Optimal. Leaf size=37 \[ -\frac {4 \left (x^4-x\right )^{3/4} \left (53 x^6-8 x^3-3\right )}{63 x^6 \left (x^3-1\right )} \]
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Rubi [A] time = 0.17, antiderivative size = 53, normalized size of antiderivative = 1.43, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2056, 1487, 453, 271, 264} \begin {gather*} -\frac {212 x}{63 \sqrt [4]{x^4-x}}+\frac {4}{21 \sqrt [4]{x^4-x} x^5}+\frac {32}{63 \sqrt [4]{x^4-x} x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rule 453
Rule 1487
Rule 2056
Rubi steps
\begin {align*} \int \frac {1+x^6}{x^6 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1+x^6}{x^{25/4} \left (-1+x^3\right )^{5/4}} \, dx}{\sqrt [4]{-x+x^4}}\\ &=-\frac {1}{3 x^2 \sqrt [4]{-x+x^4}}-\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {-3-\frac {9 x^3}{4}}{x^{25/4} \left (-1+x^3\right )^{5/4}} \, dx}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {4}{21 x^5 \sqrt [4]{-x+x^4}}-\frac {1}{3 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (53 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{x^{13/4} \left (-1+x^3\right )^{5/4}} \, dx}{28 \sqrt [4]{-x+x^4}}\\ &=\frac {4}{21 x^5 \sqrt [4]{-x+x^4}}+\frac {32}{63 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (53 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{\sqrt [4]{x} \left (-1+x^3\right )^{5/4}} \, dx}{21 \sqrt [4]{-x+x^4}}\\ &=\frac {4}{21 x^5 \sqrt [4]{-x+x^4}}+\frac {32}{63 x^2 \sqrt [4]{-x+x^4}}-\frac {212 x}{63 \sqrt [4]{-x+x^4}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 30, normalized size = 0.81 \begin {gather*} \frac {4 \left (-53 x^6+8 x^3+3\right )}{63 x^5 \sqrt [4]{x \left (x^3-1\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.49, size = 37, normalized size = 1.00 \begin {gather*} -\frac {4 \left (x^4-x\right )^{3/4} \left (53 x^6-8 x^3-3\right )}{63 x^6 \left (x^3-1\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 34, normalized size = 0.92 \begin {gather*} -\frac {4 \, {\left (53 \, x^{6} - 8 \, x^{3} - 3\right )} {\left (x^{4} - x\right )}^{\frac {3}{4}}}{63 \, {\left (x^{9} - x^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 34, normalized size = 0.92 \begin {gather*} -\frac {4}{21} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {7}{4}} + \frac {8}{9} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {3}{4}} + \frac {8}{3 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 0.73 \begin {gather*} -\frac {4 \left (53 x^{6}-8 x^{3}-3\right )}{63 \left (x^{4}-x \right )^{\frac {1}{4}} x^{5}} \end {gather*}
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*} \int \frac {x^{6} + 1}{{\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 33, normalized size = 0.89 \begin {gather*} \frac {4\,{\left (x^4-x\right )}^{3/4}\,\left (-53\,x^6+8\,x^3+3\right )}{63\,x^6\,\left (x^3-1\right )} \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 (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}{x^{6} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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