3.92.43 \(\int \frac {e^{-5+(-2 x+x^2) \log (2)} (-6+6 x) \log (2)}{-20+3 e^{-5+(-2 x+x^2) \log (2)}} \, dx\)

Optimal. Leaf size=20 \[ 3+\log \left (-5+\frac {3}{4} e^{-5+(-2+x) x \log (2)}\right ) \]

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Rubi [A]  time = 0.19, antiderivative size = 28, normalized size of antiderivative = 1.40, number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {12, 6684} \begin {gather*} \log \left (-2^{-2 x} \left (3\ 2^{x^2}-5 e^5 2^{2 x+2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-5 + (-2*x + x^2)*Log[2])*(-6 + 6*x)*Log[2])/(-20 + 3*E^(-5 + (-2*x + x^2)*Log[2])),x]

[Out]

Log[-((3*2^x^2 - 5*2^(2 + 2*x)*E^5)/2^(2*x))]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 (2) \int \frac {e^{-5+\left (-2 x+x^2\right ) \log (2)} (-6+6 x)}{-20+3 e^{-5+\left (-2 x+x^2\right ) \log (2)}} \, dx\\ &=\log \left (-2^{-2 x} \left (3\ 2^{x^2}-5\ 2^{2+2 x} e^5\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.80, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{-5+\left (-2 x+x^2\right ) \log (2)} (-6+6 x) \log (2)}{-20+3 e^{-5+\left (-2 x+x^2\right ) \log (2)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^(-5 + (-2*x + x^2)*Log[2])*(-6 + 6*x)*Log[2])/(-20 + 3*E^(-5 + (-2*x + x^2)*Log[2])),x]

[Out]

Integrate[(E^(-5 + (-2*x + x^2)*Log[2])*(-6 + 6*x)*Log[2])/(-20 + 3*E^(-5 + (-2*x + x^2)*Log[2])), x]

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fricas [A]  time = 1.27, size = 18, normalized size = 0.90 \begin {gather*} \log \left (3 \, e^{\left ({\left (x^{2} - 2 \, x\right )} \log \relax (2) - 5\right )} - 20\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x-6)*log(2)*exp((x^2-2*x)*log(2)-5)/(3*exp((x^2-2*x)*log(2)-5)-20),x, algorithm="fricas")

[Out]

log(3*e^((x^2 - 2*x)*log(2) - 5) - 20)

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giac [A]  time = 0.21, size = 20, normalized size = 1.00 \begin {gather*} \log \left ({\left | 3 \, e^{\left (x^{2} \log \relax (2) - 2 \, x \log \relax (2) - 5\right )} - 20 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x-6)*log(2)*exp((x^2-2*x)*log(2)-5)/(3*exp((x^2-2*x)*log(2)-5)-20),x, algorithm="giac")

[Out]

log(abs(3*e^(x^2*log(2) - 2*x*log(2) - 5) - 20))

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maple [A]  time = 0.14, size = 16, normalized size = 0.80

method	result	size
risch	\(5+\ln \left (2^{\left (x -2\right ) x} {\mathrm e}^{-5}-\frac {20}{3}\right )\)	\(16\)
norman	\(\ln \left (3 \,{\mathrm e}^{\left (x^{2}-2 x \right ) \ln \relax (2)-5}-20\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x-6)*ln(2)*exp((x^2-2*x)*ln(2)-5)/(3*exp((x^2-2*x)*ln(2)-5)-20),x,method=_RETURNVERBOSE)

[Out]

5+ln(2^((x-2)*x)*exp(-5)-20/3)

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maxima [A]  time = 0.47, size = 31, normalized size = 1.55 \begin {gather*} -{\left (2 \, x - \frac {\log \left (2^{\left (x^{2}\right )} - \frac {20}{3} \, e^{\left (2 \, x \log \relax (2) + 5\right )}\right )}{\log \relax (2)}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x-6)*log(2)*exp((x^2-2*x)*log(2)-5)/(3*exp((x^2-2*x)*log(2)-5)-20),x, algorithm="maxima")

[Out]

-(2*x - log(2^(x^2) - 20/3*e^(2*x*log(2) + 5))/log(2))*log(2)

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mupad [B]  time = 0.35, size = 24, normalized size = 1.20 \begin {gather*} \ln \left (3\,2^{x^2}\,{\mathrm {e}}^{-5}-20\,2^{2\,x}\right )-2\,x\,\ln \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(- log(2)*(2*x - x^2) - 5)*log(2)*(6*x - 6))/(3*exp(- log(2)*(2*x - x^2) - 5) - 20),x)

[Out]

log(3*2^(x^2)*exp(-5) - 20*2^(2*x)) - 2*x*log(2)

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sympy [A]  time = 0.15, size = 17, normalized size = 0.85 \begin {gather*} \log {\left (e^{\left (x^{2} - 2 x\right ) \log {\relax (2 )} - 5} - \frac {20}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x-6)*ln(2)*exp((x**2-2*x)*ln(2)-5)/(3*exp((x**2-2*x)*ln(2)-5)-20),x)

[Out]

log(exp((x**2 - 2*x)*log(2) - 5) - 20/3)

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