Optimal. Leaf size=33 \[ \frac {7}{4} \log \left (x^2-8 x+21\right )-\frac {13 \tan ^{-1}\left (\frac {4-x}{\sqrt {5}}\right )}{\sqrt {5}} \]
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Rubi [A] time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {634, 618, 204, 628} \begin {gather*} \frac {7}{4} \log \left (x^2-8 x+21\right )-\frac {13 \tan ^{-1}\left (\frac {4-x}{\sqrt {5}}\right )}{\sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {-2+7 x}{42-16 x+2 x^2} \, dx &=\frac {7}{4} \int \frac {-16+4 x}{42-16 x+2 x^2} \, dx+26 \int \frac {1}{42-16 x+2 x^2} \, dx\\ &=\frac {7}{4} \log \left (21-8 x+x^2\right )-52 \operatorname {Subst}\left (\int \frac {1}{-80-x^2} \, dx,x,-16+4 x\right )\\ &=-\frac {13 \tan ^{-1}\left (\frac {4-x}{\sqrt {5}}\right )}{\sqrt {5}}+\frac {7}{4} \log \left (21-8 x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 1.06 \begin {gather*} \frac {1}{2} \left (\frac {7}{2} \log \left (x^2-8 x+21\right )+\frac {26 \tan ^{-1}\left (\frac {x-4}{\sqrt {5}}\right )}{\sqrt {5}}\right ) \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+7 x}{42-16 x+2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 26, normalized size = 0.79 \begin {gather*} \frac {13}{5} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (x - 4\right )}\right ) + \frac {7}{4} \, \log \left (x^{2} - 8 \, x + 21\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 26, normalized size = 0.79 \begin {gather*} \frac {13}{5} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (x - 4\right )}\right ) + \frac {7}{4} \, \log \left (x^{2} - 8 \, x + 21\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 29, normalized size = 0.88 \begin {gather*} \frac {13 \sqrt {5}\, \arctan \left (\frac {\left (2 x -8\right ) \sqrt {5}}{10}\right )}{5}+\frac {7 \ln \left (x^{2}-8 x +21\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 26, normalized size = 0.79 \begin {gather*} \frac {13}{5} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (x - 4\right )}\right ) + \frac {7}{4} \, \log \left (x^{2} - 8 \, x + 21\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 30, normalized size = 0.91 \begin {gather*} \frac {7\,\ln \left (x^2-8\,x+21\right )}{4}+\frac {13\,\sqrt {5}\,\mathrm {atan}\left (\frac {\sqrt {5}\,x}{5}-\frac {4\,\sqrt {5}}{5}\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 39, normalized size = 1.18 \begin {gather*} \frac {7 \log {\left (x^{2} - 8 x + 21 \right )}}{4} + \frac {13 \sqrt {5} \operatorname {atan}{\left (\frac {\sqrt {5} x}{5} - \frac {4 \sqrt {5}}{5} \right )}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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