Optimal. Leaf size=299 \[ -\frac {\log \left (-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}+x^2\right )}{4 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}+\frac {\log \left (\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}+x^2\right )}{4 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2+1}-a}-\sqrt {2} x}{\sqrt {\sqrt {a^2+1}+a}}\right )}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}+a}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2+1}-a}+\sqrt {2} x}{\sqrt {\sqrt {a^2+1}+a}}\right )}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}+a}} \]
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Rubi [A] time = 0.31, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1094, 634, 618, 204, 628} \[ -\frac {\log \left (-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}+x^2\right )}{4 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}+\frac {\log \left (\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}+x^2\right )}{4 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2+1}-a}-\sqrt {2} x}{\sqrt {\sqrt {a^2+1}+a}}\right )}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}+a}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2+1}-a}+\sqrt {2} x}{\sqrt {\sqrt {a^2+1}+a}}\right )}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}+a}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1094
Rubi steps
\begin {align*} \int \frac {1}{1+a^2+2 a x^2+x^4} \, dx &=\frac {\int \frac {\sqrt {2} \sqrt {-a+\sqrt {1+a^2}}-x}{\sqrt {1+a^2}-\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} x+x^2} \, dx}{2 \sqrt {2} \sqrt {1+a^2} \sqrt {-a+\sqrt {1+a^2}}}+\frac {\int \frac {\sqrt {2} \sqrt {-a+\sqrt {1+a^2}}+x}{\sqrt {1+a^2}+\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} x+x^2} \, dx}{2 \sqrt {2} \sqrt {1+a^2} \sqrt {-a+\sqrt {1+a^2}}}\\ &=\frac {\int \frac {1}{\sqrt {1+a^2}-\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} x+x^2} \, dx}{4 \sqrt {1+a^2}}+\frac {\int \frac {1}{\sqrt {1+a^2}+\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} x+x^2} \, dx}{4 \sqrt {1+a^2}}-\frac {\int \frac {-\sqrt {2} \sqrt {-a+\sqrt {1+a^2}}+2 x}{\sqrt {1+a^2}-\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} x+x^2} \, dx}{4 \sqrt {2} \sqrt {1+a^2} \sqrt {-a+\sqrt {1+a^2}}}+\frac {\int \frac {\sqrt {2} \sqrt {-a+\sqrt {1+a^2}}+2 x}{\sqrt {1+a^2}+\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} x+x^2} \, dx}{4 \sqrt {2} \sqrt {1+a^2} \sqrt {-a+\sqrt {1+a^2}}}\\ &=-\frac {\log \left (\sqrt {1+a^2}-\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} x+x^2\right )}{4 \sqrt {2} \sqrt {1+a^2} \sqrt {-a+\sqrt {1+a^2}}}+\frac {\log \left (\sqrt {1+a^2}+\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} x+x^2\right )}{4 \sqrt {2} \sqrt {1+a^2} \sqrt {-a+\sqrt {1+a^2}}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-2 \left (a+\sqrt {1+a^2}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {-a+\sqrt {1+a^2}}+2 x\right )}{2 \sqrt {1+a^2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-2 \left (a+\sqrt {1+a^2}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {-a+\sqrt {1+a^2}}+2 x\right )}{2 \sqrt {1+a^2}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-a+\sqrt {1+a^2}}-\sqrt {2} x}{\sqrt {a+\sqrt {1+a^2}}}\right )}{2 \sqrt {2} \sqrt {1+a^2} \sqrt {a+\sqrt {1+a^2}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-a+\sqrt {1+a^2}}+\sqrt {2} x}{\sqrt {a+\sqrt {1+a^2}}}\right )}{2 \sqrt {2} \sqrt {1+a^2} \sqrt {a+\sqrt {1+a^2}}}-\frac {\log \left (\sqrt {1+a^2}-\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} x+x^2\right )}{4 \sqrt {2} \sqrt {1+a^2} \sqrt {-a+\sqrt {1+a^2}}}+\frac {\log \left (\sqrt {1+a^2}+\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} x+x^2\right )}{4 \sqrt {2} \sqrt {1+a^2} \sqrt {-a+\sqrt {1+a^2}}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 52, normalized size = 0.17 \[ -\frac {1}{2} i \left (\frac {\tan ^{-1}\left (\frac {x}{\sqrt {a-i}}\right )}{\sqrt {a-i}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {a+i}}\right )}{\sqrt {a+i}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 613, normalized size = 2.05 \[ \frac {\sqrt {2 \, a^{2} - \frac {2 \, {\left (a^{3} + a\right )}}{\sqrt {a^{2} + 1}} + 2} {\left (\frac {a}{\sqrt {a^{2} + 1}} + 1\right )} \log \left (x^{2} + \frac {\sqrt {2 \, a^{2} - \frac {2 \, {\left (a^{3} + a\right )}}{\sqrt {a^{2} + 1}} + 2} x}{{\left (a^{2} + 1\right )}^{\frac {1}{4}}} + \sqrt {a^{2} + 1}\right )}{8 \, {\left (a^{2} + 1\right )}^{\frac {1}{4}}} - \frac {\sqrt {2 \, a^{2} - \frac {2 \, {\left (a^{3} + a\right )}}{\sqrt {a^{2} + 1}} + 2} {\left (\frac {a}{\sqrt {a^{2} + 1}} + 1\right )} \log \left (x^{2} - \frac {\sqrt {2 \, a^{2} - \frac {2 \, {\left (a^{3} + a\right )}}{\sqrt {a^{2} + 1}} + 2} x}{{\left (a^{2} + 1\right )}^{\frac {1}{4}}} + \sqrt {a^{2} + 1}\right )}{8 \, {\left (a^{2} + 1\right )}^{\frac {1}{4}}} - \frac {\sqrt {2 \, a^{2} - \frac {2 \, {\left (a^{3} + a\right )}}{\sqrt {a^{2} + 1}} + 2} {\left (a^{2} + 1\right )}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {a^{4} + 2 \, a^{2} + 1} \sqrt {2 \, a^{2} - \frac {2 \, {\left (a^{3} + a\right )}}{\sqrt {a^{2} + 1}} + 2} x}{{\left (a^{2} + 1\right )}^{\frac {5}{4}}} + \frac {a^{3} + a}{\sqrt {a^{4} + 2 \, a^{2} + 1}} + \frac {\sqrt {a^{4} + 2 \, a^{2} + 1} \sqrt {2 \, a^{2} - \frac {2 \, {\left (a^{3} + a\right )}}{\sqrt {a^{2} + 1}} + 2} \sqrt {x^{2} + \frac {\sqrt {2 \, a^{2} - \frac {2 \, {\left (a^{3} + a\right )}}{\sqrt {a^{2} + 1}} + 2} x}{{\left (a^{2} + 1\right )}^{\frac {1}{4}}} + \sqrt {a^{2} + 1}}}{{\left (a^{2} + 1\right )}^{\frac {5}{4}}} - \frac {\sqrt {a^{4} + 2 \, a^{2} + 1}}{\sqrt {a^{2} + 1}}\right )}{2 \, \sqrt {a^{4} + 2 \, a^{2} + 1}} - \frac {\sqrt {2 \, a^{2} - \frac {2 \, {\left (a^{3} + a\right )}}{\sqrt {a^{2} + 1}} + 2} {\left (a^{2} + 1\right )}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {a^{4} + 2 \, a^{2} + 1} \sqrt {2 \, a^{2} - \frac {2 \, {\left (a^{3} + a\right )}}{\sqrt {a^{2} + 1}} + 2} x}{{\left (a^{2} + 1\right )}^{\frac {5}{4}}} - \frac {a^{3} + a}{\sqrt {a^{4} + 2 \, a^{2} + 1}} + \frac {\sqrt {a^{4} + 2 \, a^{2} + 1} \sqrt {2 \, a^{2} - \frac {2 \, {\left (a^{3} + a\right )}}{\sqrt {a^{2} + 1}} + 2} \sqrt {x^{2} - \frac {\sqrt {2 \, a^{2} - \frac {2 \, {\left (a^{3} + a\right )}}{\sqrt {a^{2} + 1}} + 2} x}{{\left (a^{2} + 1\right )}^{\frac {1}{4}}} + \sqrt {a^{2} + 1}}}{{\left (a^{2} + 1\right )}^{\frac {5}{4}}} + \frac {\sqrt {a^{4} + 2 \, a^{2} + 1}}{\sqrt {a^{2} + 1}}\right )}{2 \, \sqrt {a^{4} + 2 \, a^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 1073, normalized size = 3.59 \[ \frac {a^{4} \arctan \left (\frac {2 x +\sqrt {-2 a +2 \sqrt {a^{2}+1}}}{\sqrt {2 a +2 \sqrt {a^{2}+1}}}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}} \sqrt {2 a +2 \sqrt {a^{2}+1}}}-\frac {a^{4} \arctan \left (\frac {-2 x +\sqrt {-2 a +2 \sqrt {a^{2}+1}}}{\sqrt {2 a +2 \sqrt {a^{2}+1}}}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}} \sqrt {2 a +2 \sqrt {a^{2}+1}}}+\frac {\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, a^{3} \ln \left (x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, x +\sqrt {a^{2}+1}\right )}{8 \left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, a^{3} \ln \left (-x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, x -\sqrt {a^{2}+1}\right )}{8 \left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {a^{2} \arctan \left (\frac {2 x +\sqrt {-2 a +2 \sqrt {a^{2}+1}}}{\sqrt {2 a +2 \sqrt {a^{2}+1}}}\right )}{2 \sqrt {a^{2}+1}\, \sqrt {2 a +2 \sqrt {a^{2}+1}}}+\frac {3 a^{2} \arctan \left (\frac {2 x +\sqrt {-2 a +2 \sqrt {a^{2}+1}}}{\sqrt {2 a +2 \sqrt {a^{2}+1}}}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}} \sqrt {2 a +2 \sqrt {a^{2}+1}}}+\frac {a^{2} \arctan \left (\frac {-2 x +\sqrt {-2 a +2 \sqrt {a^{2}+1}}}{\sqrt {2 a +2 \sqrt {a^{2}+1}}}\right )}{2 \sqrt {a^{2}+1}\, \sqrt {2 a +2 \sqrt {a^{2}+1}}}-\frac {3 a^{2} \arctan \left (\frac {-2 x +\sqrt {-2 a +2 \sqrt {a^{2}+1}}}{\sqrt {2 a +2 \sqrt {a^{2}+1}}}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}} \sqrt {2 a +2 \sqrt {a^{2}+1}}}+\frac {\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, a^{2} \ln \left (x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, x +\sqrt {a^{2}+1}\right )}{8 a^{2}+8}-\frac {\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, a^{2} \ln \left (-x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, x -\sqrt {a^{2}+1}\right )}{8 \left (a^{2}+1\right )}+\frac {\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, a \ln \left (x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, x +\sqrt {a^{2}+1}\right )}{8 \left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, a \ln \left (-x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, x -\sqrt {a^{2}+1}\right )}{8 \left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {\arctan \left (\frac {2 x +\sqrt {-2 a +2 \sqrt {a^{2}+1}}}{\sqrt {2 a +2 \sqrt {a^{2}+1}}}\right )}{2 \sqrt {a^{2}+1}\, \sqrt {2 a +2 \sqrt {a^{2}+1}}}+\frac {\arctan \left (\frac {2 x +\sqrt {-2 a +2 \sqrt {a^{2}+1}}}{\sqrt {2 a +2 \sqrt {a^{2}+1}}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}} \sqrt {2 a +2 \sqrt {a^{2}+1}}}+\frac {\arctan \left (\frac {-2 x +\sqrt {-2 a +2 \sqrt {a^{2}+1}}}{\sqrt {2 a +2 \sqrt {a^{2}+1}}}\right )}{2 \sqrt {a^{2}+1}\, \sqrt {2 a +2 \sqrt {a^{2}+1}}}-\frac {\arctan \left (\frac {-2 x +\sqrt {-2 a +2 \sqrt {a^{2}+1}}}{\sqrt {2 a +2 \sqrt {a^{2}+1}}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}} \sqrt {2 a +2 \sqrt {a^{2}+1}}}+\frac {\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, \ln \left (x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, x +\sqrt {a^{2}+1}\right )}{8 a^{2}+8}-\frac {\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, \ln \left (-x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+1}}\, x -\sqrt {a^{2}+1}\right )}{8 \left (a^{2}+1\right )} \]
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 \[ \int \frac {1}{x^{4} + 2 \, a x^{2} + a^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.36, size = 469, normalized size = 1.57 \[ -\frac {\mathrm {atanh}\left (-\frac {2\,x\,\sqrt {\frac {a}{a^2+1}+\frac {1{}\mathrm {i}}{a^2+1}}}{\frac {2\,a}{a^2+1}+\frac {2{}\mathrm {i}}{a^2+1}}+\frac {a\,x\,\sqrt {\frac {a}{a^2+1}+\frac {1{}\mathrm {i}}{a^2+1}}\,2{}\mathrm {i}}{\frac {2\,a}{a^2+1}+\frac {2\,a^3}{a^2+1}+\frac {2{}\mathrm {i}}{a^2+1}+\frac {a^2\,2{}\mathrm {i}}{a^2+1}}+\frac {2\,a^2\,x\,\sqrt {\frac {a}{a^2+1}+\frac {1{}\mathrm {i}}{a^2+1}}}{\frac {2\,a}{a^2+1}+\frac {2\,a^3}{a^2+1}+\frac {2{}\mathrm {i}}{a^2+1}+\frac {a^2\,2{}\mathrm {i}}{a^2+1}}\right )\,\sqrt {\frac {a+1{}\mathrm {i}}{a^2+1}}}{2}+2\,\mathrm {atanh}\left (\frac {8\,x\,\sqrt {\frac {a}{16\,a^2+16}-\frac {1{}\mathrm {i}}{16\,a^2+16}}}{\frac {32\,a}{16\,a^2+16}-\frac {32{}\mathrm {i}}{16\,a^2+16}}+\frac {a\,x\,\sqrt {\frac {a}{16\,a^2+16}-\frac {1{}\mathrm {i}}{16\,a^2+16}}\,128{}\mathrm {i}}{\frac {512\,a}{16\,a^2+16}+\frac {512\,a^3}{16\,a^2+16}-\frac {512{}\mathrm {i}}{16\,a^2+16}-\frac {a^2\,512{}\mathrm {i}}{16\,a^2+16}}-\frac {128\,a^2\,x\,\sqrt {\frac {a}{16\,a^2+16}-\frac {1{}\mathrm {i}}{16\,a^2+16}}}{\frac {512\,a}{16\,a^2+16}+\frac {512\,a^3}{16\,a^2+16}-\frac {512{}\mathrm {i}}{16\,a^2+16}-\frac {a^2\,512{}\mathrm {i}}{16\,a^2+16}}\right )\,\sqrt {\frac {a-\mathrm {i}}{16\,a^2+16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 48, normalized size = 0.16 \[ \operatorname {RootSum} {\left (t^{4} \left (256 a^{2} + 256\right ) - 32 t^{2} a + 1, \left (t \mapsto t \log {\left (64 t^{3} a^{3} + 64 t^{3} a - 4 t a^{2} + 4 t + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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