Integrand size = 13, antiderivative size = 83 \[ \int \frac {x^{13}}{\left (a^4+x^4\right )^5} \, dx=-\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}-\frac {5 x^2}{128 \left (a^4+x^4\right )^2}+\frac {5 x^2}{256 a^4 \left (a^4+x^4\right )}+\frac {5 \arctan \left (\frac {x^2}{a^2}\right )}{256 a^6} \]
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Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {281, 294, 205, 209} \[ \int \frac {x^{13}}{\left (a^4+x^4\right )^5} \, dx=-\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}+\frac {5 x^2}{256 a^4 \left (a^4+x^4\right )}-\frac {5 x^2}{128 \left (a^4+x^4\right )^2}+\frac {5 \arctan \left (\frac {x^2}{a^2}\right )}{256 a^6} \]
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Rule 205
Rule 209
Rule 281
Rule 294
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^6}{\left (a^4+x^2\right )^5} \, dx,x,x^2\right ) \\ & = -\frac {x^{10}}{16 \left (a^4+x^4\right )^4}+\frac {5}{16} \text {Subst}\left (\int \frac {x^4}{\left (a^4+x^2\right )^4} \, dx,x,x^2\right ) \\ & = -\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}+\frac {5}{32} \text {Subst}\left (\int \frac {x^2}{\left (a^4+x^2\right )^3} \, dx,x,x^2\right ) \\ & = -\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}-\frac {5 x^2}{128 \left (a^4+x^4\right )^2}+\frac {5}{128} \text {Subst}\left (\int \frac {1}{\left (a^4+x^2\right )^2} \, dx,x,x^2\right ) \\ & = -\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}-\frac {5 x^2}{128 \left (a^4+x^4\right )^2}+\frac {5 x^2}{256 a^4 \left (a^4+x^4\right )}+\frac {5 \text {Subst}\left (\int \frac {1}{a^4+x^2} \, dx,x,x^2\right )}{256 a^4} \\ & = -\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}-\frac {5 x^2}{128 \left (a^4+x^4\right )^2}+\frac {5 x^2}{256 a^4 \left (a^4+x^4\right )}+\frac {5 \arctan \left (\frac {x^2}{a^2}\right )}{256 a^6} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.75 \[ \int \frac {x^{13}}{\left (a^4+x^4\right )^5} \, dx=\frac {-\frac {a^2 x^2 \left (15 a^{12}+55 a^8 x^4+73 a^4 x^8-15 x^{12}\right )}{\left (a^4+x^4\right )^4}+15 \arctan \left (\frac {x^2}{a^2}\right )}{768 a^6} \]
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Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.66
method | result | size |
risch | \(\frac {-\frac {5 a^{8} x^{2}}{256}-\frac {55 a^{4} x^{6}}{768}-\frac {73 x^{10}}{768}+\frac {5 x^{14}}{256 a^{4}}}{\left (a^{4}+x^{4}\right )^{4}}+\frac {5 \arctan \left (\frac {x^{2}}{a^{2}}\right )}{256 a^{6}}\) | \(55\) |
default | \(\frac {\frac {5 x^{14}}{128 a^{4}}-\frac {73 x^{10}}{384}-\frac {55 a^{4} x^{6}}{384}-\frac {5 a^{8} x^{2}}{128}}{2 \left (a^{4}+x^{4}\right )^{4}}+\frac {5 \arctan \left (\frac {x^{2}}{a^{2}}\right )}{256 a^{6}}\) | \(56\) |
parallelrisch | \(-\frac {-60 i \ln \left (i a^{2}+x^{2}\right ) x^{4} a^{12}+60 i \ln \left (-i a^{2}+x^{2}\right ) x^{4} a^{12}-60 i \ln \left (i a^{2}+x^{2}\right ) x^{12} a^{4}+15 i \ln \left (-i a^{2}+x^{2}\right ) x^{16}-90 i \ln \left (i a^{2}+x^{2}\right ) x^{8} a^{8}+90 i \ln \left (-i a^{2}+x^{2}\right ) x^{8} a^{8}+15 i \ln \left (-i a^{2}+x^{2}\right ) a^{16}+60 i \ln \left (-i a^{2}+x^{2}\right ) x^{12} a^{4}-15 i \ln \left (i a^{2}+x^{2}\right ) a^{16}-15 i \ln \left (i a^{2}+x^{2}\right ) x^{16}-30 x^{14} a^{2}+146 x^{10} a^{6}+110 x^{6} a^{10}+30 x^{2} a^{14}}{1536 a^{6} \left (a^{4}+x^{4}\right )^{4}}\) | \(236\) |
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Time = 0.23 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.36 \[ \int \frac {x^{13}}{\left (a^4+x^4\right )^5} \, dx=-\frac {15 \, a^{14} x^{2} + 55 \, a^{10} x^{6} + 73 \, a^{6} x^{10} - 15 \, a^{2} x^{14} - 15 \, {\left (a^{16} + 4 \, a^{12} x^{4} + 6 \, a^{8} x^{8} + 4 \, a^{4} x^{12} + x^{16}\right )} \arctan \left (\frac {x^{2}}{a^{2}}\right )}{768 \, {\left (a^{22} + 4 \, a^{18} x^{4} + 6 \, a^{14} x^{8} + 4 \, a^{10} x^{12} + a^{6} x^{16}\right )}} \]
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Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.23 \[ \int \frac {x^{13}}{\left (a^4+x^4\right )^5} \, dx=\frac {- 15 a^{12} x^{2} - 55 a^{8} x^{6} - 73 a^{4} x^{10} + 15 x^{14}}{768 a^{20} + 3072 a^{16} x^{4} + 4608 a^{12} x^{8} + 3072 a^{8} x^{12} + 768 a^{4} x^{16}} + \frac {- \frac {5 i \log {\left (- i a^{2} + x^{2} \right )}}{512} + \frac {5 i \log {\left (i a^{2} + x^{2} \right )}}{512}}{a^{6}} \]
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Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {x^{13}}{\left (a^4+x^4\right )^5} \, dx=-\frac {15 \, a^{12} x^{2} + 55 \, a^{8} x^{6} + 73 \, a^{4} x^{10} - 15 \, x^{14}}{768 \, {\left (a^{20} + 4 \, a^{16} x^{4} + 6 \, a^{12} x^{8} + 4 \, a^{8} x^{12} + a^{4} x^{16}\right )}} + \frac {5 \, \arctan \left (\frac {x^{2}}{a^{2}}\right )}{256 \, a^{6}} \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.70 \[ \int \frac {x^{13}}{\left (a^4+x^4\right )^5} \, dx=\frac {5 \, \arctan \left (\frac {x^{2}}{a^{2}}\right )}{256 \, a^{6}} - \frac {15 \, a^{12} x^{2} + 55 \, a^{8} x^{6} + 73 \, a^{4} x^{10} - 15 \, x^{14}}{768 \, {\left (a^{4} + x^{4}\right )}^{4} a^{4}} \]
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Time = 0.15 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95 \[ \int \frac {x^{13}}{\left (a^4+x^4\right )^5} \, dx=\frac {5\,\mathrm {atan}\left (\frac {x^2}{a^2}\right )}{256\,a^6}-\frac {\frac {73\,x^{10}}{768}+\frac {55\,a^4\,x^6}{768}+\frac {5\,a^8\,x^2}{256}-\frac {5\,x^{14}}{256\,a^4}}{a^{16}+4\,a^{12}\,x^4+6\,a^8\,x^8+4\,a^4\,x^{12}+x^{16}} \]
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