Optimal. Leaf size=299 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {-a+\sqrt {a^2+b}}-\sqrt {2} x}{\sqrt {a+\sqrt {a^2+b}}}\right )}{2 \sqrt {2} \sqrt {a^2+b} \sqrt {a+\sqrt {a^2+b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-a+\sqrt {a^2+b}}+\sqrt {2} x}{\sqrt {a+\sqrt {a^2+b}}}\right )}{2 \sqrt {2} \sqrt {a^2+b} \sqrt {a+\sqrt {a^2+b}}}-\frac {\log \left (\sqrt {a^2+b}-\sqrt {2} \sqrt {-a+\sqrt {a^2+b}} x+x^2\right )}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {-a+\sqrt {a^2+b}}}+\frac {\log \left (\sqrt {a^2+b}+\sqrt {2} \sqrt {-a+\sqrt {a^2+b}} x+x^2\right )}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {-a+\sqrt {a^2+b}}} \]
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Rubi [A]
time = 0.20, 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 = {1108, 648, 632,
210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {\sqrt {a^2+b}-a}-\sqrt {2} x}{\sqrt {\sqrt {a^2+b}+a}}\right )}{2 \sqrt {2} \sqrt {a^2+b} \sqrt {\sqrt {a^2+b}+a}}+\frac {\text {ArcTan}\left (\frac {\sqrt {\sqrt {a^2+b}-a}+\sqrt {2} x}{\sqrt {\sqrt {a^2+b}+a}}\right )}{2 \sqrt {2} \sqrt {a^2+b} \sqrt {\sqrt {a^2+b}+a}}-\frac {\log \left (-\sqrt {2} x \sqrt {\sqrt {a^2+b}-a}+\sqrt {a^2+b}+x^2\right )}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {\sqrt {a^2+b}-a}}+\frac {\log \left (\sqrt {2} x \sqrt {\sqrt {a^2+b}-a}+\sqrt {a^2+b}+x^2\right )}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {\sqrt {a^2+b}-a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1108
Rubi steps
\begin {align*} \int \frac {1}{a^2+b+2 a x^2+x^4} \, dx &=\frac {\int \frac {\sqrt {2} \sqrt {-a+\sqrt {a^2+b}}-x}{\sqrt {a^2+b}-\sqrt {2} \sqrt {-a+\sqrt {a^2+b}} x+x^2} \, dx}{2 \sqrt {2} \sqrt {a^2+b} \sqrt {-a+\sqrt {a^2+b}}}+\frac {\int \frac {\sqrt {2} \sqrt {-a+\sqrt {a^2+b}}+x}{\sqrt {a^2+b}+\sqrt {2} \sqrt {-a+\sqrt {a^2+b}} x+x^2} \, dx}{2 \sqrt {2} \sqrt {a^2+b} \sqrt {-a+\sqrt {a^2+b}}}\\ &=\frac {\int \frac {1}{\sqrt {a^2+b}-\sqrt {2} \sqrt {-a+\sqrt {a^2+b}} x+x^2} \, dx}{4 \sqrt {a^2+b}}+\frac {\int \frac {1}{\sqrt {a^2+b}+\sqrt {2} \sqrt {-a+\sqrt {a^2+b}} x+x^2} \, dx}{4 \sqrt {a^2+b}}-\frac {\int \frac {-\sqrt {2} \sqrt {-a+\sqrt {a^2+b}}+2 x}{\sqrt {a^2+b}-\sqrt {2} \sqrt {-a+\sqrt {a^2+b}} x+x^2} \, dx}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {-a+\sqrt {a^2+b}}}+\frac {\int \frac {\sqrt {2} \sqrt {-a+\sqrt {a^2+b}}+2 x}{\sqrt {a^2+b}+\sqrt {2} \sqrt {-a+\sqrt {a^2+b}} x+x^2} \, dx}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {-a+\sqrt {a^2+b}}}\\ &=-\frac {\log \left (\sqrt {a^2+b}-\sqrt {2} \sqrt {-a+\sqrt {a^2+b}} x+x^2\right )}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {-a+\sqrt {a^2+b}}}+\frac {\log \left (\sqrt {a^2+b}+\sqrt {2} \sqrt {-a+\sqrt {a^2+b}} x+x^2\right )}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {-a+\sqrt {a^2+b}}}-\frac {\text {Subst}\left (\int \frac {1}{-2 \left (a+\sqrt {a^2+b}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {-a+\sqrt {a^2+b}}+2 x\right )}{2 \sqrt {a^2+b}}-\frac {\text {Subst}\left (\int \frac {1}{-2 \left (a+\sqrt {a^2+b}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {-a+\sqrt {a^2+b}}+2 x\right )}{2 \sqrt {a^2+b}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-a+\sqrt {a^2+b}}-\sqrt {2} x}{\sqrt {a+\sqrt {a^2+b}}}\right )}{2 \sqrt {2} \sqrt {a^2+b} \sqrt {a+\sqrt {a^2+b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-a+\sqrt {a^2+b}}+\sqrt {2} x}{\sqrt {a+\sqrt {a^2+b}}}\right )}{2 \sqrt {2} \sqrt {a^2+b} \sqrt {a+\sqrt {a^2+b}}}-\frac {\log \left (\sqrt {a^2+b}-\sqrt {2} \sqrt {-a+\sqrt {a^2+b}} x+x^2\right )}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {-a+\sqrt {a^2+b}}}+\frac {\log \left (\sqrt {a^2+b}+\sqrt {2} \sqrt {-a+\sqrt {a^2+b}} x+x^2\right )}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {-a+\sqrt {a^2+b}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.03, size = 81, normalized size = 0.27 \begin {gather*} -\frac {i \left (\frac {\tan ^{-1}\left (\frac {x}{\sqrt {a-i \sqrt {b}}}\right )}{\sqrt {a-i \sqrt {b}}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {a+i \sqrt {b}}}\right )}{\sqrt {a+i \sqrt {b}}}\right )}{2 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(609\) vs.
\(2(223)=446\).
time = 0.09, size = 610, normalized size = 2.04
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{2}+a^{2}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R} a}\right )}{4}\) | \(37\) |
default | \(\frac {\frac {\left (\sqrt {2 \sqrt {a^{2}+b}-2 a}\, \sqrt {a^{2}+b}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b}-2 a}\, a^{3}+\sqrt {2 \sqrt {a^{2}+b}-2 a}\, \sqrt {a^{2}+b}\, b +\sqrt {2 \sqrt {a^{2}+b}-2 a}\, a b \right ) \ln \left (x^{2}+x \sqrt {2 \sqrt {a^{2}+b}-2 a}+\sqrt {a^{2}+b}\right )}{2}+\frac {2 \left (2 a^{2} b +2 b^{2}-\frac {\left (\sqrt {2 \sqrt {a^{2}+b}-2 a}\, \sqrt {a^{2}+b}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b}-2 a}\, a^{3}+\sqrt {2 \sqrt {a^{2}+b}-2 a}\, \sqrt {a^{2}+b}\, b +\sqrt {2 \sqrt {a^{2}+b}-2 a}\, a b \right ) \sqrt {2 \sqrt {a^{2}+b}-2 a}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {2 \sqrt {a^{2}+b}-2 a}}{\sqrt {2 \sqrt {a^{2}+b}+2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b}+2 a}}}{4 b \left (a^{2}+b \right )^{\frac {3}{2}}}+\frac {-\frac {\left (\sqrt {2 \sqrt {a^{2}+b}-2 a}\, \sqrt {a^{2}+b}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b}-2 a}\, a^{3}+\sqrt {2 \sqrt {a^{2}+b}-2 a}\, \sqrt {a^{2}+b}\, b +\sqrt {2 \sqrt {a^{2}+b}-2 a}\, a b \right ) \ln \left (x \sqrt {2 \sqrt {a^{2}+b}-2 a}-x^{2}-\sqrt {a^{2}+b}\right )}{2}+\frac {2 \left (-2 a^{2} b -2 b^{2}+\frac {\left (\sqrt {2 \sqrt {a^{2}+b}-2 a}\, \sqrt {a^{2}+b}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b}-2 a}\, a^{3}+\sqrt {2 \sqrt {a^{2}+b}-2 a}\, \sqrt {a^{2}+b}\, b +\sqrt {2 \sqrt {a^{2}+b}-2 a}\, a b \right ) \sqrt {2 \sqrt {a^{2}+b}-2 a}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b}-2 a}-2 x}{\sqrt {2 \sqrt {a^{2}+b}+2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b}+2 a}}}{4 b \left (a^{2}+b \right )^{\frac {3}{2}}}\) | \(610\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 583 vs.
