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

Optimal. Leaf size=20 \[ 1+x+\log \left (e^3+x+\frac {4-x^4}{x^3}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1593, 1834, 260} \begin {gather*} \log \left (e^3 x^3+4\right )+x-3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - 3*Log[x] + Log[4 + E^3*x^3]

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 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 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.80 \begin {gather*} x-3 \log (x)+\log \left (4+e^3 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - 3*Log[x] + Log[4 + E^3*x^3]

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fricas [A]  time = 0.51, size = 15, normalized size = 0.75 \begin {gather*} x + \log \left (x^{3} e^{3} + 4\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x^3*e^3 + 4) - 3*log(x)

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giac [A]  time = 0.30, size = 17, normalized size = 0.85 \begin {gather*} x + \log \left ({\left | x^{3} e^{3} + 4 \right |}\right ) - 3 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(abs(x^3*e^3 + 4)) - 3*log(abs(x))

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maple [A]  time = 0.53, size = 16, normalized size = 0.80

method	result	size
default	\(x -3 \ln \relax (x )+\ln \left (x^{3} {\mathrm e}^{3}+4\right )\)	\(16\)
norman	\(x -3 \ln \relax (x )+\ln \left (x^{3} {\mathrm e}^{3}+4\right )\)	\(16\)
risch	\(x -3 \ln \relax (x )+\ln \left (-x^{3} {\mathrm e}^{3}-4\right )\)	\(17\)
meijerg	\(-3 \ln \relax (x )-3+2 \ln \relax (2)+\ln \left (1+\frac {x^{3} {\mathrm e}^{3}}{4}\right )+\frac {2^{\frac {2}{3}} {\mathrm e}^{-1} \left (\frac {3 x \,{\mathrm e} 2^{\frac {1}{3}}}{2}-\frac {x \,{\mathrm e} 2^{\frac {1}{3}} \left (\frac {2^{\frac {2}{3}} {\mathrm e}^{-1} \ln \left (1+\frac {2^{\frac {1}{3}} {\mathrm e} \left (x^{3}\right )^{\frac {1}{3}}}{2}\right )}{\left (x^{3}\right )^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} {\mathrm e}^{-1} \ln \left (1-\frac {2^{\frac {1}{3}} {\mathrm e} \left (x^{3}\right )^{\frac {1}{3}}}{2}+\frac {2^{\frac {2}{3}} {\mathrm e}^{2} \left (x^{3}\right )^{\frac {2}{3}}}{4}\right )}{2 \left (x^{3}\right )^{\frac {1}{3}}}+\frac {2^{\frac {2}{3}} \sqrt {3}\, {\mathrm e}^{-1} \arctan \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, {\mathrm e} \left (x^{3}\right )^{\frac {1}{3}}}{4-2^{\frac {1}{3}} {\mathrm e} \left (x^{3}\right )^{\frac {1}{3}}}\right )}{\left (x^{3}\right )^{\frac {1}{3}}}\right )}{2}\right )}{3}+\frac {{\mathrm e}^{-1} 2^{\frac {2}{3}} \left (\frac {x \ln \left (1+\frac {2^{\frac {1}{3}} {\mathrm e} \left (x^{3}\right )^{\frac {1}{3}}}{2}\right )}{\left (x^{3}\right )^{\frac {1}{3}}}-\frac {x \ln \left (1-\frac {2^{\frac {1}{3}} {\mathrm e} \left (x^{3}\right )^{\frac {1}{3}}}{2}+\frac {2^{\frac {2}{3}} {\mathrm e}^{2} \left (x^{3}\right )^{\frac {2}{3}}}{4}\right )}{2 \left (x^{3}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, x \arctan \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, {\mathrm e} \left (x^{3}\right )^{\frac {1}{3}}}{4-2^{\frac {1}{3}} {\mathrm e} \left (x^{3}\right )^{\frac {1}{3}}}\right )}{\left (x^{3}\right )^{\frac {1}{3}}}\right )}{3}\)	\(264\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-3*ln(x)+ln(x^3*exp(3)+4)

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maxima [A]  time = 0.63, size = 15, normalized size = 0.75 \begin {gather*} x + \log \left (x^{3} e^{3} + 4\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x^3*e^3 + 4) - 3*log(x)

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mupad [B]  time = 1.87, size = 15, normalized size = 0.75 \begin {gather*} x+\ln \left (x^3+4\,{\mathrm {e}}^{-3}\right )-3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(4*exp(-3) + x^3) - 3*log(x)

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sympy [A]  time = 0.17, size = 15, normalized size = 0.75 \begin {gather*} x - 3 \log {\relax (x )} + \log {\left (x^{3} + \frac {4}{e^{3}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 3*log(x) + log(x**3 + 4*exp(-3))

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