Optimal. Leaf size=81 \[ \frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 (-1+x)}{\sqrt {3} \sqrt [3]{-5+7 x-3 x^2+x^3}}\right )+\frac {1}{4} \log (1-x)-\frac {3}{4} \log \left (1-x+\sqrt [3]{-5+7 x-3 x^2+x^3}\right ) \]
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Rubi [A]
time = 0.05, 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 = {2092, 2036,
335, 281, 245} \begin {gather*} \frac {\sqrt {3} \sqrt [3]{(x-1)^2+4} \sqrt [3]{x-1} \text {ArcTan}\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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 281
Rule 335
Rule 2036
Rule 2092
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{-5+7 x-3 x^2+x^3}} \, dx &=\text {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 ) \text {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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 0.01, size = 85, normalized size = 1.05 \begin {gather*} \frac {3 \sqrt [3]{(2-i)+i x} \sqrt [3]{i (-1+x)} ((-1+2 i)+x) F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};-\frac {1}{4} i ((-1+2 i)+x),-\frac {1}{2} i ((-1+2 i)+x)\right )}{4 \sqrt [3]{-5+7 x-3 x^2+x^3}} \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 = 0.90, size = 433, normalized size = 5.35
method | result | size |
trager | \(-\frac {\ln \left (92 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+624 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}}-675 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x -184 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -41 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+51 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}}+675 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}}+624 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x +82 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -583 x^{2}-624 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}}-713 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1166 x -1643\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (212 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-624 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}}-51 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x -424 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +463 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+675 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {2}{3}}+51 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}}-624 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}} x -926 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +161 x^{2}+624 \left (x^{3}-3 x^{2}+7 x -5\right )^{\frac {1}{3}}+1643 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-322 x +713\right )}{2}\) | \(433\) |
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 = 1.10, size = 120, normalized size = 1.48 \begin {gather*} -\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 ) \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 {1}{\sqrt [3]{x^{3} - 3 x^{2} + 7 x - 5}}\, 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 {1}{{\left (x^3-3\,x^2+7\,x-5\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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