3.321 \(\int \frac{8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx\)

Optimal. Leaf size=69 \[ -\frac{502 x+313}{1452 \left (2 x^2+1\right )}+\frac{2843 \log \left (2 x^2+1\right )}{7986}+\frac{5828}{9075 (2-5 x)}-\frac{59096 \log (2-5 x)}{99825}+\frac{503 \tan ^{-1}\left (\sqrt{2} x\right )}{7986 \sqrt{2}} \]

[Out]

5828/(9075*(2 - 5*x)) - (313 + 502*x)/(1452*(1 + 2*x^2)) + (503*ArcTan[Sqrt[2]*x
])/(7986*Sqrt[2]) - (59096*Log[2 - 5*x])/99825 + (2843*Log[1 + 2*x^2])/7986

_______________________________________________________________________________________

Rubi [A]  time = 0.199109, antiderivative size = 86, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 5, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{502 x+313}{1452 \left (2 x^2+1\right )}+\frac{2843 \log \left (2 x^2+1\right )}{7986}+\frac{5828}{9075 (2-5 x)}-\frac{59096 \log (2-5 x)}{99825}+\frac{272 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x\right )}{1331}-\frac{251 \tan ^{-1}\left (\sqrt{2} x\right )}{726 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(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),x]

[Out]

5828/(9075*(2 - 5*x)) - (313 + 502*x)/(1452*(1 + 2*x^2)) - (251*ArcTan[Sqrt[2]*x
])/(726*Sqrt[2]) + (272*Sqrt[2]*ArcTan[Sqrt[2]*x])/1331 - (59096*Log[2 - 5*x])/9
9825 + (2843*Log[1 + 2*x^2])/7986

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((12*x**5-7*x**3-13*x**2+8)/(100*x**6-80*x**5+116*x**4-80*x**3+41*x**2-20*x+4),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.0749282, size = 67, normalized size = 0.97 \[ \frac{142150 \log \left (2 x^2+1\right )-\frac{33 \left (36458 x^2+4675 x+2554\right )}{10 x^3-4 x^2+5 x-2}-236384 \log (2-5 x)+12575 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x\right )}{399300} \]

Antiderivative was successfully verified.

[In]  Integrate[(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),x]

[Out]

((-33*(2554 + 4675*x + 36458*x^2))/(-2 + 5*x - 4*x^2 + 10*x^3) + 12575*Sqrt[2]*A
rcTan[Sqrt[2]*x] - 236384*Log[2 - 5*x] + 142150*Log[1 + 2*x^2])/399300

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 54, normalized size = 0.8 \[{\frac{1}{3993} \left ( -{\frac{2761\,x}{4}}-{\frac{3443}{8}} \right ) \left ({x}^{2}+{\frac{1}{2}} \right ) ^{-1}}+{\frac{2843\,\ln \left ( 4\,{x}^{2}+2 \right ) }{7986}}+{\frac{503\,\arctan \left ( \sqrt{2}x \right ) \sqrt{2}}{15972}}-{\frac{5828}{45375\,x-18150}}-{\frac{59096\,\ln \left ( 5\,x-2 \right ) }{99825}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((12*x^5-7*x^3-13*x^2+8)/(100*x^6-80*x^5+116*x^4-80*x^3+41*x^2-20*x+4),x)

[Out]

1/3993*(-2761/4*x-3443/8)/(x^2+1/2)+2843/7986*ln(4*x^2+2)+503/15972*arctan(2^(1/
2)*x)*2^(1/2)-5828/9075/(5*x-2)-59096/99825*ln(5*x-2)

_______________________________________________________________________________________

Maxima [A]  time = 0.875479, size = 80, normalized size = 1.16 \[ \frac{503}{15972} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) - \frac{36458 \, x^{2} + 4675 \, x + 2554}{12100 \,{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )}} + \frac{2843}{7986} \, \log \left (2 \, x^{2} + 1\right ) - \frac{59096}{99825} \, \log \left (5 \, x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((12*x^5 - 7*x^3 - 13*x^2 + 8)/(100*x^6 - 80*x^5 + 116*x^4 - 80*x^3 + 41*x^2 - 20*x + 4),x, algorithm="maxima")

[Out]

503/15972*sqrt(2)*arctan(sqrt(2)*x) - 1/12100*(36458*x^2 + 4675*x + 2554)/(10*x^
3 - 4*x^2 + 5*x - 2) + 2843/7986*log(2*x^2 + 1) - 59096/99825*log(5*x - 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.266174, size = 155, normalized size = 2.25 \[ \frac{\sqrt{2}{\left (142150 \, \sqrt{2}{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \log \left (2 \, x^{2} + 1\right ) - 236384 \, \sqrt{2}{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \log \left (5 \, x - 2\right ) + 25150 \,{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \arctan \left (\sqrt{2} x\right ) - 33 \, \sqrt{2}{\left (36458 \, x^{2} + 4675 \, x + 2554\right )}\right )}}{798600 \,{\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((12*x^5 - 7*x^3 - 13*x^2 + 8)/(100*x^6 - 80*x^5 + 116*x^4 - 80*x^3 + 41*x^2 - 20*x + 4),x, algorithm="fricas")

[Out]

1/798600*sqrt(2)*(142150*sqrt(2)*(10*x^3 - 4*x^2 + 5*x - 2)*log(2*x^2 + 1) - 236
384*sqrt(2)*(10*x^3 - 4*x^2 + 5*x - 2)*log(5*x - 2) + 25150*(10*x^3 - 4*x^2 + 5*
x - 2)*arctan(sqrt(2)*x) - 33*sqrt(2)*(36458*x^2 + 4675*x + 2554))/(10*x^3 - 4*x
^2 + 5*x - 2)

_______________________________________________________________________________________

Sympy [A]  time = 0.565926, size = 63, normalized size = 0.91 \[ - \frac{36458 x^{2} + 4675 x + 2554}{121000 x^{3} - 48400 x^{2} + 60500 x - 24200} - \frac{59096 \log{\left (x - \frac{2}{5} \right )}}{99825} + \frac{2843 \log{\left (x^{2} + \frac{1}{2} \right )}}{7986} + \frac{503 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x \right )}}{15972} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((12*x**5-7*x**3-13*x**2+8)/(100*x**6-80*x**5+116*x**4-80*x**3+41*x**2-20*x+4),x)

[Out]

-(36458*x**2 + 4675*x + 2554)/(121000*x**3 - 48400*x**2 + 60500*x - 24200) - 590
96*log(x - 2/5)/99825 + 2843*log(x**2 + 1/2)/7986 + 503*sqrt(2)*atan(sqrt(2)*x)/
15972

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.26206, size = 80, normalized size = 1.16 \[ \frac{503}{15972} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) - \frac{36458 \, x^{2} + 4675 \, x + 2554}{12100 \,{\left (2 \, x^{2} + 1\right )}{\left (5 \, x - 2\right )}} + \frac{2843}{7986} \,{\rm ln}\left (2 \, x^{2} + 1\right ) - \frac{59096}{99825} \,{\rm ln}\left ({\left | 5 \, x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((12*x^5 - 7*x^3 - 13*x^2 + 8)/(100*x^6 - 80*x^5 + 116*x^4 - 80*x^3 + 41*x^2 - 20*x + 4),x, algorithm="giac")

[Out]

503/15972*sqrt(2)*arctan(sqrt(2)*x) - 1/12100*(36458*x^2 + 4675*x + 2554)/((2*x^
2 + 1)*(5*x - 2)) + 2843/7986*ln(2*x^2 + 1) - 59096/99825*ln(abs(5*x - 2))