Optimal. Leaf size=48 \[ \frac {\tan ^{-1}\left (\frac {2 \sqrt {42} \sqrt {x^4-3 x^3-21 x^2+83 x-60}}{19 x^2-86 x+103}\right )}{\sqrt {42}} \]
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Rubi [F] time = 0.77, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx &=\int \left (-\frac {5}{23 \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}+\frac {72 (23-9 x)}{23 \left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}\right ) \, dx\\ &=-\left (\frac {5}{23} \int \frac {1}{\sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\right )+\frac {72}{23} \int \frac {23-9 x}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\\ &=-\left (\frac {5}{23} \int \frac {1}{\sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\right )+\frac {72}{23} \int \left (\frac {-9-\frac {10 \sqrt {2}}{3}}{\left (82-48 \sqrt {2}-46 x\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}+\frac {-9+\frac {10 \sqrt {2}}{3}}{\left (82+48 \sqrt {2}-46 x\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}\right ) \, dx\\ &=-\left (\frac {5}{23} \int \frac {1}{\sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\right )-\frac {1}{23} \left (24 \left (27-10 \sqrt {2}\right )\right ) \int \frac {1}{\left (82+48 \sqrt {2}-46 x\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx-\frac {1}{23} \left (24 \left (27+10 \sqrt {2}\right )\right ) \int \frac {1}{\left (82-48 \sqrt {2}-46 x\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.46, size = 196, normalized size = 4.08 \begin {gather*} \frac {\sqrt {\frac {x-4}{x-1}} \sqrt {\frac {x-3}{x-1}} (x-1)^2 \sqrt {\frac {x+5}{x-1}} \left (-7 F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {x+5}{x-1}}}{\sqrt {3}}\right )|\frac {3}{4}\right )+\left (5+4 \sqrt {2}\right ) \Pi \left (\frac {3}{14}-\frac {3 \sqrt {2}}{7};\sin ^{-1}\left (\frac {\sqrt {\frac {x+5}{x-1}}}{\sqrt {3}}\right )|\frac {3}{4}\right )+\left (5-4 \sqrt {2}\right ) \Pi \left (\frac {3}{14}+\frac {3 \sqrt {2}}{7};\sin ^{-1}\left (\frac {\sqrt {\frac {x+5}{x-1}}}{\sqrt {3}}\right )|\frac {3}{4}\right )\right )}{7 \sqrt {6} \sqrt {x^4-3 x^3-21 x^2+83 x-60}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 48, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {2 \sqrt {42} \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}{103-86 x+19 x^2}\right )}{\sqrt {42}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 49, normalized size = 1.02 \begin {gather*} \frac {1}{42} \, \sqrt {21} \sqrt {2} \arctan \left (\frac {2 \, \sqrt {21} \sqrt {2} \sqrt {x^{4} - 3 \, x^{3} - 21 \, x^{2} + 83 \, x - 60}}{19 \, x^{2} - 86 \, x + 103}\right ) \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 {5 \, x^{2} - 46 \, x + 77}{\sqrt {x^{4} - 3 \, x^{3} - 21 \, x^{2} + 83 \, x - 60} {\left (23 \, x^{2} - 82 \, x + 23\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.47, size = 75, normalized size = 1.56
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+42\right ) \ln \left (-\frac {19 \RootOf \left (\textit {\_Z}^{2}+42\right ) x^{2}-86 \RootOf \left (\textit {\_Z}^{2}+42\right ) x +103 \RootOf \left (\textit {\_Z}^{2}+42\right )+84 \sqrt {x^{4}-3 x^{3}-21 x^{2}+83 x -60}}{23 x^{2}-82 x +23}\right )}{42}\) | \(75\) |
default | \(\text {Expression too large to display}\) | \(3042\) |
elliptic | \(\text {Expression too large to display}\) | \(3260\) |
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 {5 \, x^{2} - 46 \, x + 77}{\sqrt {x^{4} - 3 \, x^{3} - 21 \, x^{2} + 83 \, x - 60} {\left (23 \, x^{2} - 82 \, x + 23\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 = -0.02 \begin {gather*} \int -\frac {5\,x^2-46\,x+77}{\left (23\,x^2-82\,x+23\right )\,\sqrt {x^4-3\,x^3-21\,x^2+83\,x-60}} \,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 \left (- \frac {46 x}{23 x^{2} \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} - 82 x \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} + 23 \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60}}\right )\, dx - \int \frac {5 x^{2}}{23 x^{2} \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} - 82 x \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} + 23 \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60}}\, dx - \int \frac {77}{23 x^{2} \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} - 82 x \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} + 23 \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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