3.61.43 \(\int \frac {-6-12 x^2+2 x^3+e^4 (2+4 x^2)}{-6 x+2 e^4 x+x^2} \, dx\)

Optimal. Leaf size=25 \[ 4+x^2-\log \left (\frac {5 \left (-3+e^4+\frac {x}{2}\right )}{3 x}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 22, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6, 1593, 1620} \begin {gather*} x^2-\log \left (2 \left (3-e^4\right )-x\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 - 12*x^2 + 2*x^3 + E^4*(2 + 4*x^2))/(-6*x + 2*E^4*x + x^2),x]

[Out]

x^2 - Log[2*(3 - E^4) - x] + Log[x]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6-12 x^2+2 x^3+e^4 \left (2+4 x^2\right )}{\left (-6+2 e^4\right ) x+x^2} \, dx\\ &=\int \frac {-6-12 x^2+2 x^3+e^4 \left (2+4 x^2\right )}{x \left (-6+2 e^4+x\right )} \, dx\\ &=\int \left (\frac {1}{6-2 e^4-x}+\frac {1}{x}+2 x\right ) \, dx\\ &=x^2-\log \left (2 \left (3-e^4\right )-x\right )+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.24 \begin {gather*} 2 \left (\frac {x^2}{2}-\frac {1}{2} \log \left (6-2 e^4-x\right )+\frac {\log (x)}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 - 12*x^2 + 2*x^3 + E^4*(2 + 4*x^2))/(-6*x + 2*E^4*x + x^2),x]

[Out]

2*(x^2/2 - Log[6 - 2*E^4 - x]/2 + Log[x]/2)

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fricas [A]  time = 0.70, size = 16, normalized size = 0.64 \begin {gather*} x^{2} - \log \left (x + 2 \, e^{4} - 6\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+2)*exp(4)+2*x^3-12*x^2-6)/(2*x*exp(4)+x^2-6*x),x, algorithm="fricas")

[Out]

x^2 - log(x + 2*e^4 - 6) + log(x)

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giac [A]  time = 0.13, size = 18, normalized size = 0.72 \begin {gather*} x^{2} - \log \left ({\left | x + 2 \, e^{4} - 6 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+2)*exp(4)+2*x^3-12*x^2-6)/(2*x*exp(4)+x^2-6*x),x, algorithm="giac")

[Out]

x^2 - log(abs(x + 2*e^4 - 6)) + log(abs(x))

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maple [A]  time = 0.33, size = 17, normalized size = 0.68

method	result	size
default	\(x^{2}-\ln \left (2 \,{\mathrm e}^{4}+x -6\right )+\ln \relax (x )\)	\(17\)
norman	\(x^{2}-\ln \left (2 \,{\mathrm e}^{4}+x -6\right )+\ln \relax (x )\)	\(17\)
risch	\(x^{2}-\ln \left (2 \,{\mathrm e}^{4}+x -6\right )+\ln \relax (x )\)	\(17\)
meijerg	\(\left (4 \,{\mathrm e}^{4}-12\right ) \left (2 \,{\mathrm e}^{4}-6\right ) \left (\frac {x}{2 \,{\mathrm e}^{4}-6}-\ln \left (1+\frac {x}{2 \,{\mathrm e}^{4}-6}\right )\right )+8 \left ({\mathrm e}^{4}-3\right )^{2} \left (-\frac {x \left (-\frac {3 x}{2 \left ({\mathrm e}^{4}-3\right )}+6\right )}{12 \left ({\mathrm e}^{4}-3\right )}+\ln \left (1+\frac {x}{2 \,{\mathrm e}^{4}-6}\right )\right )+\frac {{\mathrm e}^{4} \left (-\ln \left (1+\frac {x}{2 \,{\mathrm e}^{4}-6}\right )+\ln \relax (x )-\ln \relax (2)-\ln \left ({\mathrm e}^{4}-3\right )\right )}{{\mathrm e}^{4}-3}-\frac {3 \left (-\ln \left (1+\frac {x}{2 \,{\mathrm e}^{4}-6}\right )+\ln \relax (x )-\ln \relax (2)-\ln \left ({\mathrm e}^{4}-3\right )\right )}{{\mathrm e}^{4}-3}\)	\(153\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^2+2)*exp(4)+2*x^3-12*x^2-6)/(2*x*exp(4)+x^2-6*x),x,method=_RETURNVERBOSE)

[Out]

x^2-ln(2*exp(4)+x-6)+ln(x)

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maxima [A]  time = 0.34, size = 16, normalized size = 0.64 \begin {gather*} x^{2} - \log \left (x + 2 \, e^{4} - 6\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+2)*exp(4)+2*x^3-12*x^2-6)/(2*x*exp(4)+x^2-6*x),x, algorithm="maxima")

[Out]

x^2 - log(x + 2*e^4 - 6) + log(x)

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mupad [B]  time = 0.10, size = 20, normalized size = 0.80 \begin {gather*} x^2-2\,\mathrm {atanh}\left (\frac {4\,x}{4\,{\mathrm {e}}^4-12}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4)*(4*x^2 + 2) - 12*x^2 + 2*x^3 - 6)/(2*x*exp(4) - 6*x + x^2),x)

[Out]

x^2 - 2*atanh((4*x)/(4*exp(4) - 12) + 1)

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sympy [A]  time = 0.31, size = 15, normalized size = 0.60 \begin {gather*} x^{2} + \log {\relax (x )} - \log {\left (x - 6 + 2 e^{4} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**2+2)*exp(4)+2*x**3-12*x**2-6)/(2*x*exp(4)+x**2-6*x),x)

[Out]

x**2 + log(x) - log(x - 6 + 2*exp(4))

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