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

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 = 0.0145243, 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, Rules used = {801, 203} \[ \frac{1}{x-1}-\frac{1}{(1-x)^2}+\tan ^{-1}(x) \]

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]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{2+2 x}{(-1+x)^3 \left (1+x^2\right )} \, dx &=\int \left (\frac{2}{(-1+x)^3}-\frac{1}{(-1+x)^2}+\frac{1}{1+x^2}\right ) \, dx\\ &=-\frac{1}{(1-x)^2}+\frac{1}{-1+x}+\int \frac{1}{1+x^2} \, dx\\ &=-\frac{1}{(1-x)^2}+\frac{1}{-1+x}+\tan ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0120315, 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 = 0.005, size = 16, normalized size = 0.9 \begin{align*} \arctan \left ( x \right ) - \left ( x-1 \right ) ^{-2}+ \left ( x-1 \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+2*x)/(x-1)^3/(x^2+1),x)

[Out]

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

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Maxima [A]  time = 1.83292, size = 23, normalized size = 1.35 \begin{align*} \frac{x - 2}{x^{2} - 2 \, x + 1} + \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x - 2)/(x^2 - 2*x + 1) + arctan(x)

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Fricas [A]  time = 0.935009, size = 72, normalized size = 4.24 \begin{align*} \frac{{\left (x^{2} - 2 \, x + 1\right )} \arctan \left (x\right ) + x - 2}{x^{2} - 2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+2*x)/(-1+x)^3/(x^2+1),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 = 0.115232, size = 14, normalized size = 0.82 \begin{align*} \frac{x - 2}{x^{2} - 2 x + 1} + \operatorname{atan}{\left (x \right )} \end{align*}

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 [A]  time = 1.18602, size = 16, normalized size = 0.94 \begin{align*} \frac{x - 2}{{\left (x - 1\right )}^{2}} + \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x - 2)/(x - 1)^2 + arctan(x)