3.7.55 \(\int \frac {9}{4} e^{-2+e^4 x} (-\frac {16}{e^4}-4 x) x^3 \, dx\)

Optimal. Leaf size=16 \[ 2-9 e^{-6+e^4 x} x^4 \]

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Rubi [A]  time = 0.18, antiderivative size = 14, normalized size of antiderivative = 0.88, number of steps used = 12, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {12, 2196, 2176, 2194} \begin {gather*} -9 e^{e^4 x-6} x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(9*E^(-2 + E^4*x)*(-16/E^4 - 4*x)*x^3)/4,x]

[Out]

-9*E^(-6 + E^4*x)*x^4

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]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {9}{4} \int e^{-2+e^4 x} \left (-\frac {16}{e^4}-4 x\right ) x^3 \, dx\\ &=\frac {9}{4} \int \left (-16 e^{-6+e^4 x} x^3-4 e^{-2+e^4 x} x^4\right ) \, dx\\ &=-\left (9 \int e^{-2+e^4 x} x^4 \, dx\right )-36 \int e^{-6+e^4 x} x^3 \, dx\\ &=-36 e^{-10+e^4 x} x^3-9 e^{-6+e^4 x} x^4+\frac {36 \int e^{-2+e^4 x} x^3 \, dx}{e^4}+\frac {108 \int e^{-6+e^4 x} x^2 \, dx}{e^4}\\ &=108 e^{-14+e^4 x} x^2-9 e^{-6+e^4 x} x^4-\frac {108 \int e^{-2+e^4 x} x^2 \, dx}{e^8}-\frac {216 \int e^{-6+e^4 x} x \, dx}{e^8}\\ &=-216 e^{-18+e^4 x} x-9 e^{-6+e^4 x} x^4+\frac {216 \int e^{-6+e^4 x} \, dx}{e^{12}}+\frac {216 \int e^{-2+e^4 x} x \, dx}{e^{12}}\\ &=216 e^{-22+e^4 x}-9 e^{-6+e^4 x} x^4-\frac {216 \int e^{-2+e^4 x} \, dx}{e^{16}}\\ &=-9 e^{-6+e^4 x} x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.88 \begin {gather*} -9 e^{-6+e^4 x} x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9*E^(-2 + E^4*x)*(-16/E^4 - 4*x)*x^3)/4,x]

[Out]

-9*E^(-6 + E^4*x)*x^4

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fricas [A]  time = 0.69, size = 49, normalized size = 3.06 \begin {gather*} -e^{\left (-2 \, {\left ({\left (\log \relax (2) + 1\right )} e^{\left (2 \, \log \relax (2) - 4\right )} - e^{\left (2 \, \log \relax (2) - 4\right )} \log \left (3 \, x^{2}\right ) - 2 \, x\right )} e^{\left (-2 \, \log \relax (2) + 4\right )} + 2 \, \log \relax (2) - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(-2*((log(2) + 1)*e^(2*log(2) - 4) - e^(2*log(2) - 4)*log(3*x^2) - 2*x)*e^(-2*log(2) + 4) + 2*log(2) - 4)

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giac [A]  time = 0.30, size = 12, normalized size = 0.75 \begin {gather*} -9 \, x^{4} e^{\left (x e^{4} - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*x^4*e^(x*e^4 - 6)

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maple [A]  time = 0.12, size = 28, normalized size = 1.75

