3.488 \(\int \frac{2+2 x}{(-1+x)^3 \left (1+x^2\right )} \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{x-1}-\frac{1}{(1-x)^2}+\tan ^{-1}(x) \]

[Out]

-(1 - x)^(-2) + (-1 + x)^(-1) + ArcTan[x]

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Rubi [A]  time = 0.0448863, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{x-1}-\frac{1}{(1-x)^2}+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-(1 - x)^(-2) + (-1 + x)^(-1) + ArcTan[x]

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Rubi in Sympy [A]  time = 5.93668, size = 14, normalized size = 0.82 \[ \operatorname{atan}{\left (x \right )} - \frac{1}{- x + 1} - \frac{1}{\left (- x + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(x) - 1/(-x + 1) - 1/(-x + 1)**2

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Mathematica [A]  time = 0.0194569, size = 17, normalized size = 1. \[ \frac{x+(x-1)^2 \tan ^{-1}(x)-2}{(x-1)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 16, normalized size = 0.9 \[ - \left ( -1+x \right ) ^{-2}+ \left ( -1+x \right ) ^{-1}+\arctan \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+2*x)/(-1+x)^3/(x^2+1),x)

[Out]

-1/(-1+x)^2+1/(-1+x)+arctan(x)

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Maxima [A]  time = 0.855404, size = 23, normalized size = 1.35 \[ \frac{x - 2}{x^{2} - 2 \, x + 1} + \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x - 2)/(x^2 - 2*x + 1) + arctan(x)

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Fricas [A]  time = 0.263458, size = 34, normalized size = 2. \[ \frac{{\left (x^{2} - 2 \, x + 1\right )} \arctan \left (x\right ) + x - 2}{x^{2} - 2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.308353, size = 14, normalized size = 0.82 \[ \frac{x - 2}{x^{2} - 2 x + 1} + \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x - 2)/(x**2 - 2*x + 1) + atan(x)

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GIAC/XCAS [A]  time = 0.261325, size = 16, normalized size = 0.94 \[ \frac{x - 2}{{\left (x - 1\right )}^{2}} + \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x - 2)/(x - 1)^2 + arctan(x)