Optimal. Leaf size=27 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {x^3+3 x-1}}\right )}{\sqrt {3}} \]
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Rubi [C] time = 3.94, antiderivative size = 1340, normalized size of antiderivative = 49.63, number of steps used = 11, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6742, 2066, 718, 419, 2080, 934, 169, 538, 537}
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Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 169
Rule 419
Rule 537
Rule 538
Rule 718
Rule 934
Rule 2066
Rule 2080
Rule 6742
Rubi steps
\begin {align*} \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx &=\int \left (\frac {1}{\sqrt {-1+3 x+x^3}}+\frac {3}{(-1+x) \sqrt {-1+3 x+x^3}}\right ) \, dx\\ &=3 \int \frac {1}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx+\int \frac {1}{\sqrt {-1+3 x+x^3}} \, dx\\ &=\frac {\left (\sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}\right ) \int \frac {1}{\sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}} \, dx}{\sqrt {-1+3 x+x^3}}+\frac {\left (3 \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}\right ) \int \frac {1}{(-1+x) \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}} \, dx}{\sqrt {-1+3 x+x^3}}\\ &=\frac {\left (3 \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}\right ) \int \frac {1}{(-1+x) \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x}} \, dx}{\sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {-1+3 x+x^3}}+\frac {\left (2 i \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )} \sqrt {\frac {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}{\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right )-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {-\frac {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}{\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right )^2-4 \left (1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}\right )}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {i \sqrt {6 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )} x^2}{\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right )-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}} \, dx,x,\sqrt {-\frac {i \left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x\right )}{\sqrt {6 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right )}{\sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {-1+3 x+x^3}}\\ &=\frac {2 i 2^{5/6} \sqrt {\frac {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}{\frac {6}{\sqrt [3]{1+\sqrt {5}}}-3 \sqrt [3]{2 \left (1+\sqrt {5}\right )}-i \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}-2 x\right )}{\sqrt {6 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right )|\frac {2 \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}{\frac {6 i}{\sqrt [3]{1+\sqrt {5}}}-3 i \sqrt [3]{2 \left (1+\sqrt {5}\right )}+\sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}\right )}{\sqrt {-1+3 x+x^3}}-\frac {\left (6 \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-x^2\right ) \sqrt {-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}+2 x^2} \sqrt {-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}+2 x^2}} \, dx,x,\sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}\right )}{\sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {-1+3 x+x^3}}\\ &=\frac {2 i 2^{5/6} \sqrt {\frac {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}{\frac {6}{\sqrt [3]{1+\sqrt {5}}}-3 \sqrt [3]{2 \left (1+\sqrt {5}\right )}-i \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}-2 x\right )}{\sqrt {6 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right )|\frac {2 \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}{\frac {6 i}{\sqrt [3]{1+\sqrt {5}}}-3 i \sqrt [3]{2 \left (1+\sqrt {5}\right )}+\sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}\right )}{\sqrt {-1+3 x+x^3}}-\frac {\left (6 \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {1+\frac {2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )}{-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-x^2\right ) \sqrt {-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}+2 x^2} \sqrt {1+\frac {2 x^2}{-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}}} \, dx,x,\sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}\right )}{\sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}+2 x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {-1+3 x+x^3}}\\ &=\frac {2 i 2^{5/6} \sqrt {\frac {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}{\frac {6}{\sqrt [3]{1+\sqrt {5}}}-3 \sqrt [3]{2 \left (1+\sqrt {5}\right )}-i \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}-2 x\right )}{\sqrt {6 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right )|\frac {2 \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}{\frac {6 i}{\sqrt [3]{1+\sqrt {5}}}-3 i \sqrt [3]{2 \left (1+\sqrt {5}\right )}+\sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}\right )}{\sqrt {-1+3 x+x^3}}-\frac {\left (6 \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {1+\frac {2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )}{-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}} \sqrt {1+\frac {2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )}{-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-x^2\right ) \sqrt {1+\frac {2 x^2}{-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}} \sqrt {1+\frac {2 x^2}{-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}}} \, dx,x,\sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}\right )}{\sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}+2 x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}+2 x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {-1+3 x+x^3}}\\ &=\frac {2 i 2^{5/6} \sqrt {\frac {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}{\frac {6}{\sqrt [3]{1+\sqrt {5}}}-3 \sqrt [3]{2 \left (1+\sqrt {5}\right )}-i \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}-2 x\right )}{\sqrt {6 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right )|\frac {2 \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}{\frac {6 i}{\sqrt [3]{1+\sqrt {5}}}-3 i \sqrt [3]{2 \left (1+\sqrt {5}\right )}+\sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}\right )}{\sqrt {-1+3 x+x^3}}-\frac {3 \sqrt [6]{\frac {2}{1+\sqrt {5}}} \sqrt {6-3 \sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}+i \sqrt [6]{2} \sqrt [3]{1+\sqrt {5}} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}} \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {1-\frac {2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )}{3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}} \sqrt {1-\frac {2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )}{3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}} \Pi \left (\frac {3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}{2 \left (1+\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right )};\sin ^{-1}\left (\frac {2^{5/6} \sqrt [6]{1+\sqrt {5}} \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}}{\sqrt {6-3 \sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}+i \sqrt [6]{2} \sqrt [3]{1+\sqrt {5}} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right )|\frac {3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}{3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}\right )}{\left (1+\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) \sqrt {-1+3 x+x^3}}\\ \end {align*}
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Mathematica [C] time = 1.06, size = 812, normalized size = 30.07 \begin {gather*} \frac {2 \sqrt {\frac {-x+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,1\right ]+1}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}} \left (\frac {3 \sqrt {-\frac {\left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]-1\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]-1\right )}{\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]\right )^2}} \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]\right ) \Pi \left (1-\frac {\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]};\sin ^{-1}\left (\sqrt {\frac {-x+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]+1}{-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}}\right )|\frac {\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}\right )}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}+\frac {F\left (\sin ^{-1}\left (\sqrt {\frac {-x+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]+1}{-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}}\right )|\frac {\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]-1\right ) \sqrt {\frac {-x+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]+1}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}}}{\sqrt {\frac {-x+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]+1}{-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}}}\right )}{\sqrt {x^3+3 x-1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.22, size = 27, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {x^3+3 x-1}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 97, normalized size = 3.59 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (\frac {x^{6} + 18 \, x^{5} + 15 \, x^{4} + 52 \, x^{3} - 4 \, \sqrt {3} {\left (x^{4} + 3 \, x^{3} + 3 \, x^{2} - x\right )} \sqrt {x^{3} + 3 \, x - 1} - 9 \, x^{2} - 6 \, x + 1}{x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, 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 {x + 2}{\sqrt {x^{3} + 3 \, x - 1} {\left (x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.42, size = 1075, normalized size = 39.81
result 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 + 2}{\sqrt {x^{3} + 3 \, x - 1} {\left (x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 1872, normalized size = 69.33
result too large to display
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 + 2}{\left (x - 1\right ) \sqrt {x^{3} + 3 x - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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