3.81.27 \(\int \frac {4+4 x}{e^9-2 x-x^2+\log (5)} \, dx\)

Optimal. Leaf size=21 \[ \log \left (\frac {1}{2 \left (e^9-2 x-x^2+\log (5)\right )^2}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 0.81, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {628} \begin {gather*} -2 \log \left (-x^2-2 x+e^9+\log (5)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + 4*x)/(E^9 - 2*x - x^2 + Log[5]),x]

[Out]

-2*Log[E^9 - 2*x - x^2 + Log[5]]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-2 \log \left (e^9-2 x-x^2+\log (5)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 0.81 \begin {gather*} -2 \log \left (e^9-2 x-x^2+\log (5)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 4*x)/(E^9 - 2*x - x^2 + Log[5]),x]

[Out]

-2*Log[E^9 - 2*x - x^2 + Log[5]]

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fricas [A]  time = 0.57, size = 18, normalized size = 0.86 \begin {gather*} -2 \, \log \left (x^{2} + 2 \, x - e^{9} - \log \relax (5)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+4)/(log(5)+exp(9)-x^2-2*x),x, algorithm="fricas")

[Out]

-2*log(x^2 + 2*x - e^9 - log(5))

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giac [A]  time = 0.18, size = 19, normalized size = 0.90 \begin {gather*} -2 \, \log \left ({\left | x^{2} + 2 \, x - e^{9} - \log \relax (5) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+4)/(log(5)+exp(9)-x^2-2*x),x, algorithm="giac")

[Out]

-2*log(abs(x^2 + 2*x - e^9 - log(5)))

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maple [A]  time = 0.39, size = 17, normalized size = 0.81

method	result	size
default	\(-2 \ln \left (\ln \relax (5)+{\mathrm e}^{9}-x^{2}-2 x \right )\)	\(17\)
norman	\(-2 \ln \left (\ln \relax (5)+{\mathrm e}^{9}-x^{2}-2 x \right )\)	\(17\)
risch	\(-2 \ln \left (-\ln \relax (5)-{\mathrm e}^{9}+x^{2}+2 x \right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x+4)/(ln(5)+exp(9)-x^2-2*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.37, size = 18, normalized size = 0.86 \begin {gather*} -2 \, \log \left (x^{2} + 2 \, x - e^{9} - \log \relax (5)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+4)/(log(5)+exp(9)-x^2-2*x),x, algorithm="maxima")

[Out]

-2*log(x^2 + 2*x - e^9 - log(5))

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mupad [B]  time = 5.53, size = 18, normalized size = 0.86 \begin {gather*} -2\,\ln \left (x^2+2\,x-{\mathrm {e}}^9-\ln \relax (5)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + 4)/(2*x - exp(9) - log(5) + x^2),x)

[Out]

-2*log(2*x - exp(9) - log(5) + x^2)

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sympy [A]  time = 0.19, size = 17, normalized size = 0.81 \begin {gather*} - 2 \log {\left (x^{2} + 2 x - e^{9} - \log {\relax (5 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+4)/(ln(5)+exp(9)-x**2-2*x),x)

[Out]

-2*log(x**2 + 2*x - exp(9) - log(5))

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