3.62.83 \(\int \frac {2+e^x-8 x+18 x^2-16 x^3+5 x^4}{-9+e^x+2 x-4 x^2+6 x^3-4 x^4+x^5} \, dx\)

Optimal. Leaf size=21 \[ \log \left (-9+e^x+x+x \left (-1+2 x-x^2\right )^2\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 31, normalized size of antiderivative = 1.48, number of steps used = 1, number of rules used = 1, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6684} \begin {gather*} \log \left (-x^5+4 x^4-6 x^3+4 x^2-2 x-e^x+9\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + E^x - 8*x + 18*x^2 - 16*x^3 + 5*x^4)/(-9 + E^x + 2*x - 4*x^2 + 6*x^3 - 4*x^4 + x^5),x]

[Out]

Log[9 - E^x - 2*x + 4*x^2 - 6*x^3 + 4*x^4 - x^5]

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 (9-e^x-2 x+4 x^2-6 x^3+4 x^4-x^5\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 27, normalized size = 1.29 \begin {gather*} \log \left (-9+e^x+2 x-4 x^2+6 x^3-4 x^4+x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + E^x - 8*x + 18*x^2 - 16*x^3 + 5*x^4)/(-9 + E^x + 2*x - 4*x^2 + 6*x^3 - 4*x^4 + x^5),x]

[Out]

Log[-9 + E^x + 2*x - 4*x^2 + 6*x^3 - 4*x^4 + x^5]

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fricas [A]  time = 0.65, size = 26, normalized size = 1.24 \begin {gather*} \log \left (x^{5} - 4 \, x^{4} + 6 \, x^{3} - 4 \, x^{2} + 2 \, x + e^{x} - 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+5*x^4-16*x^3+18*x^2-8*x+2)/(exp(x)+x^5-4*x^4+6*x^3-4*x^2+2*x-9),x, algorithm="fricas")

[Out]

log(x^5 - 4*x^4 + 6*x^3 - 4*x^2 + 2*x + e^x - 9)

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giac [A]  time = 3.12, size = 26, normalized size = 1.24 \begin {gather*} \log \left (x^{5} - 4 \, x^{4} + 6 \, x^{3} - 4 \, x^{2} + 2 \, x + e^{x} - 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+5*x^4-16*x^3+18*x^2-8*x+2)/(exp(x)+x^5-4*x^4+6*x^3-4*x^2+2*x-9),x, algorithm="giac")

[Out]

log(x^5 - 4*x^4 + 6*x^3 - 4*x^2 + 2*x + e^x - 9)

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maple [A]  time = 0.04, size = 27, normalized size = 1.29

method	result	size
derivativedivides	\(\ln \left ({\mathrm e}^{x}+x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+2 x -9\right )\)	\(27\)
default	\(\ln \left ({\mathrm e}^{x}+x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+2 x -9\right )\)	\(27\)
norman	\(\ln \left ({\mathrm e}^{x}+x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+2 x -9\right )\)	\(27\)
risch	\(\ln \left ({\mathrm e}^{x}+x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+2 x -9\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)+5*x^4-16*x^3+18*x^2-8*x+2)/(exp(x)+x^5-4*x^4+6*x^3-4*x^2+2*x-9),x,method=_RETURNVERBOSE)

[Out]

ln(exp(x)+x^5-4*x^4+6*x^3-4*x^2+2*x-9)

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maxima [A]  time = 0.34, size = 26, normalized size = 1.24 \begin {gather*} \log \left (x^{5} - 4 \, x^{4} + 6 \, x^{3} - 4 \, x^{2} + 2 \, x + e^{x} - 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+5*x^4-16*x^3+18*x^2-8*x+2)/(exp(x)+x^5-4*x^4+6*x^3-4*x^2+2*x-9),x, algorithm="maxima")

[Out]

log(x^5 - 4*x^4 + 6*x^3 - 4*x^2 + 2*x + e^x - 9)

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mupad [B]  time = 4.03, size = 26, normalized size = 1.24 \begin {gather*} \ln \left (2\,x+{\mathrm {e}}^x-4\,x^2+6\,x^3-4\,x^4+x^5-9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x) - 8*x + 18*x^2 - 16*x^3 + 5*x^4 + 2)/(2*x + exp(x) - 4*x^2 + 6*x^3 - 4*x^4 + x^5 - 9),x)

[Out]

log(2*x + exp(x) - 4*x^2 + 6*x^3 - 4*x^4 + x^5 - 9)

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sympy [A]  time = 0.15, size = 27, normalized size = 1.29 \begin {gather*} \log {\left (x^{5} - 4 x^{4} + 6 x^{3} - 4 x^{2} + 2 x + e^{x} - 9 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+5*x**4-16*x**3+18*x**2-8*x+2)/(exp(x)+x**5-4*x**4+6*x**3-4*x**2+2*x-9),x)

[Out]

log(x**5 - 4*x**4 + 6*x**3 - 4*x**2 + 2*x + exp(x) - 9)

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