3.41.49 \(\int \frac {8+16 x+e^3 (6 x-3 x^2-6 x^3)}{-8+3 e^3 x^2} \, dx\)

Optimal. Leaf size=22 \[ -x-x^2+\log \left (\frac {8}{3}-e^3 x^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1810, 260} \begin {gather*} -x^2+\log \left (8-3 e^3 x^2\right )-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8 + 16*x + E^3*(6*x - 3*x^2 - 6*x^3))/(-8 + 3*E^3*x^2),x]

[Out]

-x - x^2 + Log[8 - 3*E^3*x^2]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1-2 x+\frac {6 e^3 x}{-8+3 e^3 x^2}\right ) \, dx\\ &=-x-x^2+\left (6 e^3\right ) \int \frac {x}{-8+3 e^3 x^2} \, dx\\ &=-x-x^2+\log \left (8-3 e^3 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.91 \begin {gather*} -x-x^2+\log \left (8-3 e^3 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 + 16*x + E^3*(6*x - 3*x^2 - 6*x^3))/(-8 + 3*E^3*x^2),x]

[Out]

-x - x^2 + Log[8 - 3*E^3*x^2]

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fricas [A]  time = 0.50, size = 19, normalized size = 0.86 \begin {gather*} -x^{2} - x + \log \left (3 \, x^{2} e^{3} - 8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3-3*x^2+6*x)*exp(3)+16*x+8)/(3*x^2*exp(3)-8),x, algorithm="fricas")

[Out]

-x^2 - x + log(3*x^2*e^3 - 8)

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giac [A]  time = 0.22, size = 27, normalized size = 1.23 \begin {gather*} -{\left (x^{2} e^{6} + x e^{6}\right )} e^{\left (-6\right )} + \log \left ({\left | 3 \, x^{2} e^{3} - 8 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3-3*x^2+6*x)*exp(3)+16*x+8)/(3*x^2*exp(3)-8),x, algorithm="giac")

[Out]

-(x^2*e^6 + x*e^6)*e^(-6) + log(abs(3*x^2*e^3 - 8))

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maple [A]  time = 0.09, size = 20, normalized size = 0.91

method	result	size
default	\(-x^{2}-x +\ln \left (3 x^{2} {\mathrm e}^{3}-8\right )\)	\(20\)
norman	\(-x^{2}-x +\ln \left (3 x^{2} {\mathrm e}^{3}-8\right )\)	\(20\)
risch	\(-x^{2}-x +\ln \left (3 x^{2} {\mathrm e}^{3}-8\right )\)	\(20\)
meijerg	\(-\frac {2 \sqrt {3}\, \sqrt {2}\, {\mathrm e}^{-\frac {3}{2}} \arctanh \left (\frac {x \sqrt {3}\, \sqrt {2}\, {\mathrm e}^{\frac {3}{2}}}{4}\right )}{3}+\frac {8 \,{\mathrm e}^{-3} \left (-\frac {3 x^{2} {\mathrm e}^{3}}{8}-\ln \left (1-\frac {3 x^{2} {\mathrm e}^{3}}{8}\right )\right )}{3}+\frac {i \sqrt {6}\, {\mathrm e}^{-\frac {3}{2}} \left (\frac {i \sqrt {6}\, x \,{\mathrm e}^{\frac {3}{2}}}{2}-2 i \arctanh \left (\frac {x \sqrt {3}\, \sqrt {2}\, {\mathrm e}^{\frac {3}{2}}}{4}\right )\right )}{3}+\frac {\left (6 \,{\mathrm e}^{3}+16\right ) {\mathrm e}^{-3} \ln \left (1-\frac {3 x^{2} {\mathrm e}^{3}}{8}\right )}{6}\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^3-3*x^2+6*x)*exp(3)+16*x+8)/(3*x^2*exp(3)-8),x,method=_RETURNVERBOSE)

[Out]

-x^2-x+ln(3*x^2*exp(3)-8)

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maxima [A]  time = 0.36, size = 19, normalized size = 0.86 \begin {gather*} -x^{2} - x + \log \left (3 \, x^{2} e^{3} - 8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3-3*x^2+6*x)*exp(3)+16*x+8)/(3*x^2*exp(3)-8),x, algorithm="maxima")

[Out]

-x^2 - x + log(3*x^2*e^3 - 8)

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mupad [B]  time = 0.11, size = 18, normalized size = 0.82 \begin {gather*} \ln \left (x^2-\frac {8\,{\mathrm {e}}^{-3}}{3}\right )-x-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x - exp(3)*(3*x^2 - 6*x + 6*x^3) + 8)/(3*x^2*exp(3) - 8),x)

[Out]

log(x^2 - (8*exp(-3))/3) - x - x^2

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sympy [A]  time = 0.14, size = 15, normalized size = 0.68 \begin {gather*} - x^{2} - x + \log {\left (3 x^{2} e^{3} - 8 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**3-3*x**2+6*x)*exp(3)+16*x+8)/(3*x**2*exp(3)-8),x)

[Out]

-x**2 - x + log(3*x**2*exp(3) - 8)

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