Integrand size = 42, antiderivative size = 146 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx=\frac {3 \left (1+x^4\right )^{2/3} \left (8+15 x^3+8 x^4\right )}{80 x^5}-\frac {3 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2\ 2^{2/3} \sqrt [3]{1+x^4}}\right )}{16 \sqrt [3]{2}}+\frac {3 \log \left (-x+2^{2/3} \sqrt [3]{1+x^4}\right )}{16 \sqrt [3]{2}}-\frac {3 \log \left (x^2+2^{2/3} x \sqrt [3]{1+x^4}+2 \sqrt [3]{2} \left (1+x^4\right )^{2/3}\right )}{32 \sqrt [3]{2}} \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx=\int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \left (1+x^4\right )^{2/3}}{2 x^6}-\frac {9 \left (1+x^4\right )^{2/3}}{8 x^3}+\frac {\left (1+x^4\right )^{2/3}}{2 x^2}+\frac {3 (-3+16 x) \left (1+x^4\right )^{2/3}}{8 \left (4-x^3+4 x^4\right )}\right ) \, dx \\ & = \frac {3}{8} \int \frac {(-3+16 x) \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx+\frac {1}{2} \int \frac {\left (1+x^4\right )^{2/3}}{x^2} \, dx-\frac {9}{8} \int \frac {\left (1+x^4\right )^{2/3}}{x^3} \, dx-\frac {3}{2} \int \frac {\left (1+x^4\right )^{2/3}}{x^6} \, dx \\ & = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{10 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{2 x}+\frac {3}{8} \int \left (-\frac {3 \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4}+\frac {16 x \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4}\right ) \, dx-\frac {9}{16} \text {Subst}\left (\int \frac {\left (1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right ) \\ & = \frac {9 \left (1+x^4\right )^{2/3}}{16 x^2}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{10 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{2 x}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^2}} \, dx,x,x^2\right )-\frac {9}{8} \int \frac {\left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx+6 \int \frac {x \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx \\ & = \frac {9 \left (1+x^4\right )^{2/3}}{16 x^2}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{10 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{2 x}-\frac {9}{8} \int \frac {\left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx+6 \int \frac {x \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx-\frac {\left (9 \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{8 x^2} \\ & = \frac {9 \left (1+x^4\right )^{2/3}}{16 x^2}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{10 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{2 x}-\frac {9}{8} \int \frac {\left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx+6 \int \frac {x \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx+\frac {\left (9 \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{8 x^2}-\frac {\left (9 \left (1+\sqrt {3}\right ) \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{8 x^2} \\ & = \frac {9 \left (1+x^4\right )^{2/3}}{16 x^2}+\frac {9 x^2}{4 \left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )}-\frac {9 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1+x^4}\right ) \sqrt {\frac {1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1+x^4}}{1-\sqrt {3}-\sqrt [3]{1+x^4}}\right )|-7+4 \sqrt {3}\right )}{8 x^2 \sqrt {-\frac {1-\sqrt [3]{1+x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}}}+\frac {3\ 3^{3/4} \left (1-\sqrt [3]{1+x^4}\right ) \sqrt {\frac {1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1+x^4}}{1-\sqrt {3}-\sqrt [3]{1+x^4}}\right ),-7+4 \sqrt {3}\right )}{2 \sqrt {2} x^2 \sqrt {-\frac {1-\sqrt [3]{1+x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}}}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{10 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{2 x}-\frac {9}{8} \int \frac {\left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx+6 \int \frac {x \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx \\ \end{align*}
Time = 2.63 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx=\frac {3}{320} \left (\frac {4 \left (1+x^4\right )^{2/3} \left (8+15 x^3+8 x^4\right )}{x^5}-10\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2\ 2^{2/3} \sqrt [3]{1+x^4}}\right )+10\ 2^{2/3} \log \left (-x+2^{2/3} \sqrt [3]{1+x^4}\right )-5\ 2^{2/3} \log \left (x^2+2^{2/3} x \sqrt [3]{1+x^4}+2 \sqrt [3]{2} \left (1+x^4\right )^{2/3}\right )\right ) \]
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Time = 104.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(\frac {\frac {3 x^{5} \left (2 \arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {2}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+2 \ln \left (\frac {-2^{\frac {1}{3}} x +2 \left (x^{4}+1\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2 \,2^{\frac {1}{3}} x \left (x^{4}+1\right )^{\frac {1}{3}}+4 \left (x^{4}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right ) 2^{\frac {2}{3}}}{64}+\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}} \left (8 x^{4}+15 x^{3}+8\right )}{80}}{x^{5}}\) | \(125\) |
risch | \(\text {Expression too large to display}\) | \(812\) |
trager | \(\text {Expression too large to display}\) | \(996\) |
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Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (110) = 220\).
Time = 70.79 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.81 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx=-\frac {10 \, \sqrt {3} 2^{\frac {2}{3}} x^{5} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (64 \, x^{12} + 240 \, x^{11} + 48 \, x^{10} - x^{9} + 192 \, x^{8} + 480 \, x^{7} + 48 \, x^{6} + 192 \, x^{4} + 240 \, x^{3} + 64\right )} + 12 \, \sqrt {2} {\left (16 \, x^{10} + 28 \, x^{9} + x^{8} + 32 \, x^{6} + 28 \, x^{5} + 16 \, x^{2}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 48 \cdot 2^{\frac {1}{6}} {\left (8 \, x^{9} + 2 \, x^{8} - x^{7} + 16 \, x^{5} + 2 \, x^{4} + 8 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (64 \, x^{12} - 48 \, x^{11} - 96 \, x^{10} - x^{9} + 192 \, x^{8} - 96 \, x^{7} - 96 \, x^{6} + 192 \, x^{4} - 48 \, x^{3} + 64\right )}}\right ) - 10 \cdot 2^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (4 \, x^{4} - x^{3} + 4\right )} - 12 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x}{4 \, x^{4} - x^{3} + 4}\right ) + 5 \cdot 2^{\frac {2}{3}} x^{5} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (2 \, x^{5} + x^{4} + 2 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (16 \, x^{8} + 28 \, x^{7} + x^{6} + 32 \, x^{4} + 28 \, x^{3} + 16\right )} + 6 \, {\left (8 \, x^{6} + x^{5} + 8 \, x^{2}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{16 \, x^{8} - 8 \, x^{7} + x^{6} + 32 \, x^{4} - 8 \, x^{3} + 16}\right ) - 12 \, {\left (8 \, x^{4} + 15 \, x^{3} + 8\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}}}{320 \, x^{5}} \]
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Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + x^{3} + 2\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (4 \, x^{4} - x^{3} + 4\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + x^{3} + 2\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (4 \, x^{4} - x^{3} + 4\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx=\int \frac {{\left (x^4+1\right )}^{2/3}\,\left (x^4-3\right )\,\left (2\,x^4+x^3+2\right )}{x^6\,\left (4\,x^4-x^3+4\right )} \,d x \]
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