3.97.63 \(\int \frac {4-24 x^2}{-81-4 x+8 x^3} \, dx\)

Optimal. Leaf size=24 \[ \log \left (\frac {4}{3-3 \left (2-x-8 x^3+5 (16+x)\right )}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.54, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1587} \begin {gather*} -\log \left (-8 x^3+4 x+81\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 - 24*x^2)/(-81 - 4*x + 8*x^3),x]

[Out]

-Log[81 + 4*x - 8*x^3]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\log \left (81+4 x-8 x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 0.54 \begin {gather*} -\log \left (-81-4 x+8 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 24*x^2)/(-81 - 4*x + 8*x^3),x]

[Out]

-Log[-81 - 4*x + 8*x^3]

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fricas [A]  time = 0.58, size = 13, normalized size = 0.54 \begin {gather*} -\log \left (8 \, x^{3} - 4 \, x - 81\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24*x^2+4)/(8*x^3-4*x-81),x, algorithm="fricas")

[Out]

-log(8*x^3 - 4*x - 81)

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giac [A]  time = 0.21, size = 14, normalized size = 0.58 \begin {gather*} -\log \left ({\left | 8 \, x^{3} - 4 \, x - 81 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24*x^2+4)/(8*x^3-4*x-81),x, algorithm="giac")

[Out]

-log(abs(8*x^3 - 4*x - 81))

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maple [A]  time = 0.02, size = 14, normalized size = 0.58

method	result	size
default	\(-\ln \left (8 x^{3}-4 x -81\right )\)	\(14\)
norman	\(-\ln \left (8 x^{3}-4 x -81\right )\)	\(14\)
risch	\(-\ln \left (8 x^{3}-4 x -81\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-24*x^2+4)/(8*x^3-4*x-81),x,method=_RETURNVERBOSE)

[Out]

-ln(8*x^3-4*x-81)

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maxima [A]  time = 0.36, size = 13, normalized size = 0.54 \begin {gather*} -\log \left (8 \, x^{3} - 4 \, x - 81\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24*x^2+4)/(8*x^3-4*x-81),x, algorithm="maxima")

[Out]

-log(8*x^3 - 4*x - 81)

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mupad [B]  time = 0.05, size = 11, normalized size = 0.46 \begin {gather*} -\ln \left (x^3-\frac {x}{2}-\frac {81}{8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*x^2 - 4)/(4*x - 8*x^3 + 81),x)

[Out]

-log(x^3 - x/2 - 81/8)

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sympy [A]  time = 0.08, size = 12, normalized size = 0.50 \begin {gather*} - \log {\left (8 x^{3} - 4 x - 81 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24*x**2+4)/(8*x**3-4*x-81),x)

[Out]

-log(8*x**3 - 4*x - 81)

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