3.230 \(\int \frac{-1+3 x-3 x^2+x^3}{1+4 x+6 x^2+4 x^3+x^4} \, dx\)

Optimal. Leaf size=28 \[ \frac{6}{x+1}-\frac{6}{(x+1)^2}+\frac{8}{3 (x+1)^3}+\log (x+1) \]

[Out]

8/(3*(1 + x)^3) - 6/(1 + x)^2 + 6/(1 + x) + Log[1 + x]

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Rubi [A]  time = 0.0508693, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{6}{x+1}-\frac{6}{(x+1)^2}+\frac{8}{3 (x+1)^3}+\log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[(-1 + 3*x - 3*x^2 + x^3)/(1 + 4*x + 6*x^2 + 4*x^3 + x^4),x]

[Out]

8/(3*(1 + x)^3) - 6/(1 + x)^2 + 6/(1 + x) + Log[1 + x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int ^{x + 1} \frac{x^{3} - 6 x^{2} + 12 x - 8}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3-3*x**2+3*x-1)/(x**4+4*x**3+6*x**2+4*x+1),x)

[Out]

Integral((x**3 - 6*x**2 + 12*x - 8)/x**4, (x, x + 1))

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Mathematica [A]  time = 0.0197026, size = 24, normalized size = 0.86 \[ \frac{2 \left (9 x^2+9 x+4\right )}{3 (x+1)^3}+\log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + 3*x - 3*x^2 + x^3)/(1 + 4*x + 6*x^2 + 4*x^3 + x^4),x]

[Out]

(2*(4 + 9*x + 9*x^2))/(3*(1 + x)^3) + Log[1 + x]

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Maple [A]  time = 0.01, size = 27, normalized size = 1. \[{\frac{8}{3\, \left ( 1+x \right ) ^{3}}}-6\, \left ( 1+x \right ) ^{-2}+6\, \left ( 1+x \right ) ^{-1}+\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3-3*x^2+3*x-1)/(x^4+4*x^3+6*x^2+4*x+1),x)

[Out]

8/3/(1+x)^3-6/(1+x)^2+6/(1+x)+ln(1+x)

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Maxima [A]  time = 0.830648, size = 43, normalized size = 1.54 \[ \frac{2 \,{\left (9 \, x^{2} + 9 \, x + 4\right )}}{3 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} + \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 3*x^2 + 3*x - 1)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1),x, algorithm="maxima")

[Out]

2/3*(9*x^2 + 9*x + 4)/(x^3 + 3*x^2 + 3*x + 1) + log(x + 1)

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Fricas [A]  time = 0.264645, size = 62, normalized size = 2.21 \[ \frac{18 \, x^{2} + 3 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (x + 1\right ) + 18 \, x + 8}{3 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 3*x^2 + 3*x - 1)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1),x, algorithm="fricas")

[Out]

1/3*(18*x^2 + 3*(x^3 + 3*x^2 + 3*x + 1)*log(x + 1) + 18*x + 8)/(x^3 + 3*x^2 + 3*
x + 1)

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Sympy [A]  time = 0.231938, size = 29, normalized size = 1.04 \[ \frac{18 x^{2} + 18 x + 8}{3 x^{3} + 9 x^{2} + 9 x + 3} + \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3-3*x**2+3*x-1)/(x**4+4*x**3+6*x**2+4*x+1),x)

[Out]

(18*x**2 + 18*x + 8)/(3*x**3 + 9*x**2 + 9*x + 3) + log(x + 1)

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GIAC/XCAS [A]  time = 0.260213, size = 31, normalized size = 1.11 \[ \frac{2 \,{\left (9 \, x^{2} + 9 \, x + 4\right )}}{3 \,{\left (x + 1\right )}^{3}} +{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 3*x^2 + 3*x - 1)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1),x, algorithm="giac")

[Out]

2/3*(9*x^2 + 9*x + 4)/(x + 1)^3 + ln(abs(x + 1))