3.101.55 \(\int (-4+\log (5)+e^{\sqrt {e} (-2+2 x)} (8 \sqrt {e}-2 \sqrt {e} \log (5))) \, dx\)

Optimal. Leaf size=24 \[ \left (\frac {1}{2}-e^{2 \sqrt {e} (-1+x)}+x\right ) (-4+\log (5)) \]

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.29, number of steps used = 2, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2194} \begin {gather*} e^{-2 \sqrt {e} (1-x)} (4-\log (5))-x (4-\log (5)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-4 + Log[5] + E^(Sqrt[E]*(-2 + 2*x))*(8*Sqrt[E] - 2*Sqrt[E]*Log[5]),x]

[Out]

(4 - Log[5])/E^(2*Sqrt[E]*(1 - x)) - x*(4 - Log[5])

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 (4-\log (5))+\left (2 \sqrt {e} (4-\log (5))\right ) \int e^{\sqrt {e} (-2+2 x)} \, dx\\ &=e^{-2 \sqrt {e} (1-x)} (4-\log (5))-x (4-\log (5))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 1.17 \begin {gather*} -\left (\left (e^{-2 \sqrt {e}+2 \sqrt {e} x}-x\right ) (-4+\log (5))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-4 + Log[5] + E^(Sqrt[E]*(-2 + 2*x))*(8*Sqrt[E] - 2*Sqrt[E]*Log[5]),x]

[Out]

-((E^(-2*Sqrt[E] + 2*Sqrt[E]*x) - x)*(-4 + Log[5]))

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fricas [A]  time = 1.69, size = 22, normalized size = 0.92 \begin {gather*} -{\left (\log \relax (5) - 4\right )} e^{\left (2 \, {\left (x - 1\right )} e^{\frac {1}{2}}\right )} + x \log \relax (5) - 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(1/2)*log(5)+8*exp(1/2))*exp((2*x-2)*exp(1/2))+log(5)-4,x, algorithm="fricas")

[Out]

-(log(5) - 4)*e^(2*(x - 1)*e^(1/2)) + x*log(5) - 4*x

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giac [A]  time = 0.18, size = 30, normalized size = 1.25 \begin {gather*} -{\left (e^{\frac {1}{2}} \log \relax (5) - 4 \, e^{\frac {1}{2}}\right )} e^{\left (2 \, {\left (x - 1\right )} e^{\frac {1}{2}} - \frac {1}{2}\right )} + x \log \relax (5) - 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(1/2)*log(5)+8*exp(1/2))*exp((2*x-2)*exp(1/2))+log(5)-4,x, algorithm="giac")

[Out]

-(e^(1/2)*log(5) - 4*e^(1/2))*e^(2*(x - 1)*e^(1/2) - 1/2) + x*log(5) - 4*x

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maple [A]  time = 0.03, size = 24, normalized size = 1.00

method	result	size
norman	\(\left (-\ln \relax (5)+4\right ) {\mathrm e}^{\left (2 x -2\right ) {\mathrm e}^{\frac {1}{2}}}+\left (\ln \relax (5)-4\right ) x\)	\(24\)
risch	\(-{\mathrm e}^{2 \left (x -1\right ) {\mathrm e}^{\frac {1}{2}}} \ln \relax (5)+x \ln \relax (5)+4 \,{\mathrm e}^{2 \left (x -1\right ) {\mathrm e}^{\frac {1}{2}}}-4 x\)	\(31\)
default	\(-4 x -{\mathrm e}^{\left (2 x -2\right ) {\mathrm e}^{\frac {1}{2}}} \ln \relax (5)+4 \,{\mathrm e}^{\left (2 x -2\right ) {\mathrm e}^{\frac {1}{2}}}+x \ln \relax (5)\)	\(33\)
derivativedivides	\(\frac {{\mathrm e}^{-\frac {1}{2}} \left (\ln \relax (5)-4\right ) \left (-2 \,{\mathrm e}^{\frac {1}{2}} {\mathrm e}^{\left (2 x -2\right ) {\mathrm e}^{\frac {1}{2}}}+\ln \left ({\mathrm e}^{\left (2 x -2\right ) {\mathrm e}^{\frac {1}{2}}}\right )\right )}{2}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*exp(1/2)*ln(5)+8*exp(1/2))*exp((2*x-2)*exp(1/2))+ln(5)-4,x,method=_RETURNVERBOSE)

[Out]

(-ln(5)+4)*exp((2*x-2)*exp(1/2))+(ln(5)-4)*x

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maxima [A]  time = 0.38, size = 30, normalized size = 1.25 \begin {gather*} -{\left (e^{\frac {1}{2}} \log \relax (5) - 4 \, e^{\frac {1}{2}}\right )} e^{\left (2 \, {\left (x - 1\right )} e^{\frac {1}{2}} - \frac {1}{2}\right )} + x \log \relax (5) - 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(1/2)*log(5)+8*exp(1/2))*exp((2*x-2)*exp(1/2))+log(5)-4,x, algorithm="maxima")

[Out]

-(e^(1/2)*log(5) - 4*e^(1/2))*e^(2*(x - 1)*e^(1/2) - 1/2) + x*log(5) - 4*x

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mupad [B]  time = 0.18, size = 18, normalized size = 0.75 \begin {gather*} \left (x-{\mathrm {e}}^{\sqrt {\mathrm {e}}\,\left (2\,x-2\right )}\right )\,\left (\ln \relax (5)-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(5) + exp(exp(1/2)*(2*x - 2))*(8*exp(1/2) - 2*exp(1/2)*log(5)) - 4,x)

[Out]

(x - exp(exp(1/2)*(2*x - 2)))*(log(5) - 4)

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sympy [A]  time = 0.11, size = 22, normalized size = 0.92 \begin {gather*} x \left (-4 + \log {\relax (5 )}\right ) + \left (4 - \log {\relax (5 )}\right ) e^{\left (2 x - 2\right ) e^{\frac {1}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(1/2)*ln(5)+8*exp(1/2))*exp((2*x-2)*exp(1/2))+ln(5)-4,x)

[Out]

x*(-4 + log(5)) + (4 - log(5))*exp((2*x - 2)*exp(1/2))

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