3.12.98 \(\int \frac {-2+x^3-3 e^{-3 x} x^3-2 x^4 \log (4+e^3)}{x^3} \, dx\)

Optimal. Leaf size=21 \[ e^{-3 x}+\frac {1}{x^2}+x-x^2 \log \left (4+e^3\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14, 2194} \begin {gather*} \frac {1}{x^2}+x^2 \left (-\log \left (4+e^3\right )\right )+x+e^{-3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-3*x) + x^(-2) + x - x^2*Log[4 + E^3]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 21, normalized size = 1.00 \begin {gather*} e^{-3 x}+\frac {1}{x^2}+x-x^2 \log \left (4+e^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-3*x) + x^(-2) + x - x^2*Log[4 + E^3]

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fricas [A]  time = 0.59, size = 30, normalized size = 1.43 \begin {gather*} -\frac {x^{4} \log \left (e^{3} + 4\right ) - x^{3} - x^{2} e^{\left (-3 \, x\right )} - 1}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^4*log(e^3 + 4) - x^3 - x^2*e^(-3*x) - 1)/x^2

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giac [A]  time = 0.27, size = 30, normalized size = 1.43 \begin {gather*} -\frac {x^{4} \log \left (e^{3} + 4\right ) - x^{3} - x^{2} e^{\left (-3 \, x\right )} - 1}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^4*log(e^3 + 4) - x^3 - x^2*e^(-3*x) - 1)/x^2

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maple [A]  time = 0.03, size = 20, normalized size = 0.95

method	result	size
derivativedivides	\({\mathrm e}^{-3 x}+\frac {1}{x^{2}}-\ln \left (4+{\mathrm e}^{3}\right ) x^{2}+x\)	\(20\)
default	\({\mathrm e}^{-3 x}+\frac {1}{x^{2}}-\ln \left (4+{\mathrm e}^{3}\right ) x^{2}+x\)	\(20\)
risch	\({\mathrm e}^{-3 x}+\frac {1}{x^{2}}-\ln \left (4+{\mathrm e}^{3}\right ) x^{2}+x\)	\(20\)
norman	\(\frac {1+x^{3}+x^{2} {\mathrm e}^{-3 x}-x^{4} \ln \left (4+{\mathrm e}^{3}\right )}{x^{2}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(-3*x)+1/x^2-ln(4+exp(3))*x^2+x

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maxima [A]  time = 0.35, size = 19, normalized size = 0.90 \begin {gather*} -x^{2} \log \left (e^{3} + 4\right ) + x + \frac {1}{x^{2}} + e^{\left (-3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2*log(e^3 + 4) + x + 1/x^2 + e^(-3*x)

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mupad [B]  time = 0.09, size = 19, normalized size = 0.90 \begin {gather*} x+{\mathrm {e}}^{-3\,x}-x^2\,\ln \left ({\mathrm {e}}^3+4\right )+\frac {1}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + exp(-3*x) - x^2*log(exp(3) + 4) + 1/x^2

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sympy [A]  time = 0.13, size = 22, normalized size = 1.05 \begin {gather*} - x^{2} \log {\left (4 + e^{3} \right )} + x + e^{- 3 x} + \frac {1}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2*log(4 + exp(3)) + x + exp(-3*x) + x**(-2)

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