3.83.92 \(\int \frac {-1-e^x+e^{5+e^5 x}}{6-e^x+e^{e^5 x}-x} \, dx\)

Optimal. Leaf size=30 \[ \log \left (6-e^x+e^{-x+\left (1+e^5-x\right ) x+x^2}-x\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 18, normalized size of antiderivative = 0.60, number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {6684} \begin {gather*} \log \left (-x-e^x+e^{e^5 x}+6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - E^x + E^(5 + E^5*x))/(6 - E^x + E^(E^5*x) - x),x]

[Out]

Log[6 - E^x + E^(E^5*x) - x]

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 (6-e^x+e^{e^5 x}-x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 16, normalized size = 0.53 \begin {gather*} \log \left (-6+e^x-e^{e^5 x}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - E^x + E^(5 + E^5*x))/(6 - E^x + E^(E^5*x) - x),x]

[Out]

Log[-6 + E^x - E^(E^5*x) + x]

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fricas [A]  time = 0.63, size = 22, normalized size = 0.73 \begin {gather*} \log \left (-{\left (x - 6\right )} e^{5} + e^{\left (x e^{5} + 5\right )} - e^{\left (x + 5\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(5)*exp(x*exp(5))-exp(x)-1)/(exp(x*exp(5))-exp(x)-x+6),x, algorithm="fricas")

[Out]

log(-(x - 6)*e^5 + e^(x*e^5 + 5) - e^(x + 5))

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giac [A]  time = 0.18, size = 15, normalized size = 0.50 \begin {gather*} \log \left (-x + e^{\left (x e^{5}\right )} - e^{x} + 6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(5)*exp(x*exp(5))-exp(x)-1)/(exp(x*exp(5))-exp(x)-x+6),x, algorithm="giac")

[Out]

log(-x + e^(x*e^5) - e^x + 6)

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maple [A]  time = 0.06, size = 14, normalized size = 0.47

method	result	size
norman	\(\ln \left (-{\mathrm e}^{x \,{\mathrm e}^{5}}+{\mathrm e}^{x}+x -6\right )\)	\(14\)
derivativedivides	\(\ln \left ({\mathrm e}^{x \,{\mathrm e}^{5}}-{\mathrm e}^{x}-x +6\right )\)	\(16\)
default	\(\ln \left ({\mathrm e}^{x \,{\mathrm e}^{5}}-{\mathrm e}^{x}-x +6\right )\)	\(16\)
risch	\(\ln \left ({\mathrm e}^{x \,{\mathrm e}^{5}}-{\mathrm e}^{x}-x +6\right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(5)*exp(x*exp(5))-exp(x)-1)/(exp(x*exp(5))-exp(x)-x+6),x,method=_RETURNVERBOSE)

[Out]

ln(-exp(x*exp(5))+exp(x)+x-6)

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maxima [A]  time = 0.36, size = 13, normalized size = 0.43 \begin {gather*} \log \left (x - e^{\left (x e^{5}\right )} + e^{x} - 6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(5)*exp(x*exp(5))-exp(x)-1)/(exp(x*exp(5))-exp(x)-x+6),x, algorithm="maxima")

[Out]

log(x - e^(x*e^5) + e^x - 6)

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mupad [B]  time = 0.13, size = 13, normalized size = 0.43 \begin {gather*} \ln \left (x-{\mathrm {e}}^{x\,{\mathrm {e}}^5}+{\mathrm {e}}^x-6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x) - exp(5)*exp(x*exp(5)) + 1)/(x - exp(x*exp(5)) + exp(x) - 6),x)

[Out]

log(x - exp(x*exp(5)) + exp(x) - 6)

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sympy [A]  time = 0.22, size = 14, normalized size = 0.47 \begin {gather*} \log {\left (- x - e^{x} + e^{x e^{5}} + 6 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(5)*exp(x*exp(5))-exp(x)-1)/(exp(x*exp(5))-exp(x)-x+6),x)

[Out]

log(-x - exp(x) + exp(x*exp(5)) + 6)

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