3.21.37 \(\int \frac {-1-e^{4+x}}{e^{4+x}+x-e^4 (i \pi +\log (16))} \, dx\)

Optimal. Leaf size=24 \[ \log \left (\frac {\log (5)}{e^x-i \pi +\frac {x}{e^4}-\log (16)}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 23, normalized size of antiderivative = 0.96, number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {6684} \begin {gather*} -\log \left (x+e^{x+4}-e^4 (\log (16)+i \pi )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - E^(4 + x))/(E^(4 + x) + x - E^4*(I*Pi + Log[16])),x]

[Out]

-Log[E^(4 + x) + x - E^4*(I*Pi + Log[16])]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\log \left (e^{4+x}+x-e^4 (i \pi +\log (16))\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 23, normalized size = 0.96 \begin {gather*} -\log \left (e^{4+x}+x-e^4 (i \pi +\log (16))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - E^(4 + x))/(E^(4 + x) + x - E^4*(I*Pi + Log[16])),x]

[Out]

-Log[E^(4 + x) + x - E^4*(I*Pi + Log[16])]

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fricas [A]  time = 0.72, size = 20, normalized size = 0.83 \begin {gather*} -\log \left (-i \, \pi e^{4} - 4 \, e^{4} \log \relax (2) + x + e^{\left (x + 4\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-I*pi*e^4 - 4*e^4*log(2) + x + e^(x + 4))

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giac [A]  time = 0.26, size = 23, normalized size = 0.96 \begin {gather*} -\log \left (\pi e^{4} - 4 i \, e^{4} \log \relax (2) + i \, x + i \, e^{\left (x + 4\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(pi*e^4 - 4*I*e^4*log(2) + I*x + I*e^(x + 4))

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maple [A]  time = 0.13, size = 28, normalized size = 1.17

method	result	size
risch	\(-\ln \left ({\mathrm e}^{x}-i \left (-4 i {\mathrm e}^{4} \ln \relax (2)+\pi \,{\mathrm e}^{4}+i x \right ) {\mathrm e}^{-4}\right )\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(exp(x)-I*(-4*I*exp(4)*ln(2)+Pi*exp(4)+I*x)*exp(-4))

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maxima [A]  time = 0.36, size = 24, normalized size = 1.00 \begin {gather*} -\log \left ({\left (i \, \pi + 4 \, \log \relax (2)\right )} e^{4} - x - e^{\left (x + 4\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log((I*pi + 4*log(2))*e^4 - x - e^(x + 4))

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mupad [B]  time = 1.51, size = 20, normalized size = 0.83 \begin {gather*} -\ln \left (x+{\mathrm {e}}^{x+4}-{\mathrm {e}}^4\,\left (\ln \left (16\right )+\Pi \,1{}\mathrm {i}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4)*exp(x) + 1)/(x - exp(4)*(Pi*1i + 4*log(2)) + exp(4)*exp(x)),x)

[Out]

-log(x + exp(x + 4) - exp(4)*(Pi*1i + log(16)))

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sympy [A]  time = 0.22, size = 26, normalized size = 1.08 \begin {gather*} - \log {\left (\frac {x - 4 e^{4} \log {\relax (2 )} - i \pi e^{4}}{e^{4}} + e^{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log((x - 4*exp(4)*log(2) - I*pi*exp(4))*exp(-4) + exp(x))

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