Integrand size = 13, antiderivative size = 62 \[ \int x^{10} \sqrt [4]{1+x^4} \, dx=\frac {1}{384} \sqrt [4]{1+x^4} \left (-7 x^3+4 x^7+32 x^{11}\right )-\frac {7}{256} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {7}{256} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.31, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {285, 327, 338, 304, 209, 212} \[ \int x^{10} \sqrt [4]{1+x^4} \, dx=-\frac {7}{256} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {7}{256} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{12} \sqrt [4]{x^4+1} x^{11}+\frac {1}{96} \sqrt [4]{x^4+1} x^7-\frac {7}{384} \sqrt [4]{x^4+1} x^3 \]
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Rule 209
Rule 212
Rule 285
Rule 304
Rule 327
Rule 338
Rubi steps \begin{align*} \text {integral}& = \frac {1}{12} x^{11} \sqrt [4]{1+x^4}+\frac {1}{12} \int \frac {x^{10}}{\left (1+x^4\right )^{3/4}} \, dx \\ & = \frac {1}{96} x^7 \sqrt [4]{1+x^4}+\frac {1}{12} x^{11} \sqrt [4]{1+x^4}-\frac {7}{96} \int \frac {x^6}{\left (1+x^4\right )^{3/4}} \, dx \\ & = -\frac {7}{384} x^3 \sqrt [4]{1+x^4}+\frac {1}{96} x^7 \sqrt [4]{1+x^4}+\frac {1}{12} x^{11} \sqrt [4]{1+x^4}+\frac {7}{128} \int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx \\ & = -\frac {7}{384} x^3 \sqrt [4]{1+x^4}+\frac {1}{96} x^7 \sqrt [4]{1+x^4}+\frac {1}{12} x^{11} \sqrt [4]{1+x^4}+\frac {7}{128} \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ & = -\frac {7}{384} x^3 \sqrt [4]{1+x^4}+\frac {1}{96} x^7 \sqrt [4]{1+x^4}+\frac {1}{12} x^{11} \sqrt [4]{1+x^4}+\frac {7}{256} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {7}{256} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ & = -\frac {7}{384} x^3 \sqrt [4]{1+x^4}+\frac {1}{96} x^7 \sqrt [4]{1+x^4}+\frac {1}{12} x^{11} \sqrt [4]{1+x^4}-\frac {7}{256} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {7}{256} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int x^{10} \sqrt [4]{1+x^4} \, dx=\frac {1}{384} x^3 \sqrt [4]{1+x^4} \left (-7+4 x^4+32 x^8\right )-\frac {7}{256} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {7}{256} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.76 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.27
method | result | size |
meijerg | \(\frac {x^{11} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {11}{4}\right ], \left [\frac {15}{4}\right ], -x^{4}\right )}{11}\) | \(17\) |
risch | \(\frac {x^{3} \left (32 x^{8}+4 x^{4}-7\right ) \left (x^{4}+1\right )^{\frac {1}{4}}}{384}+\frac {7 x^{3} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -x^{4}\right )}{384}\) | \(42\) |
pseudoelliptic | \(\frac {\left (-128 x^{11}-16 x^{7}+28 x^{3}\right ) \left (x^{4}+1\right )^{\frac {1}{4}}+21 \ln \left (\frac {\left (x^{4}+1\right )^{\frac {1}{4}}-x}{x}\right )-42 \arctan \left (\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\right )-21 \ln \left (\frac {\left (x^{4}+1\right )^{\frac {1}{4}}+x}{x}\right )}{1536 {\left (-\left (x^{4}+1\right )^{\frac {1}{4}}+x \right )}^{3} \left (x^{2}+\sqrt {x^{4}+1}\right )^{3} {\left (\left (x^{4}+1\right )^{\frac {1}{4}}+x \right )}^{3}}\) | \(113\) |
trager | \(\frac {x^{3} \left (32 x^{8}+4 x^{4}-7\right ) \left (x^{4}+1\right )^{\frac {1}{4}}}{384}+\frac {7 \ln \left (-2 \left (x^{4}+1\right )^{\frac {3}{4}} x -2 \sqrt {x^{4}+1}\, x^{2}-2 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-2 x^{4}-1\right )}{512}+\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}+1\right )^{\frac {3}{4}} x -2 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{512}\) | \(139\) |
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Time = 0.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.21 \[ \int x^{10} \sqrt [4]{1+x^4} \, dx=\frac {1}{384} \, {\left (32 \, x^{11} + 4 \, x^{7} - 7 \, x^{3}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} + \frac {7}{256} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {7}{512} \, \log \left (\frac {x + {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {7}{512} \, \log \left (-\frac {x - {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) \]
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Result contains complex when optimal does not.
Time = 3.83 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.50 \[ \int x^{10} \sqrt [4]{1+x^4} \, dx=\frac {x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (50) = 100\).
Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.98 \[ \int x^{10} \sqrt [4]{1+x^4} \, dx=\frac {\frac {21 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} + \frac {18 \, {\left (x^{4} + 1\right )}^{\frac {5}{4}}}{x^{5}} - \frac {7 \, {\left (x^{4} + 1\right )}^{\frac {9}{4}}}{x^{9}}}{384 \, {\left (\frac {3 \, {\left (x^{4} + 1\right )}}{x^{4}} - \frac {3 \, {\left (x^{4} + 1\right )}^{2}}{x^{8}} + \frac {{\left (x^{4} + 1\right )}^{3}}{x^{12}} - 1\right )}} + \frac {7}{256} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {7}{512} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} + 1\right ) - \frac {7}{512} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (50) = 100\).
Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.68 \[ \int x^{10} \sqrt [4]{1+x^4} \, dx=\frac {1}{384} \, x^{12} {\left (\frac {18 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} + 1\right )}}{x} + \frac {21 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} - \frac {7 \, {\left (x^{8} + 2 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x^{9}}\right )} + \frac {7}{256} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {7}{512} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} + 1\right ) - \frac {7}{512} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} - 1\right ) \]
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Timed out. \[ \int x^{10} \sqrt [4]{1+x^4} \, dx=\int x^{10}\,{\left (x^4+1\right )}^{1/4} \,d x \]
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