Integrand size = 29, antiderivative size = 184 \[ \int \frac {1+x^2+x^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=\frac {\left (x^3+x^5\right )^{3/4}}{x^2 \left (1+x^2\right )}+\frac {3 \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{4 \sqrt [4]{2}}-\frac {3 \arctan \left (\frac {2^{3/4} x \sqrt [4]{x^3+x^5}}{\sqrt {2} x^2-\sqrt {x^3+x^5}}\right )}{4\ 2^{3/4}}+\frac {3 \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{4 \sqrt [4]{2}}+\frac {3 \text {arctanh}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^3+x^5}}{2^{3/4}}}{x \sqrt [4]{x^3+x^5}}\right )}{4\ 2^{3/4}} \]
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Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.42 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.51, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2081, 6847, 6857, 251, 1469, 541, 544, 440} \[ \int \frac {1+x^2+x^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=\frac {6 \sqrt [4]{x^2+1} x \operatorname {AppellF1}\left (\frac {1}{8},1,\frac {1}{4},\frac {9}{8},x^2,-x^2\right )}{\sqrt [4]{x^5+x^3}}-\frac {3 \sqrt [4]{x^2+1} x \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},-x^2\right )}{\sqrt [4]{x^5+x^3}}+\frac {x}{\sqrt [4]{x^5+x^3}} \]
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Rule 251
Rule 440
Rule 541
Rule 544
Rule 1469
Rule 2081
Rule 6847
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{3/4} \sqrt [4]{1+x^2}\right ) \int \frac {1+x^2+x^4}{x^{3/4} \sqrt [4]{1+x^2} \left (1-x^4\right )} \, dx}{\sqrt [4]{x^3+x^5}} \\ & = \frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1+x^8+x^{16}}{\sqrt [4]{1+x^8} \left (1-x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}} \\ & = \frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{\sqrt [4]{1+x^8}}+\frac {2+x^8}{\sqrt [4]{1+x^8} \left (1-x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}} \\ & = -\frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}+\frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {2+x^8}{\sqrt [4]{1+x^8} \left (1-x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}} \\ & = -\frac {4 x \sqrt [4]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},-x^2\right )}{\sqrt [4]{x^3+x^5}}+\frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {2+x^8}{\left (1-x^8\right ) \left (1+x^8\right )^{5/4}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}} \\ & = \frac {x}{\sqrt [4]{x^3+x^5}}-\frac {4 x \sqrt [4]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},-x^2\right )}{\sqrt [4]{x^3+x^5}}-\frac {\left (x^{3/4} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {-7+x^8}{\left (1-x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}} \\ & = \frac {x}{\sqrt [4]{x^3+x^5}}-\frac {4 x \sqrt [4]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},-x^2\right )}{\sqrt [4]{x^3+x^5}}+\frac {\left (x^{3/4} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}+\frac {\left (6 x^{3/4} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}} \\ & = \frac {x}{\sqrt [4]{x^3+x^5}}+\frac {6 x \sqrt [4]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{8},1,\frac {1}{4},\frac {9}{8},x^2,-x^2\right )}{\sqrt [4]{x^3+x^5}}-\frac {3 x \sqrt [4]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},-x^2\right )}{\sqrt [4]{x^3+x^5}} \\ \end{align*}
Time = 1.40 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.23 \[ \int \frac {1+x^2+x^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=\frac {x^{3/4} \left (8 \sqrt [4]{x}+3\ 2^{3/4} \sqrt [4]{1+x^2} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )-3 \sqrt [4]{2} \sqrt [4]{1+x^2} \arctan \left (\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {2} \sqrt {x}-\sqrt {1+x^2}}\right )+3\ 2^{3/4} \sqrt [4]{1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+3 \sqrt [4]{2} \sqrt [4]{1+x^2} \text {arctanh}\left (\frac {2 \sqrt [4]{2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{2 \sqrt {x}+\sqrt {2} \sqrt {1+x^2}}\right )\right )}{8 \sqrt [4]{x^3+x^5}} \]
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Time = 40.82 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.53
method | result | size |
pseudoelliptic | \(\frac {-6 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {3}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+3 \ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-3 \ln \left (\frac {-2^{\frac {3}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}{2^{\frac {3}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}\right ) 2^{\frac {1}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-6 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+x}{x}\right ) 2^{\frac {1}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-6 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-x}{x}\right ) 2^{\frac {1}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+16 x}{16 \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\) | \(282\) |
risch | \(\text {Expression too large to display}\) | \(730\) |
trager | \(\text {Expression too large to display}\) | \(746\) |
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Result contains complex when optimal does not.
Time = 8.31 (sec) , antiderivative size = 788, normalized size of antiderivative = 4.28 \[ \int \frac {1+x^2+x^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1+x^2+x^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=- \int \frac {x^{2}}{x^{4} \sqrt [4]{x^{5} + x^{3}} - \sqrt [4]{x^{5} + x^{3}}}\, dx - \int \frac {x^{4}}{x^{4} \sqrt [4]{x^{5} + x^{3}} - \sqrt [4]{x^{5} + x^{3}}}\, dx - \int \frac {1}{x^{4} \sqrt [4]{x^{5} + x^{3}} - \sqrt [4]{x^{5} + x^{3}}}\, dx \]
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\[ \int \frac {1+x^2+x^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=\int { -\frac {x^{4} + x^{2} + 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}} \,d x } \]
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\[ \int \frac {1+x^2+x^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=\int { -\frac {x^{4} + x^{2} + 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1+x^2+x^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx=\int -\frac {x^4+x^2+1}{{\left (x^5+x^3\right )}^{1/4}\,\left (x^4-1\right )} \,d x \]
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