\(2 (225) = 450\).
time = 0.42, size = 583, normalized size = 1.95 \begin {gather*} \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} + a}{a^{2} b + b^{2}}} \log \left ({\left ({\left (a^{3} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} + b\right )} \sqrt {\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} + a}{a^{2} b + b^{2}}} + x\right ) - \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} + a}{a^{2} b + b^{2}}} \log \left (-{\left ({\left (a^{3} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} + b\right )} \sqrt {\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} + a}{a^{2} b + b^{2}}} + x\right ) - \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} - a}{a^{2} b + b^{2}}} \log \left ({\left ({\left (a^{3} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} - b\right )} \sqrt {-\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} - a}{a^{2} b + b^{2}}} + x\right ) + \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} - a}{a^{2} b + b^{2}}} \log \left (-{\left ({\left (a^{3} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} - b\right )} \sqrt {-\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} - a}{a^{2} b + b^{2}}} + x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.47, size = 63, normalized size = 0.21 \begin {gather*} \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} b^{2} + 256 b^{3}\right ) - 32 t^{2} a b + 1, \left ( t \mapsto t \log {\left (64 t^{3} a^{3} b + 64 t^{3} a b^{2} - 4 t a^{2} + 4 t b + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.73, size = 75, normalized size = 0.25 \begin {gather*} -\frac {\sqrt {a + \sqrt {-b}} \arctan \left (\frac {x}{\sqrt {a + \sqrt {-b}}}\right )}{2 \, {\left (a \sqrt {-b} - b\right )}} + \frac {\sqrt {a - \sqrt {-b}} \arctan \left (\frac {x}{\sqrt {a - \sqrt {-b}}}\right )}{2 \, {\left (a \sqrt {-b} + b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.38, size = 872, normalized size = 2.92 \begin {gather*} -2\,\mathrm {atanh}\left (\frac {8\,x\,\sqrt {\frac {a\,b}{16\,\left (a^2\,b^2+b^3\right )}-\frac {\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}}}{\frac {2\,b\,\sqrt {-b^3}}{a^2\,b^2+b^3}-\frac {2\,a\,b^2}{a^2\,b^2+b^3}}-\frac {8\,a^2\,b^2\,x\,\sqrt {\frac {a\,b}{16\,\left (a^2\,b^2+b^3\right )}-\frac {\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}}}{\frac {2\,b^4\,\sqrt {-b^3}}{a^2\,b^2+b^3}-\frac {2\,a^3\,b^4}{a^2\,b^2+b^3}-\frac {2\,a\,b^5}{a^2\,b^2+b^3}+\frac {2\,a^2\,b^3\,\sqrt {-b^3}}{a^2\,b^2+b^3}}+\frac {8\,a\,b\,x\,\sqrt {\frac {a\,b}{16\,\left (a^2\,b^2+b^3\right )}-\frac {\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}}\,\sqrt {-b^3}}{\frac {2\,b^4\,\sqrt {-b^3}}{a^2\,b^2+b^3}-\frac {2\,a^3\,b^4}{a^2\,b^2+b^3}-\frac {2\,a\,b^5}{a^2\,b^2+b^3}+\frac {2\,a^2\,b^3\,\sqrt {-b^3}}{a^2\,b^2+b^3}}\right )\,\sqrt {\frac {a\,b-\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}}-2\,\mathrm {atanh}\left (\frac {8\,a^2\,b^2\,x\,\sqrt {\frac {\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}+\frac {a\,b}{16\,\left (a^2\,b^2+b^3\right )}}}{\frac {2\,b^4\,\sqrt {-b^3}}{a^2\,b^2+b^3}+\frac {2\,a^3\,b^4}{a^2\,b^2+b^3}+\frac {2\,a\,b^5}{a^2\,b^2+b^3}+\frac {2\,a^2\,b^3\,\sqrt {-b^3}}{a^2\,b^2+b^3}}-\frac {8\,x\,\sqrt {\frac {\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}+\frac {a\,b}{16\,\left (a^2\,b^2+b^3\right )}}}{\frac {2\,b\,\sqrt {-b^3}}{a^2\,b^2+b^3}+\frac {2\,a\,b^2}{a^2\,b^2+b^3}}+\frac {8\,a\,b\,x\,\sqrt {\frac {\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}+\frac {a\,b}{16\,\left (a^2\,b^2+b^3\right )}}\,\sqrt {-b^3}}{\frac {2\,b^4\,\sqrt {-b^3}}{a^2\,b^2+b^3}+\frac {2\,a^3\,b^4}{a^2\,b^2+b^3}+\frac {2\,a\,b^5}{a^2\,b^2+b^3}+\frac {2\,a^2\,b^3\,\sqrt {-b^3}}{a^2\,b^2+b^3}}\right )\,\sqrt {\frac {a\,b+\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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