3.366 \(\int \frac{-32+36 x-42 x^2+21 x^3-10 x^4+3 x^5}{x \left (1+x^2\right ) \left (4+x^2\right )^2} \, dx\)

Optimal. Leaf size=32 \[ \frac{1}{x^2+4}+\log \left (x^2+4\right )-2 \log (x)+\frac{1}{2} \tan ^{-1}\left (\frac{x}{2}\right )+2 \tan ^{-1}(x) \]

[Out]

(4 + x^2)^(-1) + ArcTan[x/2]/2 + 2*ArcTan[x] - 2*Log[x] + Log[4 + x^2]

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Rubi [A]  time = 0.464191, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.116 \[ \frac{1}{x^2+4}+\log \left (x^2+4\right )-2 \log (x)+\frac{1}{2} \tan ^{-1}\left (\frac{x}{2}\right )+2 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(-32 + 36*x - 42*x^2 + 21*x^3 - 10*x^4 + 3*x^5)/(x*(1 + x^2)*(4 + x^2)^2),x]

[Out]

(4 + x^2)^(-1) + ArcTan[x/2]/2 + 2*ArcTan[x] - 2*Log[x] + Log[4 + x^2]

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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((3*x**5-10*x**4+21*x**3-42*x**2+36*x-32)/x/(x**2+1)/(x**2+4)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.0290071, size = 32, normalized size = 1. \[ \frac{1}{x^2+4}+\log \left (x^2+4\right )-2 \log (x)+\frac{1}{2} \tan ^{-1}\left (\frac{x}{2}\right )+2 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[(-32 + 36*x - 42*x^2 + 21*x^3 - 10*x^4 + 3*x^5)/(x*(1 + x^2)*(4 + x^2)^2),x]

[Out]

(4 + x^2)^(-1) + ArcTan[x/2]/2 + 2*ArcTan[x] - 2*Log[x] + Log[4 + x^2]

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Maple [A]  time = 0.013, size = 29, normalized size = 0.9 \[ \left ({x}^{2}+4 \right ) ^{-1}+{\frac{1}{2}\arctan \left ({\frac{x}{2}} \right ) }+2\,\arctan \left ( x \right ) -2\,\ln \left ( x \right ) +\ln \left ({x}^{2}+4 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^5-10*x^4+21*x^3-42*x^2+36*x-32)/x/(x^2+1)/(x^2+4)^2,x)

[Out]

1/(x^2+4)+1/2*arctan(1/2*x)+2*arctan(x)-2*ln(x)+ln(x^2+4)

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Maxima [A]  time = 0.898172, size = 38, normalized size = 1.19 \[ \frac{1}{x^{2} + 4} + \frac{1}{2} \, \arctan \left (\frac{1}{2} \, x\right ) + 2 \, \arctan \left (x\right ) + \log \left (x^{2} + 4\right ) - 2 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^5 - 10*x^4 + 21*x^3 - 42*x^2 + 36*x - 32)/((x^2 + 4)^2*(x^2 + 1)*x),x, algorithm="maxima")

[Out]

1/(x^2 + 4) + 1/2*arctan(1/2*x) + 2*arctan(x) + log(x^2 + 4) - 2*log(x)

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Fricas [A]  time = 0.327498, size = 70, normalized size = 2.19 \[ \frac{{\left (x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, x\right ) + 4 \,{\left (x^{2} + 4\right )} \arctan \left (x\right ) + 2 \,{\left (x^{2} + 4\right )} \log \left (x^{2} + 4\right ) - 4 \,{\left (x^{2} + 4\right )} \log \left (x\right ) + 2}{2 \,{\left (x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^5 - 10*x^4 + 21*x^3 - 42*x^2 + 36*x - 32)/((x^2 + 4)^2*(x^2 + 1)*x),x, algorithm="fricas")

[Out]

1/2*((x^2 + 4)*arctan(1/2*x) + 4*(x^2 + 4)*arctan(x) + 2*(x^2 + 4)*log(x^2 + 4)
- 4*(x^2 + 4)*log(x) + 2)/(x^2 + 4)

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Sympy [A]  time = 0.783684, size = 29, normalized size = 0.91 \[ - 2 \log{\left (x \right )} + \log{\left (x^{2} + 4 \right )} + \frac{\operatorname{atan}{\left (\frac{x}{2} \right )}}{2} + 2 \operatorname{atan}{\left (x \right )} + \frac{1}{x^{2} + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**5-10*x**4+21*x**3-42*x**2+36*x-32)/x/(x**2+1)/(x**2+4)**2,x)

[Out]

-2*log(x) + log(x**2 + 4) + atan(x/2)/2 + 2*atan(x) + 1/(x**2 + 4)

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GIAC/XCAS [A]  time = 0.264355, size = 39, normalized size = 1.22 \[ \frac{1}{x^{2} + 4} + \frac{1}{2} \, \arctan \left (\frac{1}{2} \, x\right ) + 2 \, \arctan \left (x\right ) +{\rm ln}\left (x^{2} + 4\right ) - 2 \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^5 - 10*x^4 + 21*x^3 - 42*x^2 + 36*x - 32)/((x^2 + 4)^2*(x^2 + 1)*x),x, algorithm="giac")

[Out]

1/(x^2 + 4) + 1/2*arctan(1/2*x) + 2*arctan(x) + ln(x^2 + 4) - 2*ln(abs(x))