Optimal. Leaf size=86 \[ -\frac{502 x+313}{1452 \left (2 x^2+1\right )}+\frac{2843 \log \left (2 x^2+1\right )}{7986}+\frac{5828}{9075 (2-5 x)}-\frac{59096 \log (2-5 x)}{99825}+\frac{272 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x\right )}{1331}-\frac{251 \tan ^{-1}\left (\sqrt{2} x\right )}{726 \sqrt{2}} \]
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Rubi [A] time = 0.0978061, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2074, 639, 203, 635, 260} \[ -\frac{502 x+313}{1452 \left (2 x^2+1\right )}+\frac{2843 \log \left (2 x^2+1\right )}{7986}+\frac{5828}{9075 (2-5 x)}-\frac{59096 \log (2-5 x)}{99825}+\frac{272 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x\right )}{1331}-\frac{251 \tan ^{-1}\left (\sqrt{2} x\right )}{726 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 2074
Rule 639
Rule 203
Rule 635
Rule 260
Rubi steps
\begin{align*} \int \frac{8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx &=\int \left (\frac{5828}{1815 (-2+5 x)^2}-\frac{59096}{19965 (-2+5 x)}+\frac{-251+313 x}{363 \left (1+2 x^2\right )^2}+\frac{2 (816+2843 x)}{3993 \left (1+2 x^2\right )}\right ) \, dx\\ &=\frac{5828}{9075 (2-5 x)}-\frac{59096 \log (2-5 x)}{99825}+\frac{2 \int \frac{816+2843 x}{1+2 x^2} \, dx}{3993}+\frac{1}{363} \int \frac{-251+313 x}{\left (1+2 x^2\right )^2} \, dx\\ &=\frac{5828}{9075 (2-5 x)}-\frac{313+502 x}{1452 \left (1+2 x^2\right )}-\frac{59096 \log (2-5 x)}{99825}-\frac{251}{726} \int \frac{1}{1+2 x^2} \, dx+\frac{544 \int \frac{1}{1+2 x^2} \, dx}{1331}+\frac{5686 \int \frac{x}{1+2 x^2} \, dx}{3993}\\ &=\frac{5828}{9075 (2-5 x)}-\frac{313+502 x}{1452 \left (1+2 x^2\right )}-\frac{251 \tan ^{-1}\left (\sqrt{2} x\right )}{726 \sqrt{2}}+\frac{272 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x\right )}{1331}-\frac{59096 \log (2-5 x)}{99825}+\frac{2843 \log \left (1+2 x^2\right )}{7986}\\ \end{align*}
Mathematica [A] time = 0.0438645, size = 67, normalized size = 0.78 \[ \frac{-\frac{33 \left (36458 x^2+4675 x+2554\right )}{10 x^3-4 x^2+5 x-2}+142150 \log \left (2 x^2+1\right )-236384 \log (2-5 x)+12575 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x\right )}{399300} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 54, normalized size = 0.6 \begin{align*}{\frac{1}{3993} \left ( -{\frac{2761\,x}{4}}-{\frac{3443}{8}} \right ) \left ({x}^{2}+{\frac{1}{2}} \right ) ^{-1}}+{\frac{2843\,\ln \left ( 2\,{x}^{2}+1 \right ) }{7986}}+{\frac{503\,\arctan \left ( x\sqrt{2} \right ) \sqrt{2}}{15972}}-{\frac{5828}{45375\,x-18150}}-{\frac{59096\,\ln \left ( 5\,x-2 \right ) }{99825}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42057, size = 80, normalized size = 0.93 \begin{align*} \frac{503}{15972} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) - \frac{36458 \, x^{2} + 4675 \, x + 2554}{12100 \,{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )}} + \frac{2843}{7986} \, \log \left (2 \, x^{2} + 1\right ) - \frac{59096}{99825} \, \log \left (5 \, x - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9806, size = 312, normalized size = 3.63 \begin{align*} \frac{12575 \, \sqrt{2}{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \arctan \left (\sqrt{2} x\right ) - 1203114 \, x^{2} + 142150 \,{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \log \left (2 \, x^{2} + 1\right ) - 236384 \,{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \log \left (5 \, x - 2\right ) - 154275 \, x - 84282}{399300 \,{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.203387, size = 63, normalized size = 0.73 \begin{align*} - \frac{36458 x^{2} + 4675 x + 2554}{121000 x^{3} - 48400 x^{2} + 60500 x - 24200} - \frac{59096 \log{\left (x - \frac{2}{5} \right )}}{99825} + \frac{2843 \log{\left (x^{2} + \frac{1}{2} \right )}}{7986} + \frac{503 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x \right )}}{15972} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0753, size = 80, normalized size = 0.93 \begin{align*} \frac{503}{15972} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) - \frac{36458 \, x^{2} + 4675 \, x + 2554}{12100 \,{\left (2 \, x^{2} + 1\right )}{\left (5 \, x - 2\right )}} + \frac{2843}{7986} \, \log \left (2 \, x^{2} + 1\right ) - \frac{59096}{99825} \, \log \left ({\left | 5 \, x - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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