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

Optimal. Leaf size=11 \[ \frac {x}{1-x^2} \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {383} \begin {gather*} \frac {x}{1-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(-1 + x^2)^2,x]

[Out]

x/(1 - x^2)

Rule 383

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*x*(a + b*x^n)^(p + 1))/a, x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.91 \begin {gather*} -\frac {x}{x^2-1} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)/(-1 + x^2)^2,x]

[Out]

-(x/(-1 + x^2))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2}{\left (-1+x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 + x^2)/(-1 + x^2)^2,x]

[Out]

IntegrateAlgebraic[(1 + x^2)/(-1 + x^2)^2, x]

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fricas [A]  time = 0.41, size = 10, normalized size = 0.91 \begin {gather*} -\frac {x}{x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x^2 - 1)

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giac [A]  time = 0.43, size = 11, normalized size = 1.00 \begin {gather*} -\frac {1}{x - \frac {1}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(x - 1/x)

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maple [A]  time = 0.00, size = 16, normalized size = 1.45 \begin {gather*} -\frac {1}{2 \left (x +1\right )}-\frac {1}{2 \left (x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.03, size = 10, normalized size = 0.91 \begin {gather*} -\frac {x}{x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x^2 - 1)

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mupad [B]  time = 0.08, size = 10, normalized size = 0.91 \begin {gather*} -\frac {x}{x^2-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 1)/(x^2 - 1)^2,x)

[Out]

-x/(x^2 - 1)

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sympy [A]  time = 0.09, size = 7, normalized size = 0.64 \begin {gather*} - \frac {x}{x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x**2 - 1)

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