Optimal. Leaf size=49 \[ -\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^{12}+3 \text {$\#$1}^4-1\& ,\frac {\log \left (\sqrt [4]{x^5-x^3}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]
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Rubi [F] time = 180.00, 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*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{-1+x^2}\right ) \int \frac {1+x^6}{x^{3/4} \sqrt [4]{-1+x^2} \left (1+x^3-x^6\right )} \, dx}{\sqrt [4]{-x^3+x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1+x^{24}}{\sqrt [4]{-1+x^8} \left (1+x^{12}-x^{24}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt [4]{-1+x^8}}+\frac {2+x^{12}}{\sqrt [4]{-1+x^8} \left (1+x^{12}-x^{24}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}\\ &=-\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {2+x^{12}}{\sqrt [4]{-1+x^8} \left (1+x^{12}-x^{24}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}\\ &=-\frac {\left (4 x^{3/4} \sqrt [4]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1-\sqrt {5}}{\sqrt [4]{-1+x^8} \left (1-\sqrt {5}-2 x^{12}\right )}+\frac {1+\sqrt {5}}{\sqrt [4]{-1+x^8} \left (1+\sqrt {5}-2 x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}} \end {align*}
rest of steps removed due to Latex formating problem.
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Mathematica [F] time = 0.81, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.00, size = 49, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \text {RootSum}\left [-1+3 \text {$\#$1}^4+\text {$\#$1}^{12}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-x^3+x^5}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {x^{6} + 1}{{\left (x^{6} - x^{3} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 hanged
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^{6} - x^{3} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}}\,{d x} \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 = -0.02 \begin {gather*} \int \frac {x^6+1}{{\left (x^5-x^3\right )}^{1/4}\,\left (-x^6+x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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