Optimal. Leaf size=26 \[ \frac {\tan ^{-1}\left (\frac {x \sqrt {x^4-3}}{\sqrt {2}}\right )}{3 \sqrt {2}} \]
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Rubi [C] time = 0.65, antiderivative size = 131, normalized size of antiderivative = 5.04, number of steps used = 18, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {6725, 1215, 223, 1457, 540, 253, 538, 537} \begin {gather*} \frac {\sqrt {\sqrt {3}-x^2} \sqrt {\sqrt {3} x^2+3} \Pi \left (-\frac {\sqrt {3}}{2};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{3}}\right )\right |-1\right )}{6 \sqrt {3} \sqrt {x^4-3}}-\frac {2 \sqrt {\sqrt {3}-x^2} \sqrt {\sqrt {3} x^2+3} \Pi \left (\sqrt {3};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{3}}\right )\right |-1\right )}{3 \sqrt {3} \sqrt {x^4-3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 223
Rule 253
Rule 537
Rule 538
Rule 540
Rule 1215
Rule 1457
Rule 6725
Rubi steps
\begin {align*} \int \frac {1+x^2}{\left (-1+x^2\right ) \left (2+x^2\right ) \sqrt {-3+x^4}} \, dx &=\int \left (\frac {2}{3 \left (-1+x^2\right ) \sqrt {-3+x^4}}+\frac {1}{3 \left (2+x^2\right ) \sqrt {-3+x^4}}\right ) \, dx\\ &=\frac {1}{3} \int \frac {1}{\left (2+x^2\right ) \sqrt {-3+x^4}} \, dx+\frac {2}{3} \int \frac {1}{\left (-1+x^2\right ) \sqrt {-3+x^4}} \, dx\\ &=\frac {2 \int \frac {1}{\sqrt {-3+x^4}} \, dx}{3 \left (-1+\sqrt {3}\right )}+\frac {2 \int \frac {\sqrt {3}-x^2}{\left (-1+x^2\right ) \sqrt {-3+x^4}} \, dx}{3 \left (-1+\sqrt {3}\right )}+\frac {\int \frac {1}{\sqrt {-3+x^4}} \, dx}{3 \left (2+\sqrt {3}\right )}+\frac {\int \frac {\sqrt {3}-x^2}{\left (2+x^2\right ) \sqrt {-3+x^4}} \, dx}{3 \left (2+\sqrt {3}\right )}\\ &=-\frac {\sqrt {2} \sqrt {\frac {\sqrt {3}+x^2}{\sqrt {3}-x^2}} \sqrt {-3+\sqrt {3} x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {3} x^2}}\right )|\frac {1}{2}\right )}{3\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {-3+x^4}}+\frac {\sqrt {\frac {\sqrt {3}+x^2}{\sqrt {3}-x^2}} \sqrt {-3+\sqrt {3} x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {3} x^2}}\right )|\frac {1}{2}\right )}{3 \sqrt {2} 3^{3/4} \left (2+\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {-3+x^4}}+\frac {\left (2 \sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}\right ) \int \frac {\sqrt {\sqrt {3}-x^2}}{\sqrt {-\sqrt {3}-x^2} \left (-1+x^2\right )} \, dx}{3 \left (-1+\sqrt {3}\right ) \sqrt {-3+x^4}}+\frac {\left (\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}\right ) \int \frac {\sqrt {\sqrt {3}-x^2}}{\sqrt {-\sqrt {3}-x^2} \left (2+x^2\right )} \, dx}{3 \left (2+\sqrt {3}\right ) \sqrt {-3+x^4}}\\ &=-\frac {\sqrt {2} \sqrt {\frac {\sqrt {3}+x^2}{\sqrt {3}-x^2}} \sqrt {-3+\sqrt {3} x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {3} x^2}}\right )|\frac {1}{2}\right )}{3\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {-3+x^4}}+\frac {\sqrt {\frac {\sqrt {3}+x^2}{\sqrt {3}-x^2}} \sqrt {-3+\sqrt {3} x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {3} x^2}}\right )|\frac {1}{2}\right )}{3 \sqrt {2} 3^{3/4} \left (2+\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {-3+x^4}}+\frac {\left (\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2} \left (2+x^2\right )} \, dx}{3 \sqrt {-3+x^4}}+\frac {\left (2 \sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2} \left (-1+x^2\right )} \, dx}{3 \sqrt {-3+x^4}}-\frac {\left (2 \sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}} \, dx}{3 \left (-1+\sqrt {3}\right ) \sqrt {-3+x^4}}-\frac {\left (\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}} \, dx}{3 \left (2+\sqrt {3}\right ) \sqrt {-3+x^4}}\\ &=-\frac {\sqrt {2} \sqrt {\frac {\sqrt {3}+x^2}{\sqrt {3}-x^2}} \sqrt {-3+\sqrt {3} x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {3} x^2}}\right )|\frac {1}{2}\right )}{3\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {-3+x^4}}+\frac {\sqrt {\frac {\sqrt {3}+x^2}{\sqrt {3}-x^2}} \sqrt {-3+\sqrt {3} x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {3} x^2}}\right )|\frac {1}{2}\right )}{3 \sqrt {2} 3^{3/4} \left (2+\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {-3+x^4}}-\frac {2 \int \frac {1}{\sqrt {-3+x^4}} \, dx}{3 \left (-1+\sqrt {3}\right )}-\frac {\int \frac {1}{\sqrt {-3+x^4}} \, dx}{3 \left (2+\sqrt {3}\right )}+\frac {\left (\sqrt {\sqrt {3}-x^2} \sqrt {1+\frac {x^2}{\sqrt {3}}}\right ) \int \frac {1}{\sqrt {\sqrt {3}-x^2} \left (2+x^2\right ) \sqrt {1+\frac {x^2}{\sqrt {3}}}} \, dx}{3 \sqrt {-3+x^4}}+\frac {\left (2 \sqrt {\sqrt {3}-x^2} \sqrt {1+\frac {x^2}{\sqrt {3}}}\right ) \int \frac {1}{\sqrt {\sqrt {3}-x^2} \left (-1+x^2\right ) \sqrt {1+\frac {x^2}{\sqrt {3}}}} \, dx}{3 \sqrt {-3+x^4}}\\ &=\frac {\sqrt {\sqrt {3}-x^2} \sqrt {3+\sqrt {3} x^2} \Pi \left (-\frac {\sqrt {3}}{2};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{3}}\right )\right |-1\right )}{6 \sqrt {3} \sqrt {-3+x^4}}-\frac {2 \sqrt {\sqrt {3}-x^2} \sqrt {3+\sqrt {3} x^2} \Pi \left (\sqrt {3};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{3}}\right )\right |-1\right )}{3 \sqrt {3} \sqrt {-3+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.23, size = 74, normalized size = 2.85 \begin {gather*} \frac {\sqrt {3-x^4} \left (\Pi \left (-\frac {\sqrt {3}}{2};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{3}}\right )\right |-1\right )+4 i \Pi \left (-\sqrt {3};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt [4]{3}}\right )\right |-1\right )\right )}{6 \sqrt [4]{3} \sqrt {x^4-3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 13.44, size = 26, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {x \sqrt {x^4-3}}{\sqrt {2}}\right )}{3 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 1}{\sqrt {x^{4} - 3} {\left (x^{2} + 2\right )} {\left (x^{2} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 136, normalized size = 5.23 \begin {gather*} -\frac {2 \sqrt {1+\frac {\sqrt {3}\, x^{2}}{3}}\, \sqrt {1-\frac {\sqrt {3}\, x^{2}}{3}}\, \EllipticPi \left (\sqrt {-\frac {\sqrt {3}}{3}}\, x , -\sqrt {3}, \frac {3^{\frac {3}{4}}}{3 \sqrt {-\frac {\sqrt {3}}{3}}}\right )}{3 \sqrt {-\frac {\sqrt {3}}{3}}\, \sqrt {x^{4}-3}}+\frac {\sqrt {1+\frac {\sqrt {3}\, x^{2}}{3}}\, \sqrt {1-\frac {\sqrt {3}\, x^{2}}{3}}\, \EllipticPi \left (\sqrt {-\frac {\sqrt {3}}{3}}\, x , \frac {\sqrt {3}}{2}, \frac {3^{\frac {3}{4}}}{3 \sqrt {-\frac {\sqrt {3}}{3}}}\right )}{6 \sqrt {-\frac {\sqrt {3}}{3}}\, \sqrt {x^{4}-3}} \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} + 1}{\sqrt {x^{4} - 3} {\left (x^{2} + 2\right )} {\left (x^{2} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^2+1}{\left (x^2-1\right )\,\left (x^2+2\right )\,\sqrt {x^4-3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 2\right ) \sqrt {x^{4} - 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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