Optimal. Leaf size=185 \[ -\frac {9}{32} d \log \left (x^2-x+1\right )+\frac {9}{32} d \log \left (x^2+x+1\right )+\frac {d x \left (2-7 x^2\right )}{24 \left (x^4+x^2+1\right )}+\frac {d x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}-\frac {13 d \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {13 d \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {2 e \tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )}+\frac {e \left (2 x^2+1\right )}{12 \left (x^4+x^2+1\right )^2} \]
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Rubi [A] time = 0.12, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {1673, 12, 1092, 1178, 1169, 634, 618, 204, 628, 1107, 614} \begin {gather*} \frac {d x \left (2-7 x^2\right )}{24 \left (x^4+x^2+1\right )}+\frac {d x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}-\frac {9}{32} d \log \left (x^2-x+1\right )+\frac {9}{32} d \log \left (x^2+x+1\right )-\frac {13 d \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {13 d \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )}+\frac {e \left (2 x^2+1\right )}{12 \left (x^4+x^2+1\right )^2}+\frac {2 e \tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 614
Rule 618
Rule 628
Rule 634
Rule 1092
Rule 1107
Rule 1169
Rule 1178
Rule 1673
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (1+x^2+x^4\right )^3} \, dx &=\int \frac {d}{\left (1+x^2+x^4\right )^3} \, dx+\int \frac {e x}{\left (1+x^2+x^4\right )^3} \, dx\\ &=d \int \frac {1}{\left (1+x^2+x^4\right )^3} \, dx+e \int \frac {x}{\left (1+x^2+x^4\right )^3} \, dx\\ &=\frac {d x \left (1-x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {1}{12} d \int \frac {11-5 x^2}{\left (1+x^2+x^4\right )^2} \, dx+\frac {1}{2} e \operatorname {Subst}\left (\int \frac {1}{\left (1+x+x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac {d x \left (1-x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {d x \left (2-7 x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac {1}{72} d \int \frac {60-21 x^2}{1+x^2+x^4} \, dx+\frac {1}{2} e \operatorname {Subst}\left (\int \frac {1}{\left (1+x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {d x \left (1-x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {d x \left (2-7 x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac {e \left (1+2 x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac {1}{144} d \int \frac {60-81 x}{1-x+x^2} \, dx+\frac {1}{144} d \int \frac {60+81 x}{1+x+x^2} \, dx+\frac {1}{3} e \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^2\right )\\ &=\frac {d x \left (1-x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {d x \left (2-7 x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac {e \left (1+2 x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac {1}{96} (13 d) \int \frac {1}{1-x+x^2} \, dx+\frac {1}{96} (13 d) \int \frac {1}{1+x+x^2} \, dx-\frac {1}{32} (9 d) \int \frac {-1+2 x}{1-x+x^2} \, dx+\frac {1}{32} (9 d) \int \frac {1+2 x}{1+x+x^2} \, dx-\frac {1}{3} (2 e) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^2\right )\\ &=\frac {d x \left (1-x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {d x \left (2-7 x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac {e \left (1+2 x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac {2 e \tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {9}{32} d \log \left (1-x+x^2\right )+\frac {9}{32} d \log \left (1+x+x^2\right )-\frac {1}{48} (13 d) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )-\frac {1}{48} (13 d) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac {d x \left (1-x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {d x \left (2-7 x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac {e \left (1+2 x^2\right )}{6 \left (1+x^2+x^4\right )}-\frac {13 d \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {13 d \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {2 e \tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {9}{32} d \log \left (1-x+x^2\right )+\frac {9}{32} d \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [C] time = 0.75, size = 186, normalized size = 1.01 \begin {gather*} \frac {1}{144} \left (\frac {6 \left (d x \left (2-7 x^2\right )+e \left (8 x^2+4\right )\right )}{x^4+x^2+1}+\frac {12 \left (d \left (x-x^3\right )+2 e x^2+e\right )}{\left (x^4+x^2+1\right )^2}-\frac {\left (7 \sqrt {3}-47 i\right ) d \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}-i\right ) x\right )}{\sqrt {\frac {1}{6} \left (1+i \sqrt {3}\right )}}-\frac {\left (7 \sqrt {3}+47 i\right ) d \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}+i\right ) x\right )}{\sqrt {\frac {1}{6} \left (1-i \sqrt {3}\right )}}-32 \sqrt {3} e \tan ^{-1}\left (\frac {\sqrt {3}}{2 x^2+1}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{\left (1+x^2+x^4\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.06, size = 278, normalized size = 1.50 \begin {gather*} -\frac {84 \, d x^{7} - 96 \, e x^{6} + 60 \, d x^{5} - 144 \, e x^{4} + 84 \, d x^{3} - 192 \, e x^{2} - 2 \, \sqrt {3} {\left ({\left (13 \, d - 32 \, e\right )} x^{8} + 2 \, {\left (13 \, d - 32 \, e\right )} x^{6} + 3 \, {\left (13 \, d - 32 \, e\right )} x^{4} + 2 \, {\left (13 \, d - 32 \, e\right )} x^{2} + 13 \, d - 32 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt {3} {\left ({\left (13 \, d + 32 \, e\right )} x^{8} + 2 \, {\left (13 \, d + 32 \, e\right )} x^{6} + 3 \, {\left (13 \, d + 32 \, e\right )} x^{4} + 2 \, {\left (13 \, d + 32 \, e\right )} x^{2} + 13 \, d + 32 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 48 \, d x - 81 \, {\left (d x^{8} + 2 \, d x^{6} + 3 \, d x^{4} + 2 \, d x^{2} + d\right )} \log \left (x^{2} + x + 1\right ) + 81 \, {\left (d x^{8} + 2 \, d x^{6} + 3 \, d x^{4} + 2 \, d x^{2} + d\right )} \log \left (x^{2} - x + 1\right ) - 72 \, e}{288 \, {\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 131, normalized size = 0.71 \begin {gather*} \frac {1}{144} \, \sqrt {3} {\left (13 \, d - 32 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{144} \, \sqrt {3} {\left (13 \, d + 32 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {9}{32} \, d \log \left (x^{2} + x + 1\right ) - \frac {9}{32} \, d \log \left (x^{2} - x + 1\right ) - \frac {7 \, d x^{7} - 8 \, x^{6} e + 5 \, d x^{5} - 12 \, x^{4} e + 7 \, d x^{3} - 16 \, x^{2} e - 4 \, d x - 6 \, e}{24 \, {\left (x^{4} + x^{2} + 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 180, normalized size = 0.97 \begin {gather*} \frac {13 \sqrt {3}\, d \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{144}+\frac {13 \sqrt {3}\, d \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{144}-\frac {9 d \ln \left (x^{2}-x +1\right )}{32}+\frac {9 d \ln \left (x^{2}+x +1\right )}{32}-\frac {2 \sqrt {3}\, e \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{9}+\frac {2 \sqrt {3}\, e \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{9}+\frac {-6 d \,x^{2}+\left (-\frac {7 d}{3}-\frac {4 e}{3}\right ) x^{3}-4 d +2 e +\left (-\frac {20 d}{3}+\frac {e}{3}\right ) x}{16 \left (x^{2}+x +1\right )^{2}}-\frac {-6 d \,x^{2}+\left (\frac {7 d}{3}-\frac {4 e}{3}\right ) x^{3}-4 d -2 e +\left (\frac {20 d}{3}+\frac {e}{3}\right ) x}{16 \left (x^{2}-x +1\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.55, size = 137, normalized size = 0.74 \begin {gather*} \frac {1}{144} \, \sqrt {3} {\left (13 \, d - 32 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{144} \, \sqrt {3} {\left (13 \, d + 32 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {9}{32} \, d \log \left (x^{2} + x + 1\right ) - \frac {9}{32} \, d \log \left (x^{2} - x + 1\right ) - \frac {7 \, d x^{7} - 8 \, e x^{6} + 5 \, d x^{5} - 12 \, e x^{4} + 7 \, d x^{3} - 16 \, e x^{2} - 4 \, d x - 6 \, e}{24 \, {\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 185, normalized size = 1.00 \begin {gather*} \frac {-\frac {7\,d\,x^7}{24}+\frac {e\,x^6}{3}-\frac {5\,d\,x^5}{24}+\frac {e\,x^4}{2}-\frac {7\,d\,x^3}{24}+\frac {2\,e\,x^2}{3}+\frac {d\,x}{6}+\frac {e}{4}}{x^8+2\,x^6+3\,x^4+2\,x^2+1}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {9\,d}{32}+\frac {\sqrt {3}\,d\,13{}\mathrm {i}}{288}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}\right )+\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {9\,d}{32}-\frac {\sqrt {3}\,d\,13{}\mathrm {i}}{288}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {9\,d}{32}+\frac {\sqrt {3}\,d\,13{}\mathrm {i}}{288}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {9\,d}{32}+\frac {\sqrt {3}\,d\,13{}\mathrm {i}}{288}-\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.62, size = 1103, normalized size = 5.96
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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