Optimal. Leaf size=234 \[ -\frac {\left (a-\sqrt {2} b\right ) \log \left (x^2-\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{4 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {\left (a-\sqrt {2} b\right ) \log \left (x^2+\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{4 \sqrt {2 \left (2 \sqrt {2}-1\right )}}-\frac {1}{2} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \left (a+\sqrt {2} b\right ) \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {2}-1}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )+\frac {1}{2} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \left (a+\sqrt {2} b\right ) \tan ^{-1}\left (\frac {2 x+\sqrt {2 \sqrt {2}-1}}{\sqrt {1+2 \sqrt {2}}}\right ) \]
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Rubi [A] time = 0.23, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1169, 634, 618, 204, 628} \begin {gather*} -\frac {\left (a-\sqrt {2} b\right ) \log \left (x^2-\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{4 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {\left (a-\sqrt {2} b\right ) \log \left (x^2+\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{4 \sqrt {2 \left (2 \sqrt {2}-1\right )}}-\frac {1}{2} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \left (a+\sqrt {2} b\right ) \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {2}-1}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )+\frac {1}{2} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \left (a+\sqrt {2} b\right ) \tan ^{-1}\left (\frac {2 x+\sqrt {2 \sqrt {2}-1}}{\sqrt {1+2 \sqrt {2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1169
Rubi steps
\begin {align*} \int \frac {a+b x^2}{2+x^2+x^4} \, dx &=\frac {\int \frac {\sqrt {-1+2 \sqrt {2}} a-\left (a-\sqrt {2} b\right ) x}{\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{2 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\int \frac {\sqrt {-1+2 \sqrt {2}} a+\left (a-\sqrt {2} b\right ) x}{\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{2 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}\\ &=\frac {1}{8} \left (\sqrt {2} a+2 b\right ) \int \frac {1}{\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx+\frac {1}{8} \left (\sqrt {2} a+2 b\right ) \int \frac {1}{\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx-\frac {\left (a-\sqrt {2} b\right ) \int \frac {-\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\left (a-\sqrt {2} b\right ) \int \frac {\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}\\ &=-\frac {\left (a-\sqrt {2} b\right ) \log \left (\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\left (a-\sqrt {2} b\right ) \log \left (\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}-\frac {1}{4} \left (\sqrt {2} a+2 b\right ) \operatorname {Subst}\left (\int \frac {1}{-1-2 \sqrt {2}-x^2} \, dx,x,-\sqrt {-1+2 \sqrt {2}}+2 x\right )-\frac {1}{4} \left (\sqrt {2} a+2 b\right ) \operatorname {Subst}\left (\int \frac {1}{-1-2 \sqrt {2}-x^2} \, dx,x,\sqrt {-1+2 \sqrt {2}}+2 x\right )\\ &=-\frac {\left (a+\sqrt {2} b\right ) \tan ^{-1}\left (\frac {\sqrt {-1+2 \sqrt {2}}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )}{2 \sqrt {2 \left (1+2 \sqrt {2}\right )}}+\frac {\left (a+\sqrt {2} b\right ) \tan ^{-1}\left (\frac {\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {1+2 \sqrt {2}}}\right )}{2 \sqrt {2 \left (1+2 \sqrt {2}\right )}}-\frac {\left (a-\sqrt {2} b\right ) \log \left (\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\left (a-\sqrt {2} b\right ) \log \left (\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 111, normalized size = 0.47 \begin {gather*} \frac {\left (\left (\sqrt {7}+i\right ) b-2 i a\right ) \tan ^{-1}\left (\frac {x}{\sqrt {\frac {1}{2} \left (1-i \sqrt {7}\right )}}\right )}{\sqrt {14-14 i \sqrt {7}}}+\frac {\left (2 i a+\left (\sqrt {7}-i\right ) b\right ) \tan ^{-1}\left (\frac {x}{\sqrt {\frac {1}{2} \left (1+i \sqrt {7}\right )}}\right )}{\sqrt {14+14 i \sqrt {7}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x^2}{2+x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.28, size = 3406, normalized size = 14.56
