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

Optimal. Leaf size=19 \[ 8-e^{2 x} x-\frac {x+\log (4)}{x} \]

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.63, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {14, 2176, 2194} \begin {gather*} -\frac {1}{2} e^{2 x} (2 x+1)+\frac {e^{2 x}}{2}-\frac {\log (4)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(2*x)/2 - (E^(2*x)*(1 + 2*x))/2 - Log[4]/x

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 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.84 \begin {gather*} -e^{2 x} x-\frac {\log (4)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(2*x)*x) - Log[4]/x

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fricas [A]  time = 1.07, size = 18, normalized size = 0.95 \begin {gather*} -\frac {x^{2} e^{\left (2 \, x\right )} + 2 \, \log \relax (2)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^2*e^(2*x) + 2*log(2))/x

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giac [A]  time = 0.19, size = 18, normalized size = 0.95 \begin {gather*} -\frac {x^{2} e^{\left (2 \, x\right )} + 2 \, \log \relax (2)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^2*e^(2*x) + 2*log(2))/x

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maple [A]  time = 0.05, size = 16, normalized size = 0.84

method	result	size
derivativedivides	\(-\frac {2 \ln \relax (2)}{x}-x \,{\mathrm e}^{2 x}\)	\(16\)
default	\(-\frac {2 \ln \relax (2)}{x}-x \,{\mathrm e}^{2 x}\)	\(16\)
risch	\(-\frac {2 \ln \relax (2)}{x}-x \,{\mathrm e}^{2 x}\)	\(16\)
norman	\(\frac {-{\mathrm e}^{2 x} x^{2}-2 \ln \relax (2)}{x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*ln(2)/x-x*exp(2*x)

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maxima [A]  time = 0.37, size = 25, normalized size = 1.32 \begin {gather*} -\frac {1}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - \frac {2 \, \log \relax (2)}{x} - \frac {1}{2} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*x - 1)*e^(2*x) - 2*log(2)/x - 1/2*e^(2*x)

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mupad [B]  time = 0.06, size = 15, normalized size = 0.79 \begin {gather*} -x\,{\mathrm {e}}^{2\,x}-\frac {2\,\ln \relax (2)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- x*exp(2*x) - (2*log(2))/x

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sympy [A]  time = 0.11, size = 14, normalized size = 0.74 \begin {gather*} - x e^{2 x} - \frac {2 \log {\relax (2 )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*exp(2*x) - 2*log(2)/x

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