Optimal. Leaf size=122 \[ -\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [4]{a} x+\sqrt {a}+x^2\right )}{4 \sqrt [4]{a}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [4]{a} x+\sqrt {a}+x^2\right )}{4 \sqrt [4]{a}}-\frac {\tan ^{-1}\left (\sqrt {3}-\frac {2 x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}+\frac {\tan ^{-1}\left (\frac {2 x}{\sqrt [4]{a}}+\sqrt {3}\right )}{2 \sqrt [4]{a}} \]
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Rubi [A] time = 0.08, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1169, 634, 617, 204, 628} \begin {gather*} -\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [4]{a} x+\sqrt {a}+x^2\right )}{4 \sqrt [4]{a}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [4]{a} x+\sqrt {a}+x^2\right )}{4 \sqrt [4]{a}}-\frac {\tan ^{-1}\left (\sqrt {3}-\frac {2 x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}+\frac {\tan ^{-1}\left (\frac {2 x}{\sqrt [4]{a}}+\sqrt {3}\right )}{2 \sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 634
Rule 1169
Rubi steps
\begin {align*} \int \frac {2 \sqrt {a}-x^2}{a-\sqrt {a} x^2+x^4} \, dx &=\frac {\int \frac {2 \sqrt {3} a^{3/4}-3 \sqrt {a} x}{\sqrt {a}-\sqrt {3} \sqrt [4]{a} x+x^2} \, dx}{2 \sqrt {3} a^{3/4}}+\frac {\int \frac {2 \sqrt {3} a^{3/4}+3 \sqrt {a} x}{\sqrt {a}+\sqrt {3} \sqrt [4]{a} x+x^2} \, dx}{2 \sqrt {3} a^{3/4}}\\ &=\frac {1}{4} \int \frac {1}{\sqrt {a}-\sqrt {3} \sqrt [4]{a} x+x^2} \, dx+\frac {1}{4} \int \frac {1}{\sqrt {a}+\sqrt {3} \sqrt [4]{a} x+x^2} \, dx-\frac {\sqrt {3} \int \frac {-\sqrt {3} \sqrt [4]{a}+2 x}{\sqrt {a}-\sqrt {3} \sqrt [4]{a} x+x^2} \, dx}{4 \sqrt [4]{a}}+\frac {\sqrt {3} \int \frac {\sqrt {3} \sqrt [4]{a}+2 x}{\sqrt {a}+\sqrt {3} \sqrt [4]{a} x+x^2} \, dx}{4 \sqrt [4]{a}}\\ &=-\frac {\sqrt {3} \log \left (\sqrt {a}-\sqrt {3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt [4]{a}}+\frac {\sqrt {3} \log \left (\sqrt {a}+\sqrt {3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt [4]{a}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 x}{\sqrt {3} \sqrt [4]{a}}\right )}{2 \sqrt {3} \sqrt [4]{a}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 x}{\sqrt {3} \sqrt [4]{a}}\right )}{2 \sqrt {3} \sqrt [4]{a}}\\ &=-\frac {\tan ^{-1}\left (\sqrt {3}-\frac {2 x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}+\frac {\tan ^{-1}\left (\sqrt {3}+\frac {2 x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}-\frac {\sqrt {3} \log \left (\sqrt {a}-\sqrt {3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt [4]{a}}+\frac {\sqrt {3} \log \left (\sqrt {a}+\sqrt {3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt [4]{a}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 115, normalized size = 0.94 \begin {gather*} \frac {\sqrt [4]{-1} \left (\sqrt {\sqrt {3}-i} \left (\sqrt {3}-3 i\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {\sqrt {3}+i} \sqrt [4]{a}}\right )-\sqrt {\sqrt {3}+i} \left (\sqrt {3}+3 i\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {\sqrt {3}-i} \sqrt [4]{a}}\right )\right )}{2 \sqrt {6} \sqrt [4]{a}} \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 {2 \sqrt {a}-x^2}{a-\sqrt {a} x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.07, size = 251, normalized size = 2.06 \begin {gather*} \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {\sqrt {3} a \sqrt {-\frac {1}{a}} + \sqrt {a}}{a}} \log \left (\sqrt {\frac {1}{2}} \sqrt {a} \sqrt {\frac {\sqrt {3} a \sqrt {-\frac {1}{a}} + \sqrt {a}}{a}} + x\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {\sqrt {3} a \sqrt {-\frac {1}{a}} + \sqrt {a}}{a}} \log \left (-\sqrt {\frac {1}{2}} \sqrt {a} \sqrt {\frac {\sqrt {3} a \sqrt {-\frac {1}{a}} + \sqrt {a}}{a}} + x\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {\sqrt {3} a \sqrt {-\frac {1}{a}} - \sqrt {a}}{a}} \log \left (\sqrt {\frac {1}{2}} \sqrt {a} \sqrt {-\frac {\sqrt {3} a \sqrt {-\frac {1}{a}} - \sqrt {a}}{a}} + x\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {\sqrt {3} a \sqrt {-\frac {1}{a}} - \sqrt {a}}{a}} \log \left (-\sqrt {\frac {1}{2}} \sqrt {a} \sqrt {-\frac {\sqrt {3} a \sqrt {-\frac {1}{a}} - \sqrt {a}}{a}} + x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 96, normalized size = 0.79 \begin {gather*} \frac {\arctan \left (\frac {2 x +\sqrt {3}\, a^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{2 a^{\frac {1}{4}}}-\frac {\arctan \left (\frac {-2 x +\sqrt {3}\, a^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{2 a^{\frac {1}{4}}}+\frac {\sqrt {3}\, \ln \left (x^{2}+\sqrt {3}\, a^{\frac {1}{4}} x +\sqrt {a}\right )}{4 a^{\frac {1}{4}}}-\frac {\sqrt {3}\, \ln \left (-x^{2}+\sqrt {3}\, a^{\frac {1}{4}} x -\sqrt {a}\right )}{4 a^{\frac {1}{4}}} \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 {x^{2} - 2 \, \sqrt {a}}{x^{4} - \sqrt {a} x^{2} + a}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.06, size = 159, normalized size = 1.30 \begin {gather*} 2\,\mathrm {atanh}\left (x\,\sqrt {\frac {1}{8\,\sqrt {a}}-\frac {\sqrt {-27\,a^3}}{24\,a^2}}-\frac {9\,a^{3/2}\,x\,\sqrt {\frac {1}{8\,\sqrt {a}}-\frac {\sqrt {-27\,a^3}}{24\,a^2}}}{\sqrt {-27\,a^3}}\right )\,\sqrt {\frac {1}{8\,\sqrt {a}}-\frac {\sqrt {-27\,a^3}}{24\,a^2}}+2\,\mathrm {atanh}\left (x\,\sqrt {\frac {\sqrt {-27\,a^3}}{24\,a^2}+\frac {1}{8\,\sqrt {a}}}+\frac {9\,a^{3/2}\,x\,\sqrt {\frac {\sqrt {-27\,a^3}}{24\,a^2}+\frac {1}{8\,\sqrt {a}}}}{\sqrt {-27\,a^3}}\right )\,\sqrt {\frac {\sqrt {-27\,a^3}}{24\,a^2}+\frac {1}{8\,\sqrt {a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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