Integrand size = 21, antiderivative size = 222 \[ \int \frac {2}{(3+x) \left (2-8 x+8 x^2\right )^{2/3}} \, dx=-\frac {3 \sqrt [3]{2} \sqrt [3]{1-4 x+4 x^2}}{7 (-1+2 x)}+\frac {1}{7} \sqrt [3]{\frac {2}{7}} \sqrt {3} \arctan \left (\frac {\sqrt {3} \left (1-4 x+4 x^2\right )^{2/3}}{2 \sqrt [3]{7}-4 \sqrt [3]{7} x+\left (1-4 x+4 x^2\right )^{2/3}}\right )-\frac {2}{21} \sqrt [3]{\frac {2}{7}} \log (-1+2 x)+\frac {1}{7} \sqrt [3]{\frac {2}{7}} \log \left (-\sqrt [3]{7}+2 \sqrt [3]{7} x+\left (1-4 x+4 x^2\right )^{2/3}\right )-\frac {\log \left (\sqrt [3]{7}-2 \sqrt [3]{7} x+7^{2/3} \sqrt [3]{1-4 x+4 x^2}+\left (1-4 x+4 x^2\right )^{2/3}\right )}{7\ 2^{2/3} \sqrt [3]{7}} \]
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Time = 0.07 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {12, 660, 53, 58, 631, 210, 31} \[ \int \frac {2}{(3+x) \left (2-8 x+8 x^2\right )^{2/3}} \, dx=\frac {\sqrt [3]{\frac {2}{7}} \sqrt {3} (2 x-1)^{4/3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{2 x-1}}{\sqrt {3} \sqrt [3]{7}}\right )}{7 \left (4 x^2-4 x+1\right )^{2/3}}+\frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (4 x^2-4 x+1\right )^{2/3}}-\frac {(2 x-1)^{4/3} \log (x+3)}{7\ 2^{2/3} \sqrt [3]{7} \left (4 x^2-4 x+1\right )^{2/3}}+\frac {3 (2 x-1)^{4/3} \log \left (\sqrt [3]{8 x-4}+2^{2/3} \sqrt [3]{7}\right )}{7\ 2^{2/3} \sqrt [3]{7} \left (4 x^2-4 x+1\right )^{2/3}} \]
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Rule 12
Rule 31
Rule 53
Rule 58
Rule 210
Rule 631
Rule 660
Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {1}{(3+x) \left (2-8 x+8 x^2\right )^{2/3}} \, dx \\ & = \frac {\left (2 (-4+8 x)^{4/3}\right ) \int \frac {1}{(3+x) (-4+8 x)^{4/3}} \, dx}{\left (2-8 x+8 x^2\right )^{2/3}} \\ & = \frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (1-4 x+4 x^2\right )^{2/3}}-\frac {(-4+8 x)^{4/3} \int \frac {1}{(3+x) \sqrt [3]{-4+8 x}} \, dx}{14 \left (2-8 x+8 x^2\right )^{2/3}} \\ & = \frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (1-4 x+4 x^2\right )^{2/3}}-\frac {(-1+2 x)^{4/3} \log (3+x)}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}-\frac {\left (3 (-4+8 x)^{4/3}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt [3]{2} 7^{2/3}-2^{2/3} \sqrt [3]{7} x+x^2} \, dx,x,\sqrt [3]{-4+8 x}\right )}{28 \left (2-8 x+8 x^2\right )^{2/3}}+\frac {\left (3 (-4+8 x)^{4/3}\right ) \text {Subst}\left (\int \frac {1}{2^{2/3} \sqrt [3]{7}+x} \, dx,x,\sqrt [3]{-4+8 x}\right )}{28\ 2^{2/3} \sqrt [3]{7} \left (2-8 x+8 x^2\right )^{2/3}} \\ & = \frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (1-4 x+4 x^2\right )^{2/3}}-\frac {(-1+2 x)^{4/3} \log (3+x)}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}+\frac {3 (-1+2 x)^{4/3} \log \left (2^{2/3} \sqrt [3]{7}+\sqrt [3]{-4+8 x}\right )}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}-\frac {\left (3 (-4+8 x)^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\sqrt [3]{\frac {2}{7}} \sqrt [3]{-4+8 x}\right )}{14\ 2^{2/3} \sqrt [3]{7} \left (2-8 x+8 x^2\right )^{2/3}} \\ & = \frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (1-4 x+4 x^2\right )^{2/3}}+\frac {\sqrt [3]{\frac {2}{7}} \sqrt {3} (-1+2 x)^{4/3} \arctan \left (\frac {7-2\ 7^{2/3} \sqrt [3]{-1+2 x}}{7 \sqrt {3}}\right )}{7 \left (1-4 x+4 x^2\right )^{2/3}}-\frac {(-1+2 x)^{4/3} \log (3+x)}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}+\frac {3 (-1+2 x)^{4/3} \log \left (2^{2/3} \sqrt [3]{7}+\sqrt [3]{-4+8 x}\right )}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.72 \[ \int \frac {2}{(3+x) \left (2-8 x+8 x^2\right )^{2/3}} \, dx=\frac {(-1+2 x) \left (-42+2 \sqrt {3} 7^{2/3} \sqrt [3]{-1+2 x} \arctan \left (\frac {7-2\ 7^{2/3} \sqrt [3]{-1+2 x}}{7 \sqrt {3}}\right )+2\ 7^{2/3} \sqrt [3]{-1+2 x} \log \left (7+7^{2/3} \sqrt [3]{-1+2 x}\right )-7^{2/3} \sqrt [3]{-1+2 x} \log \left (-7+7^{2/3} \sqrt [3]{-1+2 x}-\sqrt [3]{7} (-1+2 x)^{2/3}\right )\right )}{49\ 2^{2/3} \left ((1-2 x)^2\right )^{2/3}} \]
