3.32.37 \(\int \frac {6+e^x}{8+e^x+6 x} \, dx\)

Optimal. Leaf size=13 \[ \log \left (e^x+2 x+4 (2+x)\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 9, normalized size of antiderivative = 0.69, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6684} \begin {gather*} \log \left (6 x+e^x+8\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 + E^x)/(8 + E^x + 6*x),x]

[Out]

Log[8 + E^x + 6*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 (8+e^x+6 x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 9, normalized size = 0.69 \begin {gather*} \log \left (8+e^x+6 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + E^x)/(8 + E^x + 6*x),x]

[Out]

Log[8 + E^x + 6*x]

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fricas [A]  time = 0.49, size = 8, normalized size = 0.62 \begin {gather*} \log \left (6 \, x + e^{x} + 8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+6)/(exp(x)+6*x+8),x, algorithm="fricas")

[Out]

log(6*x + e^x + 8)

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giac [A]  time = 0.30, size = 8, normalized size = 0.62 \begin {gather*} \log \left (6 \, x + e^{x} + 8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+6)/(exp(x)+6*x+8),x, algorithm="giac")

[Out]

log(6*x + e^x + 8)

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maple [A]  time = 0.02, size = 9, normalized size = 0.69

method	result	size
derivativedivides	\(\ln \left ({\mathrm e}^{x}+6 x +8\right )\)	\(9\)
default	\(\ln \left ({\mathrm e}^{x}+6 x +8\right )\)	\(9\)
norman	\(\ln \left ({\mathrm e}^{x}+6 x +8\right )\)	\(9\)
risch	\(\ln \left ({\mathrm e}^{x}+6 x +8\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)+6)/(exp(x)+6*x+8),x,method=_RETURNVERBOSE)

[Out]

ln(exp(x)+6*x+8)

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maxima [A]  time = 0.48, size = 8, normalized size = 0.62 \begin {gather*} \log \left (6 \, x + e^{x} + 8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+6)/(exp(x)+6*x+8),x, algorithm="maxima")

[Out]

log(6*x + e^x + 8)

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mupad [B]  time = 0.06, size = 8, normalized size = 0.62 \begin {gather*} \ln \left (6\,x+{\mathrm {e}}^x+8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x) + 6)/(6*x + exp(x) + 8),x)

[Out]

log(6*x + exp(x) + 8)

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sympy [A]  time = 0.09, size = 8, normalized size = 0.62 \begin {gather*} \log {\left (6 x + e^{x} + 8 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+6)/(exp(x)+6*x+8),x)

[Out]

log(6*x + exp(x) + 8)

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