3.86.83 \(\int (-64 e^{6-2 x}+2 x+2^{2+x} e^{2^{1+x}} \log (2)) \, dx\)

Optimal. Leaf size=24 \[ 4+2 \left (e^{2^{1+x}}+16 e^{6-2 x}\right )+x^2 \]

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Rubi [A]  time = 0.03, antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2194, 2282, 12} \begin {gather*} x^2+2 e^{2^{x+1}}+32 e^{6-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-64*E^(6 - 2*x) + 2*x + 2^(2 + x)*E^2^(1 + x)*Log[2],x]

[Out]

2*E^2^(1 + x) + 32*E^(6 - 2*x) + x^2

Rule 12

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

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 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x^2-64 \int e^{6-2 x} \, dx+\log (2) \int 2^{2+x} e^{2^{1+x}} \, dx\\ &=32 e^{6-2 x}+x^2+\operatorname {Subst}\left (\int 4 e^{2 x} \, dx,x,2^x\right )\\ &=32 e^{6-2 x}+x^2+4 \operatorname {Subst}\left (\int e^{2 x} \, dx,x,2^x\right )\\ &=2 e^{2^{1+x}}+32 e^{6-2 x}+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 22, normalized size = 0.92 \begin {gather*} 2 e^{2^{1+x}}+32 e^{6-2 x}+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-64*E^(6 - 2*x) + 2*x + 2^(2 + x)*E^2^(1 + x)*Log[2],x]

[Out]

2*E^2^(1 + x) + 32*E^(6 - 2*x) + x^2

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fricas [A]  time = 0.73, size = 20, normalized size = 0.83 \begin {gather*} x^{2} + 2 \, e^{\left (2 \cdot 2^{x}\right )} + 32 \, e^{\left (-2 \, x + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*log(2)*exp(x*log(2))*exp(2*exp(x*log(2)))-64*exp(3-x)^2+2*x,x, algorithm="fricas")

[Out]

x^2 + 2*e^(2*2^x) + 32*e^(-2*x + 6)

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giac [A]  time = 0.20, size = 20, normalized size = 0.83 \begin {gather*} x^{2} + 2 \, e^{\left (2 \cdot 2^{x}\right )} + 32 \, e^{\left (-2 \, x + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*log(2)*exp(x*log(2))*exp(2*exp(x*log(2)))-64*exp(3-x)^2+2*x,x, algorithm="giac")

[Out]

x^2 + 2*e^(2*2^x) + 32*e^(-2*x + 6)

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maple [A]  time = 0.07, size = 21, normalized size = 0.88

method	result	size
risch	\(x^{2}+32 \,{\mathrm e}^{6-2 x}+2 \,{\mathrm e}^{2 \,2^{x}}\)	\(21\)
default	\(x^{2}+32 \,{\mathrm e}^{6-2 x}+2 \,{\mathrm e}^{2 \,{\mathrm e}^{x \ln \relax (2)}}\)	\(25\)
norman	\(x^{2}+32 \,{\mathrm e}^{6-2 x}+2 \,{\mathrm e}^{2 \,{\mathrm e}^{x \ln \relax (2)}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*ln(2)*exp(x*ln(2))*exp(2*exp(x*ln(2)))-64*exp(3-x)^2+2*x,x,method=_RETURNVERBOSE)

[Out]

x^2+32*exp(6-2*x)+2*exp(2*2^x)

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maxima [A]  time = 0.35, size = 26, normalized size = 1.08 \begin {gather*} x^{2} + 2^{\frac {2^{x + 1}}{\log \relax (2)} + 1} + 32 \, e^{\left (-2 \, x + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*log(2)*exp(x*log(2))*exp(2*exp(x*log(2)))-64*exp(3-x)^2+2*x,x, algorithm="maxima")

[Out]

x^2 + 2^(2^(x + 1)/log(2) + 1) + 32*e^(-2*x + 6)

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mupad [B]  time = 0.11, size = 20, normalized size = 0.83 \begin {gather*} 2\,{\mathrm {e}}^{2\,2^x}+32\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^6+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x - 64*exp(6 - 2*x) + 4*exp(2*exp(x*log(2)))*exp(x*log(2))*log(2),x)

[Out]

2*exp(2*2^x) + 32*exp(-2*x)*exp(6) + x^2

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sympy [A]  time = 0.16, size = 22, normalized size = 0.92 \begin {gather*} x^{2} + 32 e^{6 - 2 x} + 2 e^{2 e^{x \log {\relax (2 )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*ln(2)*exp(x*ln(2))*exp(2*exp(x*ln(2)))-64*exp(3-x)**2+2*x,x)

[Out]

x**2 + 32*exp(6 - 2*x) + 2*exp(2*exp(x*log(2)))

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