Optimal. Leaf size=499 \[ \frac {(-1)^{2/3} \sqrt [3]{-54+35 i \sqrt {3}} \log \left (6 \sqrt [3]{x^2-1}-i 2^{2/3} \sqrt {3} x+3\ 2^{2/3}\right )}{6\ 6^{2/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-54-35 i \sqrt {3}} \log \left (6 \sqrt [3]{x^2-1}+i 2^{2/3} \sqrt {3} x+3\ 2^{2/3}\right )}{6\ 6^{2/3}}+\frac {\sqrt [3]{54+35 i \sqrt {3}} \log \left (-\sqrt [3]{2} x^2-i 2^{2/3} \sqrt {3} \sqrt [3]{x^2-1} x+6 \left (x^2-1\right )^{2/3}-3\ 2^{2/3} \sqrt [3]{x^2-1}+2 i \sqrt [3]{2} \sqrt {3} x+3 \sqrt [3]{2}\right )}{12\ 6^{2/3}}+\frac {\sqrt [3]{54-35 i \sqrt {3}} \log \left (-\sqrt [3]{2} x^2+i 2^{2/3} \sqrt {3} \sqrt [3]{x^2-1} x+6 \left (x^2-1\right )^{2/3}-3\ 2^{2/3} \sqrt [3]{x^2-1}-2 i \sqrt [3]{2} \sqrt {3} x+3 \sqrt [3]{2}\right )}{12\ 6^{2/3}}-\frac {\sqrt [6]{-253+1260 i \sqrt {3}} \tan ^{-1}\left (\frac {3 \sqrt [3]{x^2-1}}{\sqrt {3} \sqrt [3]{x^2-1}-i 2^{2/3} x-2^{2/3} \sqrt {3}}\right )}{6\ 2^{2/3}}-\frac {\sqrt [6]{-253-1260 i \sqrt {3}} \tan ^{-1}\left (\frac {3 \sqrt [3]{x^2-1}}{\sqrt {3} \sqrt [3]{x^2-1}+i 2^{2/3} x-2^{2/3} \sqrt {3}}\right )}{6\ 2^{2/3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 213, normalized size of antiderivative = 0.43, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1010, 393, 444, 56, 617, 204, 31} \begin {gather*} \frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}-\frac {3 \log \left (\sqrt [3]{x^2-1}+2^{2/3}\right )}{2\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{x^2-1}}{\sqrt {3}}\right )}{2^{2/3}}-\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{x^2-1}+1\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{2} \sqrt [3]{x^2-1}+\sqrt [3]{-1}}\right )-\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 56
Rule 204
Rule 393
Rule 444
Rule 617
Rule 1010
Rubi steps
\begin {align*} \int \frac {1+2 x}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx &=2 \int \frac {x}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx+\int \frac {1}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx\\ &=-\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1+(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{-1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}(x)-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{-1}+\sqrt [3]{2} \sqrt [3]{-1+x^2}}\right )+\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} (3+x)} \, dx,x,x^2\right )\\ &=-\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1+(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{-1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}(x)-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{-1}+\sqrt [3]{2} \sqrt [3]{-1+x^2}}\right )+\frac {\log \left (3+x^2\right )}{2\ 2^{2/3}}+\frac {3}{2} \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 )}{2\ 2^{2/3}}\\ &=-\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1+(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{-1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}(x)-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{-1}+\sqrt [3]{2} \sqrt [3]{-1+x^2}}\right )+\frac {\log \left (3+x^2\right )}{2\ 2^{2/3}}-\frac {3 \log \left (2^{2/3}+\sqrt [3]{-1+x^2}\right )}{2\ 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^{2/3}}\\ &=-\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{-1+x^2}}{\sqrt {3}}\right )}{2^{2/3}}-\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1+(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{-1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}(x)-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{-1}+\sqrt [3]{2} \sqrt [3]{-1+x^2}}\right )+\frac {\log \left (3+x^2\right )}{2\ 2^{2/3}}-\frac {3 \log \left (2^{2/3}+\sqrt [3]{-1+x^2}\right )}{2\ 2^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.17, size = 151, normalized size = 0.30 \begin {gather*} \frac {x \left (x \sqrt [3]{1-x^2} F_1\left (1;\frac {1}{3},1;2;x^2,-\frac {x^2}{3}\right )-\frac {27 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 ) \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 )}\right )}{3 \sqrt [3]{x^2-1}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 7.99, size = 499, normalized size = 1.00 \begin {gather*} \frac {(-1)^{2/3} \sqrt [3]{-54+35 i \sqrt {3}} \log \left (6 \sqrt [3]{x^2-1}-i 2^{2/3} \sqrt {3} x+3\ 2^{2/3}\right )}{6\ 6^{2/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-54-35 i \sqrt {3}} \log \left (6 \sqrt [3]{x^2-1}+i 2^{2/3} \sqrt {3} x+3\ 2^{2/3}\right )}{6\ 6^{2/3}}+\frac {\sqrt [3]{54+35 i \sqrt {3}} \log \left (-\sqrt [3]{2} x^2-i 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 i \sqrt [3]{2} \sqrt {3} x+3 \sqrt [3]{2}\right )}{12\ 6^{2/3}}+\frac {\sqrt [3]{54-35 i \sqrt {3}} \log \left (-\sqrt [3]{2} x^2+i 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 i \sqrt [3]{2} \sqrt {3} x+3 \sqrt [3]{2}\right )}{12\ 6^{2/3}}-\frac {\sqrt [6]{-253+1260 i \sqrt {3}} \tan ^{-1}\left (\frac {3 \sqrt [3]{x^2-1}}{\sqrt {3} \sqrt [3]{x^2-1}-i 2^{2/3} x-2^{2/3} \sqrt {3}}\right )}{6\ 2^{2/3}}-\frac {\sqrt [6]{-253-1260 i \sqrt {3}} \tan ^{-1}\left (\frac {3 \sqrt [3]{x^2-1}}{\sqrt {3} \sqrt [3]{x^2-1}+i 2^{2/3} x-2^{2/3} \sqrt {3}}\right )}{6\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x + 1}{{\left (x^{2} + 3\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+2 x}{\left (x^{2}-1\right )^{\frac {1}{3}} \left (x^{2}+3\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x + 1}{{\left (x^{2} + 3\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {2\,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.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x + 1}{\sqrt [3]{\left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________