3.4.50 \(\int \frac {1-4 e^{2 x} x}{x} \, dx\)

Optimal. Leaf size=22 \[ \log \left (\frac {1}{5} e^{-2-2 e^{2 x}} (1+e)^2 x\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 0.45, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {14, 2194} \begin {gather*} \log (x)-2 e^{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 4*E^(2*x)*x)/x,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.45 \begin {gather*} -2 e^{2 x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 4*E^(2*x)*x)/x,x]

[Out]

-2*E^(2*x) + Log[x]

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fricas [A]  time = 1.15, size = 9, normalized size = 0.41 \begin {gather*} -2 \, e^{\left (2 \, x\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(x)^2+1)/x,x, algorithm="fricas")

[Out]

-2*e^(2*x) + log(x)

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giac [A]  time = 0.35, size = 9, normalized size = 0.41 \begin {gather*} -2 \, e^{\left (2 \, x\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(x)^2+1)/x,x, algorithm="giac")

[Out]

-2*e^(2*x) + log(x)

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maple [A]  time = 0.02, size = 10, normalized size = 0.45

method	result	size
default	\(\ln \relax (x )-2 \,{\mathrm e}^{2 x}\)	\(10\)
norman	\(\ln \relax (x )-2 \,{\mathrm e}^{2 x}\)	\(10\)
risch	\(\ln \relax (x )-2 \,{\mathrm e}^{2 x}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x*exp(x)^2+1)/x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.74, size = 9, normalized size = 0.41 \begin {gather*} -2 \, e^{\left (2 \, x\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(x)^2+1)/x,x, algorithm="maxima")

[Out]

-2*e^(2*x) + log(x)

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mupad [B]  time = 0.33, size = 9, normalized size = 0.41 \begin {gather*} \ln \relax (x)-2\,{\mathrm {e}}^{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x*exp(2*x) - 1)/x,x)

[Out]

log(x) - 2*exp(2*x)

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sympy [A]  time = 0.08, size = 8, normalized size = 0.36 \begin {gather*} - 2 e^{2 x} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(x)**2+1)/x,x)

[Out]

-2*exp(2*x) + log(x)

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