3.3.17 \(\int (-1+2 e^{2 x}+4 \log (4 e^9)) \, dx\)

Optimal. Leaf size=29 \[ -e^9-x+x \left (\frac {e^{2 x}}{x}+4 \log \left (4 e^9\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 0.41, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2194} \begin {gather*} e^{2 x}+x (35+\log (256)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 + 2*E^(2*x) + 4*Log[4*E^9],x]

[Out]

E^(2*x) + x*(35 + Log[256])

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} &=x (35+\log (256))+2 \int e^{2 x} \, dx\\ &=e^{2 x}+x (35+\log (256))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 0.45 \begin {gather*} e^{2 x}+35 x+x \log (256) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + 2*E^(2*x) + 4*Log[4*E^9],x]

[Out]

E^(2*x) + 35*x + x*Log[256]

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fricas [A]  time = 0.83, size = 13, normalized size = 0.45 \begin {gather*} 8 \, x \log \relax (2) + 35 \, x + e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*log(4*exp(9))+2*exp(x)^2-1,x, algorithm="fricas")

[Out]

8*x*log(2) + 35*x + e^(2*x)

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giac [A]  time = 0.65, size = 16, normalized size = 0.55 \begin {gather*} 4 \, x \log \left (4 \, e^{9}\right ) - x + e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*log(4*exp(9))+2*exp(x)^2-1,x, algorithm="giac")

[Out]

4*x*log(4*e^9) - x + e^(2*x)

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maple [A]  time = 0.03, size = 14, normalized size = 0.48

method	result	size
norman	\({\mathrm e}^{2 x}+\left (8 \ln \relax (2)+35\right ) x\)	\(14\)
risch	\({\mathrm e}^{2 x}+8 x \ln \relax (2)+35 x\)	\(14\)
default	\({\mathrm e}^{2 x}-x +4 \ln \left (4 \,{\mathrm e}^{9}\right ) x\)	\(17\)
derivativedivides	\({\mathrm e}^{2 x}+\left (4 \ln \left (4 \,{\mathrm e}^{9}\right )-1\right ) \ln \left ({\mathrm e}^{x}\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*ln(4*exp(9))+2*exp(x)^2-1,x,method=_RETURNVERBOSE)

[Out]

exp(x)^2+(8*ln(2)+35)*x

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maxima [A]  time = 0.48, size = 16, normalized size = 0.55 \begin {gather*} 4 \, x \log \left (4 \, e^{9}\right ) - x + e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*log(4*exp(9))+2*exp(x)^2-1,x, algorithm="maxima")

[Out]

4*x*log(4*e^9) - x + e^(2*x)

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mupad [B]  time = 0.08, size = 11, normalized size = 0.38 \begin {gather*} {\mathrm {e}}^{2\,x}+x\,\left (\ln \left (256\right )+35\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*log(4*exp(9)) + 2*exp(2*x) - 1,x)

[Out]

exp(2*x) + x*(log(256) + 35)

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sympy [A]  time = 0.08, size = 12, normalized size = 0.41 \begin {gather*} x \left (8 \log {\relax (2 )} + 35\right ) + e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*ln(4*exp(9))+2*exp(x)**2-1,x)

[Out]

x*(8*log(2) + 35) + exp(2*x)

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