\(\int \frac {-2 e^4 x+e^x x-6 e^{-6+3 x} x}{2 x} \, dx\) [7161]

Optimal result

Integrand size = 29, antiderivative size = 23 \[ \int \frac {-2 e^4 x+e^x x-6 e^{-6+3 x} x}{2 x} \, dx=\frac {e^x}{2}-e^{-6+3 x}-e^4 x \]

[Out]

1/2*exp(x)-exp(ln(x)+4)-exp(-6+3*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 14, 2225} \[ \int \frac {-2 e^4 x+e^x x-6 e^{-6+3 x} x}{2 x} \, dx=-e^4 x+\frac {e^x}{2}-e^{3 x-6} \]

[In]

Int[(-2*E^4*x + E^x*x - 6*E^(-6 + 3*x)*x)/(2*x),x]

[Out]

E^x/2 - E^(-6 + 3*x) - E^4*x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2225

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{align*} \text {integral}& = \frac {1}{2} \int \frac {-2 e^4 x+e^x x-6 e^{-6+3 x} x}{x} \, dx \\ & = \frac {1}{2} \int \left (-2 e^4+e^x-6 e^{-6+3 x}\right ) \, dx \\ & = -e^4 x+\frac {\int e^x \, dx}{2}-3 \int e^{-6+3 x} \, dx \\ & = \frac {e^x}{2}-e^{-6+3 x}-e^4 x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-2 e^4 x+e^x x-6 e^{-6+3 x} x}{2 x} \, dx=\frac {e^x}{2}-e^{-6+3 x}-e^4 x \]

[In]

Integrate[(-2*E^4*x + E^x*x - 6*E^(-6 + 3*x)*x)/(2*x),x]

[Out]

E^x/2 - E^(-6 + 3*x) - E^4*x

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

method	result	size
risch	\(-x \,{\mathrm e}^{4}-{\mathrm e}^{-6+3 x}+\frac {{\mathrm e}^{x}}{2}\)	\(19\)
default	\(\frac {{\mathrm e}^{x}}{2}-{\mathrm e}^{\ln \left (x \right )+4}-{\mathrm e}^{-6+3 x}\)	\(21\)
norman	\(-x \,{\mathrm e}^{4}-{\mathrm e}^{-6} {\mathrm e}^{3 x}+\frac {{\mathrm e}^{x}}{2}\)	\(21\)
parts	\(\frac {{\mathrm e}^{x}}{2}-{\mathrm e}^{\ln \left (x \right )+4}-{\mathrm e}^{-6+3 x}\)	\(21\)
parallelrisch	\(\frac {{\mathrm e}^{x} x^{2}-2 x^{2} {\mathrm e}^{-6+3 x}-2 \,{\mathrm e}^{\ln \left (x \right )+4} x^{2}}{2 x^{2}}\)	\(34\)

[In]

int(1/2*(-2*exp(ln(x)+4)-6*x*exp(-6+3*x)+exp(x)*x)/x,x,method=_RETURNVERBOSE)

[Out]

-x*exp(4)-exp(-6+3*x)+1/2*exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-2 e^4 x+e^x x-6 e^{-6+3 x} x}{2 x} \, dx=-\frac {1}{2} \, {\left (2 \, x e^{10} + 2 \, e^{\left (3 \, x\right )} - e^{\left (x + 6\right )}\right )} e^{\left (-6\right )} \]

[In]

integrate(1/2*(-2*exp(log(x)+4)-6*x*exp(-6+3*x)+exp(x)*x)/x,x, algorithm="fricas")

[Out]

-1/2*(2*x*e^10 + 2*e^(3*x) - e^(x + 6))*e^(-6)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-2 e^4 x+e^x x-6 e^{-6+3 x} x}{2 x} \, dx=- x e^{4} + \frac {- 2 e^{3 x} + e^{6} e^{x}}{2 e^{6}} \]

[In]

integrate(1/2*(-2*exp(ln(x)+4)-6*x*exp(-6+3*x)+exp(x)*x)/x,x)

[Out]

-x*exp(4) + (-2*exp(3*x) + exp(6)*exp(x))*exp(-6)/2

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {-2 e^4 x+e^x x-6 e^{-6+3 x} x}{2 x} \, dx=-x e^{4} - e^{\left (3 \, x - 6\right )} + \frac {1}{2} \, e^{x} \]

[In]

integrate(1/2*(-2*exp(log(x)+4)-6*x*exp(-6+3*x)+exp(x)*x)/x,x, algorithm="maxima")

[Out]

-x*e^4 - e^(3*x - 6) + 1/2*e^x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-2 e^4 x+e^x x-6 e^{-6+3 x} x}{2 x} \, dx=-\frac {1}{2} \, {\left (2 \, x e^{10} + 2 \, e^{\left (3 \, x\right )} - e^{\left (x + 6\right )}\right )} e^{\left (-6\right )} \]

[In]

integrate(1/2*(-2*exp(log(x)+4)-6*x*exp(-6+3*x)+exp(x)*x)/x,x, algorithm="giac")

[Out]

-1/2*(2*x*e^10 + 2*e^(3*x) - e^(x + 6))*e^(-6)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {-2 e^4 x+e^x x-6 e^{-6+3 x} x}{2 x} \, dx=\frac {{\mathrm {e}}^x}{2}-{\mathrm {e}}^{3\,x-6}-x\,{\mathrm {e}}^4 \]

[In]

int(-(exp(log(x) + 4) + 3*x*exp(3*x - 6) - (x*exp(x))/2)/x,x)

[Out]

exp(x)/2 - exp(3*x - 6) - x*exp(4)