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

Optimal. Leaf size=25 \[ \frac{2}{1-x}-\frac{1}{(1-x)^2}+\log (1-x) \]

[Out]

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

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Rubi [A]  time = 0.0105989, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {697} \[ \frac{2}{1-x}-\frac{1}{(1-x)^2}+\log (1-x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(-1 + x)^3,x]

[Out]

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

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0089271, size = 16, normalized size = 0.64 \[ \frac{1-2 x}{(x-1)^2}+\log (x-1) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)/(-1 + x)^3,x]

[Out]

(1 - 2*x)/(-1 + x)^2 + Log[-1 + x]

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Maple [A]  time = 0.004, size = 20, normalized size = 0.8 \begin{align*} - \left ( -1+x \right ) ^{-2}+\ln \left ( -1+x \right ) -2\, \left ( -1+x \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.926384, size = 30, normalized size = 1.2 \begin{align*} -\frac{2 \, x - 1}{x^{2} - 2 \, x + 1} + \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.1045, size = 76, normalized size = 3.04 \begin{align*} \frac{{\left (x^{2} - 2 \, x + 1\right )} \log \left (x - 1\right ) - 2 \, x + 1}{x^{2} - 2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.088809, size = 17, normalized size = 0.68 \begin{align*} - \frac{2 x - 1}{x^{2} - 2 x + 1} + \log{\left (x - 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.04723, size = 24, normalized size = 0.96 \begin{align*} -\frac{2 \, x - 1}{{\left (x - 1\right )}^{2}} + \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*x - 1)/(x - 1)^2 + log(abs(x - 1))