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Time = 1.71 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.52
method | result | size |
risch | \(-\frac {3 \left (-1+2 x \right ) 2^{\frac {1}{3}}}{7 \left (\left (-1+2 x \right )^{2}\right )^{\frac {2}{3}}}+\frac {\left (\frac {7^{\frac {2}{3}} \ln \left (\left (-1+2 x \right )^{\frac {1}{3}}+7^{\frac {1}{3}}\right )}{49}-\frac {7^{\frac {2}{3}} \ln \left (\left (-1+2 x \right )^{\frac {2}{3}}-7^{\frac {1}{3}} \left (-1+2 x \right )^{\frac {1}{3}}+7^{\frac {2}{3}}\right )}{98}-\frac {\sqrt {3}\, 7^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,7^{\frac {2}{3}} \left (-1+2 x \right )^{\frac {1}{3}}}{7}-1\right )}{3}\right )}{49}\right ) 2^{\frac {1}{3}} \left (-1+2 x \right )^{\frac {4}{3}}}{\left (\left (-1+2 x \right )^{2}\right )^{\frac {2}{3}}}\) | \(116\) |
trager | \(\text {Expression too large to display}\) | \(1509\) |
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Time = 0.26 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.99 \[ \int \frac {2}{(3+x) \left (2-8 x+8 x^2\right )^{2/3}} \, dx=\frac {2 \cdot 7^{\frac {2}{3}} \sqrt {3} 2^{\frac {1}{3}} {\left (2 \, x - 1\right )} \arctan \left (-\frac {7^{\frac {1}{6}} \sqrt {3} {\left (7^{\frac {5}{6}} {\left (2 \, x - 1\right )} - 7 \cdot 7^{\frac {1}{6}} 2^{\frac {2}{3}} {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}}\right )}}{21 \, {\left (2 \, x - 1\right )}}\right ) - 7^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (2 \, x - 1\right )} \log \left (-\frac {7^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}} {\left (2 \, x - 1\right )} - 7^{\frac {1}{3}} 2^{\frac {2}{3}} {\left (4 \, x^{2} - 4 \, x + 1\right )} - 7 \, {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {2}{3}}}{4 \, x^{2} - 4 \, x + 1}\right ) + 2 \cdot 7^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (2 \, x - 1\right )} \log \left (\frac {7^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (2 \, x - 1\right )} + 7 \, {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}}}{2 \, x - 1}\right ) - 42 \, {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}}}{98 \, {\left (2 \, x - 1\right )}} \]
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\[ \int \frac {2}{(3+x) \left (2-8 x+8 x^2\right )^{2/3}} \, dx=\sqrt [3]{2} \int \frac {1}{x \left (4 x^{2} - 4 x + 1\right )^{\frac {2}{3}} + 3 \left (4 x^{2} - 4 x + 1\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {2}{(3+x) \left (2-8 x+8 x^2\right )^{2/3}} \, dx=\int { \frac {2}{{\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {2}{3}} {\left (x + 3\right )}} \,d x } \]
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\[ \int \frac {2}{(3+x) \left (2-8 x+8 x^2\right )^{2/3}} \, dx=\int { \frac {2}{{\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {2}{3}} {\left (x + 3\right )}} \,d x } \]
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Timed out. \[ \int \frac {2}{(3+x) \left (2-8 x+8 x^2\right )^{2/3}} \, dx=\int \frac {2}{\left (x+3\right )\,{\left (8\,x^2-8\,x+2\right )}^{2/3}} \,d x \]
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