Optimal. Leaf size=110 \[ \frac {1}{4} \log \left (2 \sqrt [3]{x^3-x}+x-1\right )+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^3-x}}{\sqrt [3]{x^3-x}-x+1}\right )-\frac {1}{8} \log \left (4 \left (x^3-x\right )^{2/3}+(2-2 x) \sqrt [3]{x^3-x}+x^2-2 x+1\right ) \]
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Rubi [C] time = 0.21, antiderivative size = 220, normalized size of antiderivative = 2.00, number of steps used = 14, number of rules used = 13, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {2056, 959, 466, 465, 377, 200, 31, 634, 618, 204, 628, 511, 510} \begin {gather*} -\frac {9 \sqrt [3]{1-x^2} x^2 F_1\left (\frac {5}{6};1,\frac {1}{3};\frac {11}{6};9 x^2,x^2\right )}{5 \sqrt [3]{x^3-x}}-\frac {\sqrt [3]{x^2-1} \sqrt [3]{x} \log \left (\frac {4 x^{4/3}}{\left (x^2-1\right )^{2/3}}-\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1\right )}{8 \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x^2-1} \sqrt [3]{x} \log \left (\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1\right )}{4 \sqrt [3]{x^3-x}}-\frac {\sqrt {3} \sqrt [3]{x^2-1} \sqrt [3]{x} \tan ^{-1}\left (\frac {1-\frac {4 x^{2/3}}{\sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{4 \sqrt [3]{x^3-x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 31
Rule 200
Rule 204
Rule 377
Rule 465
Rule 466
Rule 510
Rule 511
Rule 618
Rule 628
Rule 634
Rule 959
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{(1+3 x) \sqrt [3]{-x+x^3}} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} (1+3 x) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x+x^3}}\\ &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \left (1-9 x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x+x^3}}-\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \int \frac {x^{2/3}}{\left (1-9 x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-9 x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}-\frac {\left (9 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1-9 x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}\\ &=-\frac {\left (9 \sqrt [3]{x} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1-9 x^6\right ) \sqrt [3]{1-x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-9 x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=-\frac {9 x^2 \sqrt [3]{1-x^2} F_1\left (\frac {5}{6};1,\frac {1}{3};\frac {11}{6};9 x^2,x^2\right )}{5 \sqrt [3]{-x+x^3}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+8 x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=-\frac {9 x^2 \sqrt [3]{1-x^2} F_1\left (\frac {5}{6};1,\frac {1}{3};\frac {11}{6};9 x^2,x^2\right )}{5 \sqrt [3]{-x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+2 x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{-x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {2-2 x}{1-2 x+4 x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=-\frac {9 x^2 \sqrt [3]{1-x^2} F_1\left (\frac {5}{6};1,\frac {1}{3};\frac {11}{6};9 x^2,x^2\right )}{5 \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \sqrt [3]{-x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {-2+8 x}{1-2 x+4 x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{8 \sqrt [3]{-x+x^3}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 x+4 x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \sqrt [3]{-x+x^3}}\\ &=-\frac {9 x^2 \sqrt [3]{1-x^2} F_1\left (\frac {5}{6};1,\frac {1}{3};\frac {11}{6};9 x^2,x^2\right )}{5 \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+\frac {4 x^{4/3}}{\left (-1+x^2\right )^{2/3}}-\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{8 \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \sqrt [3]{-x+x^3}}-\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,-2+\frac {8 x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=-\frac {9 x^2 \sqrt [3]{1-x^2} F_1\left (\frac {5}{6};1,\frac {1}{3};\frac {11}{6};9 x^2,x^2\right )}{5 \sqrt [3]{-x+x^3}}-\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {4 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+\frac {4 x^{4/3}}{\left (-1+x^2\right )^{2/3}}-\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{8 \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \sqrt [3]{-x+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.27, size = 81, normalized size = 0.74 \begin {gather*} \frac {3 \sqrt [3]{\frac {\frac {1}{x}-1}{\frac {1}{x}+3}} \sqrt [3]{\frac {\frac {1}{x}+1}{\frac {1}{x}+3}} x F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {2}{3+\frac {1}{x}},\frac {4}{3+\frac {1}{x}}\right )}{2 \sqrt [3]{x \left (x^2-1\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.29, size = 110, normalized size = 1.00 \begin {gather*} \frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-x+x^3}}{1-x+\sqrt [3]{-x+x^3}}\right )+\frac {1}{4} \log \left (-1+x+2 \sqrt [3]{-x+x^3}\right )-\frac {1}{8} \log \left (1-2 x+x^2+(2-2 x) \sqrt [3]{-x+x^3}+4 \left (-x+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 117, normalized size = 1.06 \begin {gather*} \frac {1}{4} \, \sqrt {3} \arctan \left (\frac {286273 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (635653 \, x^{2} + 434719 \, x + 66978\right )} + 539695 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{1293894 \, x^{2} + 1974837 \, x - 226981}\right ) + \frac {1}{8} \, \log \left (\frac {9 \, x^{2} + 6 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 6 \, x + 12 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1}{9 \, x^{2} + 6 \, x + 1}\right ) \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 {1}{{\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.84, size = 656, normalized size = 5.96
method | result | size |
trager | \(\frac {\ln \left (-\frac {7297600 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}-18973760 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x +7258272 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-19359420 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x +32081060 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+8757120 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+19359420 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}+40119368 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x -19359420 \left (x^{3}-x \right )^{\frac {2}{3}}-11494278 x \left (x^{3}-x \right )^{\frac {1}{3}}-10880597 x^{2}-2679436 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+11494278 \left (x^{3}-x \right )^{\frac {1}{3}}-10794014 x -28861}{\left (1+3 x \right )^{2}}\right )}{4}-\frac {\ln \left (\frac {11915360 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}-30979936 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x +7258272 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+22988556 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -47884574 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+14298432 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-22988556 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-77197364 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +22988556 \left (x^{3}-x \right )^{\frac {2}{3}}+9679710 x \left (x^{3}-x \right )^{\frac {1}{3}}-9726157 x^{2}+9770930 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-9679710 \left (x^{3}-x \right )^{\frac {1}{3}}-13795558 x +1356467}{\left (1+3 x \right )^{2}}\right )}{4}-\frac {\ln \left (\frac {11915360 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}-30979936 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x +7258272 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+22988556 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -47884574 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+14298432 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-22988556 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-77197364 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +22988556 \left (x^{3}-x \right )^{\frac {2}{3}}+9679710 x \left (x^{3}-x \right )^{\frac {1}{3}}-9726157 x^{2}+9770930 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-9679710 \left (x^{3}-x \right )^{\frac {1}{3}}-13795558 x +1356467}{\left (1+3 x \right )^{2}}\right ) \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{2}\) | \(656\) |
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 {1}{{\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x + 1\right )}}\,{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 = -0.01 \begin {gather*} \int \frac {1}{{\left (x^3-x\right )}^{1/3}\,\left (3\,x+1\right )} \,d x \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 \left (x - 1\right ) \left (x + 1\right )} \left (3 x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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