3.2.45 \(\int (2+2^{2 e^x x} e^x (2+2 x) \log (2)+2^{e^x x} e^x (8+8 x) \log (2)) \, dx\)

Optimal. Leaf size=20 \[ 1+\left (4+2^{e^x x}\right )^2+2 x-4 \log (5) \]

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Rubi [A]  time = 0.17, antiderivative size = 21, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6706} \begin {gather*} 2 x+2^{2 e^x x}+2^{e^x x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2 + 2^(2*E^x*x)*E^x*(2 + 2*x)*Log[2] + 2^(E^x*x)*E^x*(8 + 8*x)*Log[2],x]

[Out]

2^(2*E^x*x) + 2^(3 + E^x*x) + 2*x

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 x+\log (2) \int 2^{2 e^x x} e^x (2+2 x) \, dx+\log (2) \int 2^{e^x x} e^x (8+8 x) \, dx\\ &=2^{2 e^x x}+2^{3+e^x x}+2 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 15, normalized size = 0.75 \begin {gather*} \left (4+2^{e^x x}\right )^2+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2 + 2^(2*E^x*x)*E^x*(2 + 2*x)*Log[2] + 2^(E^x*x)*E^x*(8 + 8*x)*Log[2],x]

[Out]

(4 + 2^(E^x*x))^2 + 2*x

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fricas [A]  time = 1.13, size = 19, normalized size = 0.95 \begin {gather*} 2^{2 \, x e^{x}} + 8 \cdot 2^{x e^{x}} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+2)*log(2)*exp(x)*exp(x*log(2)*exp(x))^2+(8*x+8)*log(2)*exp(x)*exp(x*log(2)*exp(x))+2,x, algorit
hm="fricas")

[Out]

2^(2*x*e^x) + 8*2^(x*e^x) + 2*x

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giac [A]  time = 0.40, size = 19, normalized size = 0.95 \begin {gather*} 2^{2 \, x e^{x}} + 8 \cdot 2^{x e^{x}} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+2)*log(2)*exp(x)*exp(x*log(2)*exp(x))^2+(8*x+8)*log(2)*exp(x)*exp(x*log(2)*exp(x))+2,x, algorit
hm="giac")

[Out]

2^(2*x*e^x) + 8*2^(x*e^x) + 2*x

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maple [A]  time = 0.10, size = 21, normalized size = 1.05

method	result	size
risch	\(2 x +2^{2 \,{\mathrm e}^{x} x}+8 \,2^{{\mathrm e}^{x} x}\)	\(21\)
default	\(2 x +{\mathrm e}^{2 x \ln \relax (2) {\mathrm e}^{x}}+8 \,{\mathrm e}^{x \ln \relax (2) {\mathrm e}^{x}}\)	\(23\)
norman	\(2 x +{\mathrm e}^{2 x \ln \relax (2) {\mathrm e}^{x}}+8 \,{\mathrm e}^{x \ln \relax (2) {\mathrm e}^{x}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x+2)*ln(2)*exp(x)*exp(x*ln(2)*exp(x))^2+(8*x+8)*ln(2)*exp(x)*exp(x*ln(2)*exp(x))+2,x,method=_RETURNVERB
OSE)

[Out]

2*x+(2^(exp(x)*x))^2+8*2^(exp(x)*x)

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maxima [A]  time = 0.92, size = 19, normalized size = 0.95 \begin {gather*} 2^{2 \, x e^{x}} + 8 \cdot 2^{x e^{x}} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+2)*log(2)*exp(x)*exp(x*log(2)*exp(x))^2+(8*x+8)*log(2)*exp(x)*exp(x*log(2)*exp(x))+2,x, algorit
hm="maxima")

[Out]

2^(2*x*e^x) + 8*2^(x*e^x) + 2*x

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mupad [B]  time = 0.25, size = 19, normalized size = 0.95 \begin {gather*} 2\,x+8\,2^{x\,{\mathrm {e}}^x}+2^{2\,x\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x*exp(x)*log(2))*exp(x)*log(2)*(2*x + 2) + exp(x*exp(x)*log(2))*exp(x)*log(2)*(8*x + 8) + 2,x)

[Out]

2*x + 8*2^(x*exp(x)) + 2^(2*x*exp(x))

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sympy [A]  time = 0.19, size = 26, normalized size = 1.30 \begin {gather*} 2 x + e^{2 x e^{x} \log {\relax (2 )}} + 8 e^{x e^{x} \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+2)*ln(2)*exp(x)*exp(x*ln(2)*exp(x))**2+(8*x+8)*ln(2)*exp(x)*exp(x*ln(2)*exp(x))+2,x)

[Out]

2*x + exp(2*x*exp(x)*log(2)) + 8*exp(x*exp(x)*log(2))

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