3.16.46 \(\int (e^x+\log (\frac {1+4 e^{166}}{e^{166}}) (-1+\log (\frac {4}{x}))) \, dx\)

Optimal. Leaf size=18 \[ e^x+x \log \left (4+\frac {1}{e^{166}}\right ) \log \left (\frac {4}{x}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2194, 2295} \begin {gather*} e^x+x \log \left (4+\frac {1}{e^{166}}\right ) \log \left (\frac {4}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x + Log[(1 + 4*E^166)/E^166]*(-1 + Log[4/x]),x]

[Out]

E^x + x*Log[4 + E^(-166)]*Log[4/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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (4+\frac {1}{e^{166}}\right ) \int \left (-1+\log \left (\frac {4}{x}\right )\right ) \, dx+\int e^x \, dx\\ &=e^x-x \log \left (4+\frac {1}{e^{166}}\right )+\log \left (4+\frac {1}{e^{166}}\right ) \int \log \left (\frac {4}{x}\right ) \, dx\\ &=e^x+x \log \left (4+\frac {1}{e^{166}}\right ) \log \left (\frac {4}{x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} e^x+x \log \left (4+\frac {1}{e^{166}}\right ) \log \left (\frac {4}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x + Log[(1 + 4*E^166)/E^166]*(-1 + Log[4/x]),x]

[Out]

E^x + x*Log[4 + E^(-166)]*Log[4/x]

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fricas [A]  time = 0.79, size = 21, normalized size = 1.17 \begin {gather*} x \log \left ({\left (4 \, e^{166} + 1\right )} e^{\left (-166\right )}\right ) \log \left (\frac {4}{x}\right ) + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(4/x)-1)*log((4*exp(166)+1)/exp(166))+exp(x),x, algorithm="fricas")

[Out]

x*log((4*e^166 + 1)*e^(-166))*log(4/x) + e^x

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giac [A]  time = 0.15, size = 21, normalized size = 1.17 \begin {gather*} x \log \left ({\left (4 \, e^{166} + 1\right )} e^{\left (-166\right )}\right ) \log \left (\frac {4}{x}\right ) + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(4/x)-1)*log((4*exp(166)+1)/exp(166))+exp(x),x, algorithm="giac")

[Out]

x*log((4*e^166 + 1)*e^(-166))*log(4/x) + e^x

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maple [A]  time = 0.04, size = 21, normalized size = 1.17

method	result	size
norman	\(\left (\ln \left (4 \,{\mathrm e}^{166}+1\right )-166\right ) x \ln \left (\frac {4}{x}\right )+{\mathrm e}^{x}\)	\(21\)
risch	\(\left (\ln \left (4 \,{\mathrm e}^{166}+1\right )-166\right ) x \ln \left (\frac {4}{x}\right )+{\mathrm e}^{x}\)	\(21\)
default	\(\ln \left (\left (4 \,{\mathrm e}^{166}+1\right ) {\mathrm e}^{-166}\right ) \ln \left (\frac {4}{x}\right ) x +{\mathrm e}^{x}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(4/x)-1)*ln((4*exp(166)+1)/exp(166))+exp(x),x,method=_RETURNVERBOSE)

[Out]

(ln(4*exp(166)+1)-166)*x*ln(4/x)+exp(x)

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maxima [A]  time = 0.42, size = 21, normalized size = 1.17 \begin {gather*} x \log \left ({\left (4 \, e^{166} + 1\right )} e^{\left (-166\right )}\right ) \log \left (\frac {4}{x}\right ) + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(4/x)-1)*log((4*exp(166)+1)/exp(166))+exp(x),x, algorithm="maxima")

[Out]

x*log((4*e^166 + 1)*e^(-166))*log(4/x) + e^x

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mupad [B]  time = 1.07, size = 20, normalized size = 1.11 \begin {gather*} {\mathrm {e}}^x+x\,\ln \left (\frac {4}{x}\right )\,\left (\ln \left (4\,{\mathrm {e}}^{166}+1\right )-166\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x) + log(exp(-166)*(4*exp(166) + 1))*(log(4/x) - 1),x)

[Out]

exp(x) + x*log(4/x)*(log(4*exp(166) + 1) - 166)

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sympy [A]  time = 0.27, size = 20, normalized size = 1.11 \begin {gather*} \left (- 166 x + x \log {\left (1 + 4 e^{166} \right )}\right ) \log {\left (\frac {4}{x} \right )} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(4/x)-1)*ln((4*exp(166)+1)/exp(166))+exp(x),x)

[Out]

(-166*x + x*log(1 + 4*exp(166)))*log(4/x) + exp(x)

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