Optimal. Leaf size=83 \[ -\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}+\frac {5 x^2}{256 a^4 \left (a^4+x^4\right )}-\frac {5 x^2}{128 \left (a^4+x^4\right )^2}+\frac {5 \tan ^{-1}\left (\frac {x^2}{a^2}\right )}{256 a^6} \]
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Rubi [A] time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {275, 288, 199, 203} \[ -\frac {x^{10}}{16 \left (a^4+x^4\right )^4}-\frac {5 x^6}{96 \left (a^4+x^4\right )^3}+\frac {5 x^2}{256 a^4 \left (a^4+x^4\right )}-\frac {5 x^2}{128 \left (a^4+x^4\right )^2}+\frac {5 \tan ^{-1}\left (\frac {x^2}{a^2}\right )}{256 a^6} \]
Antiderivative was successfully verified.
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Rule 199
Rule 203
Rule 275
Rule 288
Rubi steps
\begin {align*} \int \frac {x^{13}}{\left (a^4+x^4\right )^5} \, dx &=\frac {1}{2} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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 \operatorname {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 \tan ^{-1}\left (\frac {x^2}{a^2}\right )}{256 a^6}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 0.75 \[ \frac {15 \tan ^{-1}\left (\frac {x^2}{a^2}\right )-\frac {a^2 x^2 \left (15 a^{12}+55 a^8 x^4+73 a^4 x^8-15 x^{12}\right )}{\left (a^4+x^4\right )^4}}{768 a^6} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 62, normalized size = 0.75 \[ \frac {5 \tan ^{-1}\left (\frac {x^2}{a^2}\right )}{256 a^6}-\frac {x^2 \left (15 a^{12}+55 a^8 x^4+73 a^4 x^8-15 x^{12}\right )}{768 a^4 \left (a^4+x^4\right )^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 113, normalized size = 1.36 \[ -\frac {15 \, a^{14} x^{2} + 55 \, a^{10} x^{6} + 73 \, a^{6} x^{10} - 15 \, a^{2} x^{14} - 15 \, {\left (a^{16} + 4 \, a^{12} x^{4} + 6 \, a^{8} x^{8} + 4 \, a^{4} x^{12} + x^{16}\right )} \arctan \left (\frac {x^{2}}{a^{2}}\right )}{768 \, {\left (a^{22} + 4 \, a^{18} x^{4} + 6 \, a^{14} x^{8} + 4 \, a^{10} x^{12} + a^{6} x^{16}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 58, normalized size = 0.70 \[ \frac {5 \, \arctan \left (\frac {x^{2}}{a^{2}}\right )}{256 \, a^{6}} - \frac {15 \, a^{12} x^{2} + 55 \, a^{8} x^{6} + 73 \, a^{4} x^{10} - 15 \, x^{14}}{768 \, {\left (a^{4} + x^{4}\right )}^{4} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 55, normalized size = 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 | \(\frac {\frac {5 x^{14}}{128 a^{4}}-\frac {73 x^{10}}{384}-\frac {55 a^{4} x^{6}}{384}-\frac {5 a^{8} x^{2}}{128}}{2 \left (a^{4}+x^{4}\right )^{4}}+\frac {5 \arctan \left (\frac {x^{2}}{a^{2}}\right )}{256 a^{6}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 83, normalized size = 1.00 \[ -\frac {15 \, a^{12} x^{2} + 55 \, a^{8} x^{6} + 73 \, a^{4} x^{10} - 15 \, x^{14}}{768 \, {\left (a^{20} + 4 \, a^{16} x^{4} + 6 \, a^{12} x^{8} + 4 \, a^{8} x^{12} + a^{4} x^{16}\right )}} + \frac {5 \, \arctan \left (\frac {x^{2}}{a^{2}}\right )}{256 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 79, normalized size = 0.95 \[ \frac {5\,\mathrm {atan}\left (\frac {x^2}{a^2}\right )}{256\,a^6}-\frac {\frac {73\,x^{10}}{768}+\frac {55\,a^4\,x^6}{768}+\frac {5\,a^8\,x^2}{256}-\frac {5\,x^{14}}{256\,a^4}}{a^{16}+4\,a^{12}\,x^4+6\,a^8\,x^8+4\,a^4\,x^{12}+x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.69, size = 102, normalized size = 1.23 \[ \frac {- 15 a^{12} x^{2} - 55 a^{8} x^{6} - 73 a^{4} x^{10} + 15 x^{14}}{768 a^{20} + 3072 a^{16} x^{4} + 4608 a^{12} x^{8} + 3072 a^{8} x^{12} + 768 a^{4} x^{16}} + \frac {- \frac {5 i \log {\left (- i a^{2} + x^{2} \right )}}{512} + \frac {5 i \log {\left (i a^{2} + x^{2} \right )}}{512}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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