3.115 \(\int \frac{-20+8 x+5 x^3}{(-4+x)^3 \left (8-4 x+x^2\right )} \, dx\)

Optimal. Leaf size=58 \[ \frac{45}{32} \log \left (x^2-4 x+8\right )+\frac{41}{4 (4-x)}-\frac{83}{4 (4-x)^2}-\frac{45}{16} \log (4-x)-\frac{3}{16} \tan ^{-1}\left (1-\frac{x}{2}\right ) \]

[Out]

-83/(4*(4 - x)^2) + 41/(4*(4 - x)) - (3*ArcTan[1 - x/2])/16 - (45*Log[4 - x])/16
+ (45*Log[8 - 4*x + x^2])/32

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Rubi [A]  time = 0.0966032, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{45}{32} \log \left (x^2-4 x+8\right )+\frac{41}{4 (4-x)}-\frac{83}{4 (4-x)^2}-\frac{45}{16} \log (4-x)-\frac{3}{16} \tan ^{-1}\left (1-\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-20 + 8*x + 5*x^3)/((-4 + x)^3*(8 - 4*x + x^2)),x]

[Out]

-83/(4*(4 - x)^2) + 41/(4*(4 - x)) - (3*ArcTan[1 - x/2])/16 - (45*Log[4 - x])/16
+ (45*Log[8 - 4*x + x^2])/32

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Rubi in Sympy [A]  time = 20.5429, size = 46, normalized size = 0.79 \[ - \frac{45 \log{\left (- x + 4 \right )}}{16} + \frac{45 \log{\left (x^{2} - 4 x + 8 \right )}}{32} + \frac{3 \operatorname{atan}{\left (\frac{x}{2} - 1 \right )}}{16} + \frac{41}{4 \left (- x + 4\right )} - \frac{83}{4 \left (- x + 4\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**3+8*x-20)/(-4+x)**3/(x**2-4*x+8),x)

[Out]

-45*log(-x + 4)/16 + 45*log(x**2 - 4*x + 8)/32 + 3*atan(x/2 - 1)/16 + 41/(4*(-x
+ 4)) - 83/(4*(-x + 4)**2)

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Mathematica [A]  time = 0.0400513, size = 46, normalized size = 0.79 \[ \frac{1}{32} \left (45 \log \left (x^2-4 x+8\right )-\frac{328}{x-4}-\frac{664}{(x-4)^2}-90 \log (x-4)+6 \tan ^{-1}\left (\frac{x-2}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-20 + 8*x + 5*x^3)/((-4 + x)^3*(8 - 4*x + x^2)),x]

[Out]

(-664/(-4 + x)^2 - 328/(-4 + x) + 6*ArcTan[(-2 + x)/2] - 90*Log[-4 + x] + 45*Log
[8 - 4*x + x^2])/32

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Maple [A]  time = 0.013, size = 41, normalized size = 0.7 \[{\frac{45\,\ln \left ({x}^{2}-4\,x+8 \right ) }{32}}+{\frac{3}{16}\arctan \left ( -1+{\frac{x}{2}} \right ) }-{\frac{83}{4\, \left ( x-4 \right ) ^{2}}}-{\frac{41}{4\,x-16}}-{\frac{45\,\ln \left ( x-4 \right ) }{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^3+8*x-20)/(x-4)^3/(x^2-4*x+8),x)

[Out]

45/32*ln(x^2-4*x+8)+3/16*arctan(-1+1/2*x)-83/4/(x-4)^2-41/4/(x-4)-45/16*ln(x-4)

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Maxima [A]  time = 1.49062, size = 58, normalized size = 1. \[ -\frac{41 \, x - 81}{4 \,{\left (x^{2} - 8 \, x + 16\right )}} + \frac{3}{16} \, \arctan \left (\frac{1}{2} \, x - 1\right ) + \frac{45}{32} \, \log \left (x^{2} - 4 \, x + 8\right ) - \frac{45}{16} \, \log \left (x - 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^3 + 8*x - 20)/((x^2 - 4*x + 8)*(x - 4)^3),x, algorithm="maxima")

[Out]

-1/4*(41*x - 81)/(x^2 - 8*x + 16) + 3/16*arctan(1/2*x - 1) + 45/32*log(x^2 - 4*x
+ 8) - 45/16*log(x - 4)

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Fricas [A]  time = 0.217226, size = 89, normalized size = 1.53 \[ \frac{6 \,{\left (x^{2} - 8 \, x + 16\right )} \arctan \left (\frac{1}{2} \, x - 1\right ) + 45 \,{\left (x^{2} - 8 \, x + 16\right )} \log \left (x^{2} - 4 \, x + 8\right ) - 90 \,{\left (x^{2} - 8 \, x + 16\right )} \log \left (x - 4\right ) - 328 \, x + 648}{32 \,{\left (x^{2} - 8 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^3 + 8*x - 20)/((x^2 - 4*x + 8)*(x - 4)^3),x, algorithm="fricas")

[Out]

1/32*(6*(x^2 - 8*x + 16)*arctan(1/2*x - 1) + 45*(x^2 - 8*x + 16)*log(x^2 - 4*x +
8) - 90*(x^2 - 8*x + 16)*log(x - 4) - 328*x + 648)/(x^2 - 8*x + 16)

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Sympy [A]  time = 0.226978, size = 46, normalized size = 0.79 \[ - \frac{41 x - 81}{4 x^{2} - 32 x + 64} - \frac{45 \log{\left (x - 4 \right )}}{16} + \frac{45 \log{\left (x^{2} - 4 x + 8 \right )}}{32} + \frac{3 \operatorname{atan}{\left (\frac{x}{2} - 1 \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**3+8*x-20)/(-4+x)**3/(x**2-4*x+8),x)

[Out]

-(41*x - 81)/(4*x**2 - 32*x + 64) - 45*log(x - 4)/16 + 45*log(x**2 - 4*x + 8)/32
+ 3*atan(x/2 - 1)/16

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GIAC/XCAS [A]  time = 0.202825, size = 53, normalized size = 0.91 \[ -\frac{41 \, x - 81}{4 \,{\left (x - 4\right )}^{2}} + \frac{3}{16} \, \arctan \left (\frac{1}{2} \, x - 1\right ) + \frac{45}{32} \,{\rm ln}\left (x^{2} - 4 \, x + 8\right ) - \frac{45}{16} \,{\rm ln}\left ({\left | x - 4 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^3 + 8*x - 20)/((x^2 - 4*x + 8)*(x - 4)^3),x, algorithm="giac")

[Out]

-1/4*(41*x - 81)/(x - 4)^2 + 3/16*arctan(1/2*x - 1) + 45/32*ln(x^2 - 4*x + 8) -
45/16*ln(abs(x - 4))