Optimal. Leaf size=468 \[ \frac {\sqrt [3]{\frac {1}{2} \left (9+5 \sqrt {3}\right )} \log \left (-6 \sqrt [3]{x^2+1}+2^{2/3} \sqrt {3} x+3\ 2^{2/3}\right )}{6\ 3^{2/3}}+\frac {\sqrt [3]{\frac {1}{2} \left (9-5 \sqrt {3}\right )} \log \left (6 \sqrt [3]{x^2+1}+2^{2/3} \sqrt {3} x-3\ 2^{2/3}\right )}{6\ 3^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (9-5 \sqrt {3}\right )} \log \left (-\sqrt [3]{2} x^2+2^{2/3} \sqrt {3} \sqrt [3]{x^2+1} x-6 \left (x^2+1\right )^{2/3}-3\ 2^{2/3} \sqrt [3]{x^2+1}+2 \sqrt [3]{2} \sqrt {3} x-3 \sqrt [3]{2}\right )}{12\ 3^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (9+5 \sqrt {3}\right )} \log \left (\sqrt [3]{2} x^2+2^{2/3} \sqrt {3} \sqrt [3]{x^2+1} x+6 \left (x^2+1\right )^{2/3}+3\ 2^{2/3} \sqrt [3]{x^2+1}+2 \sqrt [3]{2} \sqrt {3} x+3 \sqrt [3]{2}\right )}{12\ 3^{2/3}}+\frac {1}{6} \sqrt [3]{\frac {1}{2} \left (3 \sqrt {3}-5\right )} \tan ^{-1}\left (\frac {-\frac {\sqrt [3]{x^2+1}}{\sqrt {3}}+\frac {1}{3} 2^{2/3} x-\frac {2^{2/3}}{\sqrt {3}}}{\sqrt [3]{x^2+1}}\right )-\frac {1}{6} \sqrt [3]{\frac {1}{2} \left (5+3 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+1}}{\sqrt {3}}+\frac {1}{3} 2^{2/3} x+\frac {2^{2/3}}{\sqrt {3}}}{\sqrt [3]{x^2+1}}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 191, normalized size of antiderivative = 0.41, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1010, 392, 444, 55, 617, 204, 31} \begin {gather*} -\frac {\log \left (3-x^2\right )}{4\ 2^{2/3}}+\frac {3 \log \left (2^{2/3}-\sqrt [3]{x^2+1}\right )}{4\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{2} \sqrt [3]{x^2+1}+1}{\sqrt {3}}\right )}{2\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}(x)}{6\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 55
Rule 204
Rule 392
Rule 444
Rule 617
Rule 1010
Rubi steps
\begin {align*} \int \frac {1+x}{\left (-3+x^2\right ) \sqrt [3]{1+x^2}} \, dx &=\int \frac {1}{\left (-3+x^2\right ) \sqrt [3]{1+x^2}} \, dx+\int \frac {x}{\left (-3+x^2\right ) \sqrt [3]{1+x^2}} \, dx\\ &=\frac {\tan ^{-1}(x)}{6\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(-3+x) \sqrt [3]{1+x}} \, dx,x,x^2\right )\\ &=\frac {\tan ^{-1}(x)}{6\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\log \left (3-x^2\right )}{4\ 2^{2/3}}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\sqrt [3]{1+x^2}\right )-\frac {3 \operatorname {Subst}\left (\int \frac {1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1+x^2}\right )}{4\ 2^{2/3}}\\ &=\frac {\tan ^{-1}(x)}{6\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\log \left (3-x^2\right )}{4\ 2^{2/3}}+\frac {3 \log \left (2^{2/3}-\sqrt [3]{1+x^2}\right )}{4\ 2^{2/3}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{2\ 2^{2/3}}\\ &=\frac {\tan ^{-1}(x)}{6\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt [3]{2} \sqrt [3]{1+x^2}}{\sqrt {3}}\right )}{2\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\log \left (3-x^2\right )}{4\ 2^{2/3}}+\frac {3 \log \left (2^{2/3}-\sqrt [3]{1+x^2}\right )}{4\ 2^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.27, size = 151, normalized size = 0.32 \begin {gather*} \frac {9 x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-x^2,\frac {x^2}{3}\right )}{\left (x^2-3\right ) \sqrt [3]{x^2+1} \left (2 x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};-x^2,\frac {x^2}{3}\right )-F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};-x^2,\frac {x^2}{3}\right )\right )+9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-x^2,\frac {x^2}{3}\right )\right )}-\frac {1}{6} x^2 F_1\left (1;\frac {1}{3},1;2;-x^2,\frac {x^2}{3}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 6.42, size = 468, normalized size = 1.00 \begin {gather*} \frac {1}{6} \sqrt [3]{\frac {1}{2} \left (-5+3 \sqrt {3}\right )} \tan ^{-1}\left (\frac {-\frac {2^{2/3}}{\sqrt {3}}+\frac {1}{3} 2^{2/3} x-\frac {\sqrt [3]{1+x^2}}{\sqrt {3}}}{\sqrt [3]{1+x^2}}\right )-\frac {1}{6} \sqrt [3]{\frac {1}{2} \left (5+3 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\frac {2^{2/3}}{\sqrt {3}}+\frac {1}{3} 2^{2/3} x+\frac {\sqrt [3]{1+x^2}}{\sqrt {3}}}{\sqrt [3]{1+x^2}}\right )+\frac {\sqrt [3]{\frac {1}{2} \left (9+5 \sqrt {3}\right )} \log \left (3\ 2^{2/3}+2^{2/3} \sqrt {3} x-6 \sqrt [3]{1+x^2}\right )}{6\ 3^{2/3}}+\frac {\sqrt [3]{\frac {1}{2} \left (9-5 \sqrt {3}\right )} \log \left (-3 2^{2/3}+2^{2/3} \sqrt {3} x+6 \sqrt [3]{1+x^2}\right )}{6\ 3^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (9-5 \sqrt {3}\right )} \log \left (-3 \sqrt [3]{2}+2 \sqrt [3]{2} \sqrt {3} x-\sqrt [3]{2} x^2-3\ 2^{2/3} \sqrt [3]{1+x^2}+2^{2/3} \sqrt {3} x \sqrt [3]{1+x^2}-6 \left (1+x^2\right )^{2/3}\right )}{12\ 3^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (9+5 \sqrt {3}\right )} \log \left (3 \sqrt [3]{2}+2 \sqrt [3]{2} \sqrt {3} x+\sqrt [3]{2} x^2+3\ 2^{2/3} \sqrt [3]{1+x^2}+2^{2/3} \sqrt {3} x \sqrt [3]{1+x^2}+6 \left (1+x^2\right )^{2/3}\right )}{12\ 3^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 + 1}{{\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 3\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 117.77, size = 17124, normalized size = 36.59 \[\text {output too large to display}\]
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 + 1}{{\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 3\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 = -0.00 \begin {gather*} \int \frac {x+1}{{\left (x^2+1\right )}^{1/3}\,\left (x^2-3\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 {x + 1}{\left (x^{2} - 3\right ) \sqrt [3]{x^{2} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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