3.53.58 \(\int \frac {e^{e^3+x-\log ^2(\frac {1}{2} (x+2 \log (4)))} (x+2 \log (4)-2 \log (\frac {1}{2} (x+2 \log (4))))}{x+2 \log (4)} \, dx\)

Optimal. Leaf size=25 \[ e^{e^3+x-\log ^2\left (\frac {x}{2}+\log (4)\right )}+\log ^2(3) \]

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Rubi [A]  time = 0.28, antiderivative size = 20, normalized size of antiderivative = 0.80, number of steps used = 1, number of rules used = 1, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6706} \begin {gather*} e^{x-\log ^2\left (\frac {1}{2} (x+\log (16))\right )+e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(E^3 + x - Log[(x + Log[16])/2]^2)

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} &=e^{e^3+x-\log ^2\left (\frac {1}{2} (x+\log (16))\right )}\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.45, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{e^3+x-\log ^2\left (\frac {1}{2} (x+2 \log (4))\right )} \left (x+2 \log (4)-2 \log \left (\frac {1}{2} (x+2 \log (4))\right )\right )}{x+2 \log (4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 18, normalized size = 0.72 \begin {gather*} e^{\left (-\log \left (\frac {1}{2} \, x + 2 \, \log \relax (2)\right )^{2} + x + e^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(2*log(2)+1/2*x)+4*log(2)+x)*exp(-log(2*log(2)+1/2*x)^2+exp(3)+x)/(x+4*log(2)),x, algorithm="
fricas")

[Out]

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

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giac [A]  time = 0.18, size = 18, normalized size = 0.72 \begin {gather*} e^{\left (-\log \left (\frac {1}{2} \, x + 2 \, \log \relax (2)\right )^{2} + x + e^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(2*log(2)+1/2*x)+4*log(2)+x)*exp(-log(2*log(2)+1/2*x)^2+exp(3)+x)/(x+4*log(2)),x, algorithm="
giac")

[Out]

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

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maple [A]  time = 0.17, size = 19, normalized size = 0.76

method	result	size
norman	\({\mathrm e}^{-\ln \left (2 \ln \relax (2)+\frac {x}{2}\right )^{2}+{\mathrm e}^{3}+x}\)	\(19\)
risch	\({\mathrm e}^{-\ln \left (2 \ln \relax (2)+\frac {x}{2}\right )^{2}+{\mathrm e}^{3}+x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*ln(2*ln(2)+1/2*x)+4*ln(2)+x)*exp(-ln(2*ln(2)+1/2*x)^2+exp(3)+x)/(x+4*ln(2)),x,method=_RETURNVERBOSE)

[Out]

exp(-ln(2*ln(2)+1/2*x)^2+exp(3)+x)

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maxima [A]  time = 0.55, size = 33, normalized size = 1.32 \begin {gather*} e^{\left (-\log \relax (2)^{2} + 2 \, \log \relax (2) \log \left (x + 4 \, \log \relax (2)\right ) - \log \left (x + 4 \, \log \relax (2)\right )^{2} + x + e^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(2*log(2)+1/2*x)+4*log(2)+x)*exp(-log(2*log(2)+1/2*x)^2+exp(3)+x)/(x+4*log(2)),x, algorithm="
maxima")

[Out]

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

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mupad [B]  time = 0.28, size = 18, normalized size = 0.72 \begin {gather*} {\mathrm {e}}^{-{\ln \left (\frac {x}{2}+\ln \relax (4)\right )}^2}\,{\mathrm {e}}^{{\mathrm {e}}^3}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.32, size = 17, normalized size = 0.68 \begin {gather*} e^{x - \log {\left (\frac {x}{2} + 2 \log {\relax (2 )} \right )}^{2} + e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(2*ln(2)+1/2*x)+4*ln(2)+x)*exp(-ln(2*ln(2)+1/2*x)**2+exp(3)+x)/(x+4*ln(2)),x)

[Out]

exp(x - log(x/2 + 2*log(2))**2 + exp(3))

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