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.88, size = 544, normalized size = 2.32 \begin {gather*} -\frac {1}{14336} \, \sqrt {7} {\left (32 \, \sqrt {7} 2^{\frac {1}{4}} b {\left (\sqrt {2} + 4\right )}^{\frac {3}{2}} + 96 \, \sqrt {7} 2^{\frac {1}{4}} b \sqrt {\sqrt {2} + 4} {\left (\sqrt {2} - 4\right )} - 24 \cdot 2^{\frac {3}{4}} b {\left (\sqrt {2} + 4\right )} \sqrt {-8 \, \sqrt {2} + 32} + 2^{\frac {3}{4}} b {\left (-8 \, \sqrt {2} + 32\right )}^{\frac {3}{2}} - 128 \, \sqrt {7} 2^{\frac {3}{4}} a \sqrt {\sqrt {2} + 4} + 64 \cdot 2^{\frac {1}{4}} a \sqrt {-8 \, \sqrt {2} + 32}\right )} \arctan \left (\frac {2 \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} {\left (x + 2^{\frac {1}{4}} \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}}\right )}}{\sqrt {\sqrt {2} + 4}}\right ) - \frac {1}{14336} \, \sqrt {7} {\left (32 \, \sqrt {7} 2^{\frac {1}{4}} b {\left (\sqrt {2} + 4\right )}^{\frac {3}{2}} + 96 \, \sqrt {7} 2^{\frac {1}{4}} b \sqrt {\sqrt {2} + 4} {\left (\sqrt {2} - 4\right )} - 24 \cdot 2^{\frac {3}{4}} b {\left (\sqrt {2} + 4\right )} \sqrt {-8 \, \sqrt {2} + 32} + 2^{\frac {3}{4}} b {\left (-8 \, \sqrt {2} + 32\right )}^{\frac {3}{2}} - 128 \, \sqrt {7} 2^{\frac {3}{4}} a \sqrt {\sqrt {2} + 4} + 64 \cdot 2^{\frac {1}{4}} a \sqrt {-8 \, \sqrt {2} + 32}\right )} \arctan \left (\frac {2 \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} {\left (x - 2^{\frac {1}{4}} \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}}\right )}}{\sqrt {\sqrt {2} + 4}}\right ) - \frac {1}{28672} \, \sqrt {7} {\left (24 \, \sqrt {7} 2^{\frac {3}{4}} b {\left (\sqrt {2} + 4\right )} \sqrt {-8 \, \sqrt {2} + 32} - \sqrt {7} 2^{\frac {3}{4}} b {\left (-8 \, \sqrt {2} + 32\right )}^{\frac {3}{2}} + 32 \cdot 2^{\frac {1}{4}} b {\left (\sqrt {2} + 4\right )}^{\frac {3}{2}} + 96 \cdot 2^{\frac {1}{4}} b \sqrt {\sqrt {2} + 4} {\left (\sqrt {2} - 4\right )} - 128 \cdot 2^{\frac {3}{4}} a \sqrt {\sqrt {2} + 4} - 64 \, \sqrt {7} 2^{\frac {1}{4}} a \sqrt {-8 \, \sqrt {2} + 32}\right )} \log \left (x^{2} + 2 \cdot 2^{\frac {1}{4}} x \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}} + \sqrt {2}\right ) + \frac {1}{28672} \, \sqrt {7} {\left (24 \, \sqrt {7} 2^{\frac {3}{4}} b {\left (\sqrt {2} + 4\right )} \sqrt {-8 \, \sqrt {2} + 32} - \sqrt {7} 2^{\frac {3}{4}} b {\left (-8 \, \sqrt {2} + 32\right )}^{\frac {3}{2}} + 32 \cdot 2^{\frac {1}{4}} b {\left (\sqrt {2} + 4\right )}^{\frac {3}{2}} + 96 \cdot 2^{\frac {1}{4}} b \sqrt {\sqrt {2} + 4} {\left (\sqrt {2} - 4\right )} - 128 \cdot 2^{\frac {3}{4}} a \sqrt {\sqrt {2} + 4} - 64 \, \sqrt {7} 2^{\frac {1}{4}} a \sqrt {-8 \, \sqrt {2} + 32}\right )} \log \left (x^{2} - 2 \cdot 2^{\frac {1}{4}} x \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}} + \sqrt {2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 710, normalized size = 3.03 \begin {gather*} -\frac {\left (-1+2 \sqrt {2}\right ) \sqrt {2}\, a \arctan \left (\frac {2 x -\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{28 \sqrt {1+2 \sqrt {2}}}-\frac {\left (-1+2 \sqrt {2}\right ) a \arctan \left (\frac {2 x -\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{7 \sqrt {1+2 \sqrt {2}}}+\frac {\sqrt {2}\, a \arctan \left (\frac {2 x -\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{2 \sqrt {1+2 \sqrt {2}}}-\frac {\left (-1+2 \sqrt {2}\right ) \sqrt {2}\, a \arctan \left (\frac {2 x +\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{28 \sqrt {1+2 \sqrt {2}}}-\frac {\left (-1+2 \sqrt {2}\right ) a \arctan \left (\frac {2 x +\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{7 \sqrt {1+2 \sqrt {2}}}+\frac {\sqrt {2}\, a \arctan \left (\frac {2 x +\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{2 \sqrt {1+2 \sqrt {2}}}-\frac {\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a \ln \left (x^{2}-\sqrt {-1+2 \sqrt {2}}\, x +\sqrt {2}\right )}{56}-\frac {\sqrt {-1+2 \sqrt {2}}\, a \ln \left (x^{2}-\sqrt {-1+2 \sqrt {2}}\, x +\sqrt {2}\right )}{14}+\frac {\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a \ln \left (x^{2}+\sqrt {-1+2 \sqrt {2}}\, x +\sqrt {2}\right )}{56}+\frac {\sqrt {-1+2 \sqrt {2}}\, a \ln \left (x^{2}+\sqrt {-1+2 \sqrt {2}}\, x +\sqrt {2}\right )}{14}+\frac {\left (-1+2 \sqrt {2}\right ) \sqrt {2}\, b \arctan \left (\frac {2 x -\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{7 \sqrt {1+2 \sqrt {2}}}+\frac {\left (-1+2 \sqrt {2}\right ) b \arctan \left (\frac {2 x -\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{14 \sqrt {1+2 \sqrt {2}}}+\frac {\left (-1+2 \sqrt {2}\right ) \sqrt {2}\, b \arctan \left (\frac {2 x +\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{7 \sqrt {1+2 \sqrt {2}}}+\frac {\left (-1+2 \sqrt {2}\right ) b \arctan \left (\frac {2 x +\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{14 \sqrt {1+2 \sqrt {2}}}+\frac {\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b \ln \left (x^{2}-\sqrt {-1+2 \sqrt {2}}\, x +\sqrt {2}\right )}{14}+\frac {\sqrt {-1+2 \sqrt {2}}\, b \ln \left (x^{2}-\sqrt {-1+2 \sqrt {2}}\, x +\sqrt {2}\right )}{28}-\frac {\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b \ln \left (x^{2}+\sqrt {-1+2 \sqrt {2}}\, x +\sqrt {2}\right )}{14}-\frac {\sqrt {-1+2 \sqrt {2}}\, b \ln \left (x^{2}+\sqrt {-1+2 \sqrt {2}}\, x +\sqrt {2}\right )}{28} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b x^{2} + a}{x^{4} + x^{2} + 2}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.49, size = 771, normalized size = 3.29 \begin {gather*} -\mathrm {atan}\left (\frac {a^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}+\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}-\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}\,7{}\mathrm {i}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}-a\,b^2-2\,a^2\,b+\frac {a^3}{2}+4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}-\frac {b^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}+\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}-\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}\,14{}\mathrm {i}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}-a\,b^2-2\,a^2\,b+\frac {a^3}{2}+4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}+\frac {\sqrt {7}\,a^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}+\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}-\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}-a\,b^2-2\,a^2\,b+\frac {a^3}{2}+4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}-\frac {2\,\sqrt {7}\,b^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}+\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}-\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}-a\,b^2-2\,a^2\,b+\frac {a^3}{2}+4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}\right )\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}+\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}-\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}\,2{}\mathrm {i}-2\,\mathrm {atanh}\left (\frac {7\,a^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}-\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}+\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}+a\,b^2+2\,a^2\,b-\frac {a^3}{2}-4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}-\frac {14\,b^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}-\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}+\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}+a\,b^2+2\,a^2\,b-\frac {a^3}{2}-4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}+\frac {\sqrt {7}\,a^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}-\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}+\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}\,1{}\mathrm {i}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}+a\,b^2+2\,a^2\,b-\frac {a^3}{2}-4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}-\frac {\sqrt {7}\,b^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}-\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}+\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}\,2{}\mathrm {i}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}+a\,b^2+2\,a^2\,b-\frac {a^3}{2}-4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}\right )\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}-\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}+\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.32, size = 122, normalized size = 0.52 \begin {gather*} \operatorname {RootSum} {\left (1568 t^{4} + t^{2} \left (- 28 a^{2} + 224 a b - 56 b^{2}\right ) + a^{4} - 2 a^{3} b + 5 a^{2} b^{2} - 4 a b^{3} + 4 b^{4}, \left (t \mapsto t \log {\left (x + \frac {112 t^{3} a - 448 t^{3} b + 6 t a^{3} + 12 t a^{2} b - 48 t a b^{2} + 8 t b^{3}}{a^{4} - a^{3} b + 2 a b^{3} - 4 b^{4}} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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