3.51.98 \(\int \frac {-1-x-4 e^{1024-4 x} x}{x} \, dx\)

Optimal. Leaf size=16 \[ 4+e^{1024-4 x}-x-\log (x) \]

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {14, 2194, 43} \begin {gather*} -x+e^{1024-4 x}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - x - 4*E^(1024 - 4*x)*x)/x,x]

[Out]

E^(1024 - 4*x) - 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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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^{1024-4 x}+\frac {-1-x}{x}\right ) \, dx\\ &=-\left (4 \int e^{1024-4 x} \, dx\right )+\int \frac {-1-x}{x} \, dx\\ &=e^{1024-4 x}+\int \left (-1-\frac {1}{x}\right ) \, dx\\ &=e^{1024-4 x}-x-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 15, normalized size = 0.94 \begin {gather*} e^{1024-4 x}-x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - x - 4*E^(1024 - 4*x)*x)/x,x]

[Out]

E^(1024 - 4*x) - x - Log[x]

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fricas [A]  time = 0.68, size = 14, normalized size = 0.88 \begin {gather*} -x + e^{\left (-4 \, x + 1024\right )} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(256-x)^4-x-1)/x,x, algorithm="fricas")

[Out]

-x + e^(-4*x + 1024) - log(x)

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giac [A]  time = 0.14, size = 14, normalized size = 0.88 \begin {gather*} -x + e^{\left (-4 \, x + 1024\right )} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(256-x)^4-x-1)/x,x, algorithm="giac")

[Out]

-x + e^(-4*x + 1024) - log(x)

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maple [A]  time = 0.07, size = 15, normalized size = 0.94

method	result	size
risch	\({\mathrm e}^{1024-4 x}-x -\ln \relax (x )\)	\(15\)
norman	\({\mathrm e}^{1024-4 x}-x -\ln \relax (x )\)	\(17\)
derivativedivides	\(256-x +256 \ln \left (-x \right )-257 \ln \relax (x )+{\mathrm e}^{1024-4 x}\)	\(24\)
default	\(256-x +256 \ln \left (-x \right )-257 \ln \relax (x )+{\mathrm e}^{1024-4 x}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x*exp(256-x)^4-x-1)/x,x,method=_RETURNVERBOSE)

[Out]

exp(1024-4*x)-x-ln(x)

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maxima [A]  time = 0.34, size = 14, normalized size = 0.88 \begin {gather*} -x + e^{\left (-4 \, x + 1024\right )} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(256-x)^4-x-1)/x,x, algorithm="maxima")

[Out]

-x + e^(-4*x + 1024) - log(x)

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mupad [B]  time = 3.19, size = 15, normalized size = 0.94 \begin {gather*} {\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^{1024}-\ln \relax (x)-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + 4*x*exp(1024 - 4*x) + 1)/x,x)

[Out]

exp(-4*x)*exp(1024) - log(x) - x

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sympy [A]  time = 0.09, size = 10, normalized size = 0.62 \begin {gather*} - x + e^{1024 - 4 x} - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(256-x)**4-x-1)/x,x)

[Out]

-x + exp(1024 - 4*x) - log(x)

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