Optimal. Leaf size=105 \[ \frac {3 \left (1+x^3+x^8\right )^{2/3}}{2 x^2}-\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3+x^8}}\right )+\log \left (-x+\sqrt [3]{1+x^3+x^8}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^3+x^8}+\left (1+x^3+x^8\right )^{2/3}\right ) \]
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Rubi [F]
time = 1.92, 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 {\left (1+x^3+x^8\right )^{2/3} \left (-3+5 x^8\right )}{x^3 \left (1+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (1+x^3+x^8\right )^{2/3} \left (-3+5 x^8\right )}{x^3 \left (1+x^8\right )} \, dx &=\int \left (-\frac {3 \left (1+x^3+x^8\right )^{2/3}}{x^3}+\frac {8 x^5 \left (1+x^3+x^8\right )^{2/3}}{1+x^8}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )+8 \int \frac {x^5 \left (1+x^3+x^8\right )^{2/3}}{1+x^8} \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )+8 \int \left (\frac {x \left (1+x^3+x^8\right )^{2/3}}{2 \left (-i+x^4\right )}+\frac {x \left (1+x^3+x^8\right )^{2/3}}{2 \left (i+x^4\right )}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )+4 \int \frac {x \left (1+x^3+x^8\right )^{2/3}}{-i+x^4} \, dx+4 \int \frac {x \left (1+x^3+x^8\right )^{2/3}}{i+x^4} \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )+4 \int \left (-\frac {(-1)^{3/4} x \left (1+x^3+x^8\right )^{2/3}}{2 \left (-\sqrt [4]{-1}+x^2\right )}+\frac {(-1)^{3/4} x \left (1+x^3+x^8\right )^{2/3}}{2 \left (\sqrt [4]{-1}+x^2\right )}\right ) \, dx+4 \int \left (-\frac {\sqrt [4]{-1} x \left (1+x^3+x^8\right )^{2/3}}{2 \left (-(-1)^{3/4}+x^2\right )}+\frac {\sqrt [4]{-1} x \left (1+x^3+x^8\right )^{2/3}}{2 \left ((-1)^{3/4}+x^2\right )}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )-\left (2 \sqrt [4]{-1}\right ) \int \frac {x \left (1+x^3+x^8\right )^{2/3}}{-(-1)^{3/4}+x^2} \, dx+\left (2 \sqrt [4]{-1}\right ) \int \frac {x \left (1+x^3+x^8\right )^{2/3}}{(-1)^{3/4}+x^2} \, dx-\left (2 (-1)^{3/4}\right ) \int \frac {x \left (1+x^3+x^8\right )^{2/3}}{-\sqrt [4]{-1}+x^2} \, dx+\left (2 (-1)^{3/4}\right ) \int \frac {x \left (1+x^3+x^8\right )^{2/3}}{\sqrt [4]{-1}+x^2} \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )-\left (2 \sqrt [4]{-1}\right ) \int \left (-\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left ((-1)^{3/8}-x\right )}+\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left ((-1)^{3/8}+x\right )}\right ) \, dx+\left (2 \sqrt [4]{-1}\right ) \int \left (-\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left (-(-1)^{7/8}-x\right )}+\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left (-(-1)^{7/8}+x\right )}\right ) \, dx-\left (2 (-1)^{3/4}\right ) \int \left (-\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left (\sqrt [8]{-1}-x\right )}+\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left (\sqrt [8]{-1}+x\right )}\right ) \, dx+\left (2 (-1)^{3/4}\right ) \int \left (-\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left (-(-1)^{5/8}-x\right )}+\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left (-(-1)^{5/8}+x\right )}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )+\sqrt [4]{-1} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{(-1)^{3/8}-x} \, dx-\sqrt [4]{-1} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{-(-1)^{7/8}-x} \, dx-\sqrt [4]{-1} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{(-1)^{3/8}+x} \, dx+\sqrt [4]{-1} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{-(-1)^{7/8}+x} \, dx+(-1)^{3/4} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{\sqrt [8]{-1}-x} \, dx-(-1)^{3/4} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{-(-1)^{5/8}-x} \, dx-(-1)^{3/4} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{\sqrt [8]{-1}+x} \, dx+(-1)^{3/4} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{-(-1)^{5/8}+x} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.38, size = 105, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^3+x^8\right )^{2/3}}{2 x^2}-\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3+x^8}}\right )+\log \left (-x+\sqrt [3]{1+x^3+x^8}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^3+x^8}+\left (1+x^3+x^8\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 28.50, size = 280, normalized size = 2.67
method | result | size |
risch | \(\frac {3 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}-x^{8}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+2 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} x -\left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{8}+1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}+x^{8}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-\left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} x -\left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{8}+1}\right )\) | \(280\) |
trager | \(\text {Expression too large to display}\) | \(632\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 6.54, size = 142, normalized size = 1.35 \begin {gather*} -\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {137873421075913623962723091849713877803864238548587911957688 \, \sqrt {3} {\left (x^{8} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 404258375252242985308203241426570926701619857965304026905546 \, \sqrt {3} {\left (x^{8} + x^{3} + 1\right )}^{\frac {2}{3}} x - \sqrt {3} {\left (82882407811392064917283059085655866224123024545593970500905 \, x^{8} + 133192477088164680672740074788428524448877809708358057473929 \, x^{3} + 82882407811392064917283059085655866224123024545593970500905\right )}}{3 \, {\left (260722961671046910462256771296925520157489755605248242108289 \, x^{8} + 271065898164078304635463166638142402252742048256945969431617 \, x^{3} + 260722961671046910462256771296925520157489755605248242108289\right )}}\right ) - x^{2} \log \left (\frac {x^{8} + 3 \, {\left (x^{8} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{8} + x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{8} + 1}\right ) - 3 \, {\left (x^{8} + x^{3} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (5 x^{8} - 3\right ) \left (x^{8} + x^{3} + 1\right )^{\frac {2}{3}}}{x^{3} \left (x^{8} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (5\,x^8-3\right )\,{\left (x^8+x^3+1\right )}^{2/3}}{x^3\,\left (x^8+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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