3.360 \(\int \frac{5+x^3}{\left (10-6 x+x^2\right ) \left (\frac{1}{2}-x+x^2\right )} \, dx\)

Optimal. Leaf size=49 \[ \frac{56}{221} \log \left (x^2-6 x+10\right )+\frac{109}{442} \log \left (2 x^2-2 x+1\right )-\frac{261}{221} \tan ^{-1}(1-2 x)-\frac{1026}{221} \tan ^{-1}(3-x) \]

[Out]

(-261*ArcTan[1 - 2*x])/221 - (1026*ArcTan[3 - x])/221 + (56*Log[10 - 6*x + x^2])
/221 + (109*Log[1 - 2*x + 2*x^2])/442

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Rubi [A]  time = 0.257859, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{56}{221} \log \left (x^2-6 x+10\right )+\frac{109}{442} \log \left (2 x^2-2 x+1\right )-\frac{261}{221} \tan ^{-1}(1-2 x)-\frac{1026}{221} \tan ^{-1}(3-x) \]

Antiderivative was successfully verified.

[In]  Int[(5 + x^3)/((10 - 6*x + x^2)*(1/2 - x + x^2)),x]

[Out]

(-261*ArcTan[1 - 2*x])/221 - (1026*ArcTan[3 - x])/221 + (56*Log[10 - 6*x + x^2])
/221 + (109*Log[1 - 2*x + 2*x^2])/442

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Rubi in Sympy [A]  time = 54.7232, size = 46, normalized size = 0.94 \[ \frac{56 \log{\left (x^{2} - 6 x + 10 \right )}}{221} + \frac{109 \log{\left (2 x^{2} - 2 x + 1 \right )}}{442} + \frac{1026 \operatorname{atan}{\left (x - 3 \right )}}{221} + \frac{261 \operatorname{atan}{\left (2 x - 1 \right )}}{221} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3+5)/(x**2-6*x+10)/(1/2-x+x**2),x)

[Out]

56*log(x**2 - 6*x + 10)/221 + 109*log(2*x**2 - 2*x + 1)/442 + 1026*atan(x - 3)/2
21 + 261*atan(2*x - 1)/221

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Mathematica [A]  time = 0.0229265, size = 49, normalized size = 1. \[ \frac{56}{221} \log \left (x^2-6 x+10\right )+\frac{109}{442} \log \left (2 x^2-2 x+1\right )-\frac{261}{221} \tan ^{-1}(1-2 x)-\frac{1026}{221} \tan ^{-1}(3-x) \]

Antiderivative was successfully verified.

[In]  Integrate[(5 + x^3)/((10 - 6*x + x^2)*(1/2 - x + x^2)),x]

[Out]

(-261*ArcTan[1 - 2*x])/221 - (1026*ArcTan[3 - x])/221 + (56*Log[10 - 6*x + x^2])
/221 + (109*Log[1 - 2*x + 2*x^2])/442

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Maple [A]  time = 0.01, size = 40, normalized size = 0.8 \[{\frac{261\,\arctan \left ( 2\,x-1 \right ) }{221}}+{\frac{1026\,\arctan \left ( -3+x \right ) }{221}}+{\frac{56\,\ln \left ({x}^{2}-6\,x+10 \right ) }{221}}+{\frac{109\,\ln \left ( 2\,{x}^{2}-2\,x+1 \right ) }{442}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+5)/(x^2-6*x+10)/(1/2-x+x^2),x)

[Out]

261/221*arctan(2*x-1)+1026/221*arctan(-3+x)+56/221*ln(x^2-6*x+10)+109/442*ln(2*x
^2-2*x+1)

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Maxima [A]  time = 0.883293, size = 53, normalized size = 1.08 \[ \frac{261}{221} \, \arctan \left (2 \, x - 1\right ) + \frac{1026}{221} \, \arctan \left (x - 3\right ) + \frac{109}{442} \, \log \left (2 \, x^{2} - 2 \, x + 1\right ) + \frac{56}{221} \, \log \left (x^{2} - 6 \, x + 10\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(x^3 + 5)/((2*x^2 - 2*x + 1)*(x^2 - 6*x + 10)),x, algorithm="maxima")

[Out]

261/221*arctan(2*x - 1) + 1026/221*arctan(x - 3) + 109/442*log(2*x^2 - 2*x + 1)
+ 56/221*log(x^2 - 6*x + 10)

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Fricas [A]  time = 0.277207, size = 50, normalized size = 1.02 \[ \frac{261}{221} \, \arctan \left (2 \, x - 1\right ) + \frac{1026}{221} \, \arctan \left (x - 3\right ) + \frac{109}{442} \, \log \left (x^{2} - x + \frac{1}{2}\right ) + \frac{56}{221} \, \log \left (x^{2} - 6 \, x + 10\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(x^3 + 5)/((2*x^2 - 2*x + 1)*(x^2 - 6*x + 10)),x, algorithm="fricas")

[Out]

261/221*arctan(2*x - 1) + 1026/221*arctan(x - 3) + 109/442*log(x^2 - x + 1/2) +
56/221*log(x^2 - 6*x + 10)

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Sympy [A]  time = 0.63495, size = 44, normalized size = 0.9 \[ \frac{56 \log{\left (x^{2} - 6 x + 10 \right )}}{221} + \frac{109 \log{\left (x^{2} - x + \frac{1}{2} \right )}}{442} + \frac{1026 \operatorname{atan}{\left (x - 3 \right )}}{221} + \frac{261 \operatorname{atan}{\left (2 x - 1 \right )}}{221} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+5)/(x**2-6*x+10)/(1/2-x+x**2),x)

[Out]

56*log(x**2 - 6*x + 10)/221 + 109*log(x**2 - x + 1/2)/442 + 1026*atan(x - 3)/221
+ 261*atan(2*x - 1)/221

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GIAC/XCAS [A]  time = 0.25985, size = 53, normalized size = 1.08 \[ \frac{261}{221} \, \arctan \left (2 \, x - 1\right ) + \frac{1026}{221} \, \arctan \left (x - 3\right ) + \frac{109}{442} \,{\rm ln}\left (2 \, x^{2} - 2 \, x + 1\right ) + \frac{56}{221} \,{\rm ln}\left (x^{2} - 6 \, x + 10\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(x^3 + 5)/((2*x^2 - 2*x + 1)*(x^2 - 6*x + 10)),x, algorithm="giac")

[Out]

261/221*arctan(2*x - 1) + 1026/221*arctan(x - 3) + 109/442*ln(2*x^2 - 2*x + 1) +
56/221*ln(x^2 - 6*x + 10)