Optimal. Leaf size=81 \[ -\frac {3}{4} \log \left (\sqrt [3]{x^3-3 x^2+7 x-5}-x+1\right )+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {2 (x-1)}{\sqrt {3} \sqrt [3]{x^3-3 x^2+7 x-5}}+\frac {1}{\sqrt {3}}\right )+\frac {1}{4} \log (1-x) \]
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Rubi [A] time = 0.07, antiderivative size = 131, normalized size of antiderivative = 1.62, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2067, 2011, 329, 275, 239} \[ \frac {\sqrt {3} \sqrt [3]{(x-1)^2+4} \sqrt [3]{x-1} \tan ^{-1}\left (\frac {\frac {2 (x-1)^{2/3}}{\sqrt [3]{(x-1)^2+4}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{(x-1)^3+4 (x-1)}}-\frac {3 \sqrt [3]{(x-1)^2+4} \sqrt [3]{x-1} \log \left ((x-1)^{2/3}-\sqrt [3]{(x-1)^2+4}\right )}{4 \sqrt [3]{(x-1)^3+4 (x-1)}} \]
Antiderivative was successfully verified.
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Rule 239
Rule 275
Rule 329
Rule 2011
Rule 2067
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{-5+7 x-3 x^2+x^3}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{4 x+x^3}} \, dx,x,-1+x\right )\\ &=\frac {\left (\sqrt [3]{4+(-1+x)^2} \sqrt [3]{-1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{x} \sqrt [3]{4+x^2}} \, dx,x,-1+x\right )}{\sqrt [3]{4 (-1+x)+(-1+x)^3}}\\ &=\frac {\left (3 \sqrt [3]{4+(-1+x)^2} \sqrt [3]{-1+x}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{4+x^6}} \, dx,x,\sqrt [3]{-1+x}\right )}{\sqrt [3]{4 (-1+x)+(-1+x)^3}}\\ &=\frac {\left (3 \sqrt [3]{4+(-1+x)^2} \sqrt [3]{-1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{4+x^3}} \, dx,x,(-1+x)^{2/3}\right )}{2 \sqrt [3]{4 (-1+x)+(-1+x)^3}}\\ &=\frac {\sqrt {3} \sqrt [3]{4+(-1+x)^2} \sqrt [3]{-1+x} \tan ^{-1}\left (\frac {1+\frac {2 (-1+x)^{2/3}}{\sqrt [3]{4+(-1+x)^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-4 (1-x)+(-1+x)^3}}-\frac {3 \sqrt [3]{4+(-1+x)^2} \sqrt [3]{-1+x} \log \left (\sqrt [3]{4+(-1+x)^2}-(-1+x)^{2/3}\right )}{4 \sqrt [3]{-4 (1-x)+(-1+x)^3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 85, normalized size = 1.05 \[ \frac {3 \sqrt [3]{i x+(2-i)} \sqrt [3]{i (x-1)} (x-(1-2 i)) F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};-\frac {1}{4} i (x-(1-2 i)),-\frac {1}{2} i (x-(1-2 i))\right )}{4 \sqrt [3]{x^3-3 x^2+7 x-5}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.75, size = 120, normalized size = 1.48 \[ -\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {22791076 \, \sqrt {3} {\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (20389537 \, x^{2} - 40779074 \, x + 53222437\right )} + 17987998 \, \sqrt {3} {\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac {2}{3}}}{7204617 \, x^{2} - 14409234 \, x - 20666867}\right ) - \frac {1}{4} \, \log \left (3 \, {\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 3 \, {\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac {2}{3}} + 4\right ) \]
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 \[ \int \frac {1}{{\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.35, size = 653, normalized size = 8.06 \[ -\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-304 x^{2} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-320 x^{2} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+608 x \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+371 x^{2}-624 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+640 x \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+675 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x -742 x -624 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+624 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2356 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+675 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}}-675 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}}+1643\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-304 x^{2} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+928 x^{2} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+608 x \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-253 x^{2}+624 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1856 x \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+51 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x +506 x +624 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-624 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2356 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+51 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}}-51 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}}-713\right )}{2}+\frac {\ln \left (-304 x^{2} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-320 x^{2} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+608 x \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+371 x^{2}-624 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+640 x \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+675 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x -742 x -624 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+624 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2356 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+675 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}}-675 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}}+1643\right )}{2} \]
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 \[ \int \frac {1}{{\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (x^3-3\,x^2+7\,x-5\right )}^{1/3}} \,d x \]
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 \[ \int \frac {1}{\sqrt [3]{x^{3} - 3 x^{2} + 7 x - 5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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