3.74.2 \(\int \frac {1}{16} e^{\frac {8 e^4-3 x}{4 e^4}} (4 e^4-3 x) \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{4} e^{6-\frac {3 x}{4 e^4}} x \]

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Rubi [B]  time = 0.03, antiderivative size = 58, normalized size of antiderivative = 3.41, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {12, 2176, 2194} \begin {gather*} \frac {1}{3} e^{\frac {8 e^4-3 x}{4 e^4}+8}-\frac {1}{12} e^{\frac {8 e^4-3 x}{4 e^4}+4} \left (4 e^4-3 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((8*E^4 - 3*x)/(4*E^4))*(4*E^4 - 3*x))/16,x]

[Out]

E^(8 + (8*E^4 - 3*x)/(4*E^4))/3 - (E^(4 + (8*E^4 - 3*x)/(4*E^4))*(4*E^4 - 3*x))/12

Rule 12

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

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} &=\frac {1}{16} \int e^{\frac {8 e^4-3 x}{4 e^4}} \left (4 e^4-3 x\right ) \, dx\\ &=-\frac {1}{12} e^{4+\frac {8 e^4-3 x}{4 e^4}} \left (4 e^4-3 x\right )-\frac {1}{4} e^4 \int e^{\frac {8 e^4-3 x}{4 e^4}} \, dx\\ &=\frac {1}{3} e^{8+\frac {8 e^4-3 x}{4 e^4}}-\frac {1}{12} e^{4+\frac {8 e^4-3 x}{4 e^4}} \left (4 e^4-3 x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{4} e^{6-\frac {3 x}{4 e^4}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((8*E^4 - 3*x)/(4*E^4))*(4*E^4 - 3*x))/16,x]

[Out]

(E^(6 - (3*x)/(4*E^4))*x)/4

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fricas [A]  time = 0.75, size = 18, normalized size = 1.06 \begin {gather*} \frac {1}{4} \, x e^{\left (-\frac {1}{4} \, {\left (3 \, x - 8 \, e^{4}\right )} e^{\left (-4\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(4*exp(4)-3*x)*exp(1/4*(8*exp(4)-3*x)/exp(4)),x, algorithm="fricas")

[Out]

1/4*x*e^(-1/4*(3*x - 8*e^4)*e^(-4) + 4)

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giac [B]  time = 0.17, size = 41, normalized size = 2.41 \begin {gather*} \frac {1}{12} \, {\left (3 \, x e^{4} + 4 \, e^{8}\right )} e^{\left (-\frac {1}{4} \, {\left (3 \, x - 8 \, e^{4}\right )} e^{\left (-4\right )}\right )} - \frac {1}{3} \, e^{\left (-\frac {3}{4} \, {\left (x - 8 \, e^{4}\right )} e^{\left (-4\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(4*exp(4)-3*x)*exp(1/4*(8*exp(4)-3*x)/exp(4)),x, algorithm="giac")

[Out]

1/12*(3*x*e^4 + 4*e^8)*e^(-1/4*(3*x - 8*e^4)*e^(-4)) - 1/3*e^(-3/4*(x - 8*e^4)*e^(-4) + 4)

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maple [A]  time = 0.04, size = 18, normalized size = 1.06

method	result	size
risch	\(\frac {x \,{\mathrm e}^{4+2 \,{\mathrm e}^{-4} {\mathrm e}^{4}-\frac {3 x \,{\mathrm e}^{-4}}{4}}}{4}\)	\(18\)
gosper	\(\frac {{\mathrm e}^{4+\frac {\left (8 \,{\mathrm e}^{4}-3 x \right ) {\mathrm e}^{-4}}{4}} x}{4}\)	\(21\)
norman	\(\frac {{\mathrm e}^{4} x \,{\mathrm e}^{\frac {\left (8 \,{\mathrm e}^{4}-3 x \right ) {\mathrm e}^{-4}}{4}}}{4}\)	\(21\)
meijerg	\(\frac {{\mathrm e}^{10} \left (1-{\mathrm e}^{-\frac {3 x \,{\mathrm e}^{-4}}{4}}\right )}{3}-\frac {{\mathrm e}^{10} \left (1-\frac {\left (2+\frac {3 x \,{\mathrm e}^{-4}}{2}\right ) {\mathrm e}^{-\frac {3 x \,{\mathrm e}^{-4}}{4}}}{2}\right )}{3}\)	\(37\)
derivativedivides	\(-\frac {{\mathrm e}^{4} \left ({\mathrm e}^{4} \left ({\mathrm e}^{-\frac {3 x \,{\mathrm e}^{-4}}{4}+2} \left (-\frac {3 x \,{\mathrm e}^{-4}}{4}+2\right )-{\mathrm e}^{-\frac {3 x \,{\mathrm e}^{-4}}{4}+2}\right )-{\mathrm e}^{-\frac {3 x \,{\mathrm e}^{-4}}{4}+2} {\mathrm e}^{4}\right )}{3}\)	\(56\)
default	\(-\frac {{\mathrm e}^{4} \left ({\mathrm e}^{4} \left ({\mathrm e}^{-\frac {3 x \,{\mathrm e}^{-4}}{4}+2} \left (-\frac {3 x \,{\mathrm e}^{-4}}{4}+2\right )-{\mathrm e}^{-\frac {3 x \,{\mathrm e}^{-4}}{4}+2}\right )-{\mathrm e}^{-\frac {3 x \,{\mathrm e}^{-4}}{4}+2} {\mathrm e}^{4}\right )}{3}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/16*(4*exp(4)-3*x)*exp(1/4*(8*exp(4)-3*x)/exp(4)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.35, size = 29, normalized size = 1.71 \begin {gather*} \frac {1}{12} \, {\left (3 \, x e^{6} + 4 \, e^{10}\right )} e^{\left (-\frac {3}{4} \, x e^{\left (-4\right )}\right )} - \frac {1}{3} \, e^{\left (-\frac {3}{4} \, x e^{\left (-4\right )} + 10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(4*exp(4)-3*x)*exp(1/4*(8*exp(4)-3*x)/exp(4)),x, algorithm="maxima")

[Out]

1/12*(3*x*e^6 + 4*e^10)*e^(-3/4*x*e^(-4)) - 1/3*e^(-3/4*x*e^(-4) + 10)

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mupad [B]  time = 0.07, size = 11, normalized size = 0.65 \begin {gather*} \frac {x\,{\mathrm {e}}^6\,{\mathrm {e}}^{-\frac {3\,x\,{\mathrm {e}}^{-4}}{4}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-exp(-4)*((3*x)/4 - 2*exp(4)))*(3*x - 4*exp(4)))/16,x)

[Out]

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

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sympy [A]  time = 0.12, size = 20, normalized size = 1.18 \begin {gather*} \frac {x e^{4} e^{\frac {- \frac {3 x}{4} + 2 e^{4}}{e^{4}}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(4*exp(4)-3*x)*exp(1/4*(8*exp(4)-3*x)/exp(4)),x)

[Out]

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

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