method	result	size
gosper	\(-9 \,{\mathrm e}^{x \,{\mathrm e}^{4}} x^{4} {\mathrm e}^{-6}\)	\(28\)
risch	\(-9 x^{4} {\mathrm e}^{x \,{\mathrm e}^{4}-6-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}}\)	\(62\)
meijerg	\(-36 \,{\mathrm e}^{-22} \left (6-\frac {\left (-4 x^{3} {\mathrm e}^{12}+12 x^{2} {\mathrm e}^{8}-24 x \,{\mathrm e}^{4}+24\right ) {\mathrm e}^{x \,{\mathrm e}^{4}}}{4}\right )+36 \,{\mathrm e}^{-22-2 \ln \relax (2)} \left (24-\frac {\left (5 x^{4} {\mathrm e}^{16}-20 x^{3} {\mathrm e}^{12}+60 x^{2} {\mathrm e}^{8}-120 x \,{\mathrm e}^{4}+120\right ) {\mathrm e}^{x \,{\mathrm e}^{4}}}{5}\right )\)	\(82\)
norman	\(\left (-216 \,{\mathrm e}^{-16} \left ({\mathrm e}^{-4} {\mathrm e}^{4}-1\right ) {\mathrm e}^{-3} {\mathrm e}^{x \,{\mathrm e}^{4}}+36 \,{\mathrm e}^{-4} \left ({\mathrm e}^{-4} {\mathrm e}^{4}-1\right ) {\mathrm e}^{-3} x^{3} {\mathrm e}^{x \,{\mathrm e}^{4}}-108 \,{\mathrm e}^{-8} \left ({\mathrm e}^{-4} {\mathrm e}^{4}-1\right ) {\mathrm e}^{-3} x^{2} {\mathrm e}^{x \,{\mathrm e}^{4}}+216 \,{\mathrm e}^{-12} \left ({\mathrm e}^{-4} {\mathrm e}^{4}-1\right ) {\mathrm e}^{-3} x \,{\mathrm e}^{x \,{\mathrm e}^{4}}-9 \,{\mathrm e}^{-3} {\mathrm e}^{4} {\mathrm e}^{-4} x^{4} {\mathrm e}^{x \,{\mathrm e}^{4}}\right ) {\mathrm e}^{-3}\)	\(163\)
default	\({\mathrm e}^{4} {\mathrm e}^{-16} \left (-2160 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} {\mathrm e}^{-4}-1296 \,{\mathrm e}^{-4} \left ({\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )-{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}\right )-288 \,{\mathrm e}^{-4} \left (\left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )^{2} {\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}-2 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )+2 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}\right )-28 \,{\mathrm e}^{-4} \left (\left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )^{3} {\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}-3 \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )^{2} {\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}+6 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )-6 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}\right )-{\mathrm e}^{-4} \left ({\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )^{4}-4 \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )^{3} {\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}+12 \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )^{2} {\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}-24 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )+24 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}\right )+224 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} {\mathrm e}^{-4} \left (\ln \left (3 x^{2}\right )-2 \ln \relax (x )\right )^{3}-1152 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} {\mathrm e}^{-4} \left (\ln \left (3 x^{2}\right )-2 \ln \relax (x )\right )^{2}+2592 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} {\mathrm e}^{-4} \left (\ln \left (3 x^{2}\right )-2 \ln \relax (x )\right )+1152 \,{\mathrm e}^{-4} \left (\ln \left (3 x^{2}\right )-2 \ln \relax (x )\right ) \left ({\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )-{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}\right )-336 \,{\mathrm e}^{-4} \left (\ln \left (3 x^{2}\right )-2 \ln \relax (x )\right )^{2} \left ({\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )-{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}\right )+168 \,{\mathrm e}^{-4} \left (\ln \left (3 x^{2}\right )-2 \ln \relax (x )\right ) \left (\left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )^{2} {\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}-2 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )+2 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}\right )-24 \,{\mathrm e}^{-4} \left (\ln \left (3 x^{2}\right )-2 \ln \relax (x )\right )^{2} \left (\left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )^{2} {\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}-2 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )+2 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}\right )+8 \,{\mathrm e}^{-4} \left (\ln \left (3 x^{2}\right )-2 \ln \relax (x )\right ) \left (\left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )^{3} {\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}-3 \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )^{2} {\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}+6 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )-6 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}\right )-16 \,{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} {\mathrm e}^{-4} \left (\ln \left (3 x^{2}\right )-2 \ln \relax (x )\right )^{4}+32 \,{\mathrm e}^{-4} \left (\ln \left (3 x^{2}\right )-2 \ln \relax (x )\right )^{3} \left ({\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )} \left (x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )\right )-{\mathrm e}^{x \,{\mathrm e}^{4}-6+2 \ln \left (3 x^{2}\right )-4 \ln \relax (x )}\right )\right )\)	\(2046\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*exp(2*ln(2)-4)-4*x)*exp(x/exp(2*ln(2)-4))^4*exp(ln(3*x^2)-3)^2/x/exp(2*ln(2)-4),x,method=_RETURNVERBOS
E)

[Out]

-exp(x/exp(2*ln(2)-4))^4*exp(ln(3*x^2)-3)^2

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maxima [B]  time = 0.52, size = 66, normalized size = 4.12 \begin {gather*} -9 \, {\left (x^{4} e^{16} - 4 \, x^{3} e^{12} + 12 \, x^{2} e^{8} - 24 \, x e^{4} + 24\right )} e^{\left (x e^{4} - 22\right )} - 36 \, {\left (x^{3} e^{12} - 3 \, x^{2} e^{8} + 6 \, x e^{4} - 6\right )} e^{\left (x e^{4} - 22\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*(x^4*e^16 - 4*x^3*e^12 + 12*x^2*e^8 - 24*x*e^4 + 24)*e^(x*e^4 - 22) - 36*(x^3*e^12 - 3*x^2*e^8 + 6*x*e^4 -
6)*e^(x*e^4 - 22)

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mupad [B]  time = 0.64, size = 12, normalized size = 0.75 \begin {gather*} -9\,x^4\,{\mathrm {e}}^{-6}\,{\mathrm {e}}^{x\,{\mathrm {e}}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*x^4*exp(-6)*exp(x*exp(4))

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sympy [A]  time = 0.12, size = 15, normalized size = 0.94 \begin {gather*} - \frac {9 x^{4} e^{x e^{4}}}{e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*x**4*exp(-6)*exp(x*exp(4))

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