3]{-1+x^6} \left (14 x^2+2 x^5+14 x^8+14 x^{11}\right )}{x^7} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt [3]{-1+x^6} \left (14 x+2 x^4+14 x^7+14 x^{10}\right )}{x^6} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt [3]{-1+x^6} \left (14+2 x^3+14 x^6+14 x^9\right )}{x^5} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {\left (-1+x^6\right )^{4/3}}{4 x^4}+\frac {1}{112} \int \frac {\sqrt [3]{-1+x^6} \left (16 x^2+112 x^8\right )}{x^4} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {\left (-1+x^6\right )^{4/3}}{4 x^4}+\frac {1}{112} \int \frac {\sqrt [3]{-1+x^6} \left (16+112 x^6\right )}{x^2} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {\left (-1+x^6\right )^{4/3}}{4 x^4}+\frac {\left (-1+x^6\right )^{4/3}}{7 x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.46, size = 28, normalized size = 0.74 \begin {gather*} \frac {\left (-1+x^6\right )^{4/3} \left (-4+7 x^3+4 x^6\right )}{28 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.29, size = 35, normalized size = 0.92
method | result | size |
trager | \(\frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (4 x^{12}+7 x^{9}-8 x^{6}-7 x^{3}+4\right )}{28 x^{7}}\) | \(35\) |
gosper | \(\frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (4 x^{6}+7 x^{3}-4\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{28 x^{7}}\) | \(45\) |
risch | \(\frac {4 x^{18}+7 x^{15}-12 x^{12}-14 x^{9}+12 x^{6}+7 x^{3}-4}{28 \left (x^{6}-1\right )^{\frac {2}{3}} x^{7}}\) | \(45\) |
meijerg | \(\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} x^{5} \hypergeom \left (\left [-\frac {1}{3}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], x^{6}\right )}{5 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}}}+\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} x^{2} \hypergeom \left (\left [-\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{2 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}}}-\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {2}{3}, -\frac {1}{3}\right ], \left [\frac {1}{3}\right ], x^{6}\right )}{4 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{4}}+\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {7}{6}, -\frac {1}{3}\right ], \left [-\frac {1}{6}\right ], x^{6}\right )}{7 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{7}}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 55, normalized size = 1.45 \begin {gather*} \frac {{\left (4 \, x^{12} + 7 \, x^{9} - 8 \, x^{6} - 7 \, x^{3} + 4\right )} {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {1}{3}}}{28 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 34, normalized size = 0.89 \begin {gather*} \frac {{\left (4 \, x^{12} + 7 \, x^{9} - 8 \, x^{6} - 7 \, x^{3} + 4\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{28 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 2.67, size = 148, normalized size = 3.89 \begin {gather*} \frac {x^{5} e^{\frac {i \pi }{3}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} + \frac {x^{2} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} - \frac {e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{4} \Gamma \left (\frac {1}{3}\right )} + \frac {e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, - \frac {1}{3} \\ - \frac {1}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{7} \Gamma \left (- \frac {1}{6}\right )} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.46, size = 56, normalized size = 1.47 \begin {gather*} {\left (x^6-1\right )}^{1/3}\,\left (\frac {x^5}{7}+\frac {x^2}{4}\right )-\frac {2\,{\left (x^6-1\right )}^{1/3}}{7\,x}-\frac {{\left (x^6-1\right )}^{1/3}}{4\,x^4}+\frac {{\left (x^6-1\right )}^{1/3}}{7\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________