3.12.80 \(\int \frac {-11+6 x}{(-1+2 x) (-1+x^2)} \, dx\)

Optimal. Leaf size=29 \[ \frac {16}{3} \log (1-2 x)-\frac {5}{2} \log (1-x)-\frac {17}{6} \log (x+1) \]

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Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {801} \begin {gather*} \frac {16}{3} \log (1-2 x)-\frac {5}{2} \log (1-x)-\frac {17}{6} \log (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-11 + 6*x)/((-1 + 2*x)*(-1 + x^2)),x]

[Out]

(16*Log[1 - 2*x])/3 - (5*Log[1 - x])/2 - (17*Log[1 + x])/6

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]

Rubi steps

\begin {align*} \int \frac {-11+6 x}{(-1+2 x) \left (-1+x^2\right )} \, dx &=\int \left (-\frac {5}{2 (-1+x)}-\frac {17}{6 (1+x)}+\frac {32}{3 (-1+2 x)}\right ) \, dx\\ &=\frac {16}{3} \log (1-2 x)-\frac {5}{2} \log (1-x)-\frac {17}{6} \log (1+x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.07 \begin {gather*} -\frac {5}{2} \log (2-2 x)+\frac {16}{3} \log (2 x-1)-\frac {17}{6} \log (2 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-11 + 6*x)/((-1 + 2*x)*(-1 + x^2)),x]

[Out]

(-5*Log[2 - 2*x])/2 + (16*Log[-1 + 2*x])/3 - (17*Log[2 + 2*x])/6

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-11+6 x}{(-1+2 x) \left (-1+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(-11 + 6*x)/((-1 + 2*x)*(-1 + x^2)),x]

[Out]

IntegrateAlgebraic[(-11 + 6*x)/((-1 + 2*x)*(-1 + x^2)), x]

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fricas [A]  time = 0.42, size = 21, normalized size = 0.72 \begin {gather*} \frac {16}{3} \, \log \left (2 \, x - 1\right ) - \frac {17}{6} \, \log \left (x + 1\right ) - \frac {5}{2} \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-11+6*x)/(-1+2*x)/(x^2-1),x, algorithm="fricas")

[Out]

16/3*log(2*x - 1) - 17/6*log(x + 1) - 5/2*log(x - 1)

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giac [A]  time = 0.17, size = 24, normalized size = 0.83 \begin {gather*} \frac {16}{3} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) - \frac {17}{6} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {5}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-11+6*x)/(-1+2*x)/(x^2-1),x, algorithm="giac")

[Out]

16/3*log(abs(2*x - 1)) - 17/6*log(abs(x + 1)) - 5/2*log(abs(x - 1))

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maple [A]  time = 0.05, size = 22, normalized size = 0.76 \begin {gather*} -\frac {5 \ln \left (x -1\right )}{2}-\frac {17 \ln \left (x +1\right )}{6}+\frac {16 \ln \left (2 x -1\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-11+6*x)/(2*x-1)/(x^2-1),x)

[Out]

-5/2*ln(x-1)+16/3*ln(2*x-1)-17/6*ln(x+1)

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maxima [A]  time = 0.57, size = 21, normalized size = 0.72 \begin {gather*} \frac {16}{3} \, \log \left (2 \, x - 1\right ) - \frac {17}{6} \, \log \left (x + 1\right ) - \frac {5}{2} \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-11+6*x)/(-1+2*x)/(x^2-1),x, algorithm="maxima")

[Out]

16/3*log(2*x - 1) - 17/6*log(x + 1) - 5/2*log(x - 1)

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mupad [B]  time = 0.07, size = 19, normalized size = 0.66 \begin {gather*} \frac {16\,\ln \left (x-\frac {1}{2}\right )}{3}-\frac {17\,\ln \left (x+1\right )}{6}-\frac {5\,\ln \left (x-1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x - 11)/((2*x - 1)*(x^2 - 1)),x)

[Out]

(16*log(x - 1/2))/3 - (17*log(x + 1))/6 - (5*log(x - 1))/2

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sympy [A]  time = 0.14, size = 26, normalized size = 0.90 \begin {gather*} - \frac {5 \log {\left (x - 1 \right )}}{2} + \frac {16 \log {\left (x - \frac {1}{2} \right )}}{3} - \frac {17 \log {\left (x + 1 \right )}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-11+6*x)/(-1+2*x)/(x**2-1),x)

[Out]

-5*log(x - 1)/2 + 16*log(x - 1/2)/3 - 17*log(x + 1)/6

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