3.15.40 \(\int \frac {e^{\frac {4 x+4 x^2}{\log (4)}} (-4-8 x)+\log (4)}{2 e^{\frac {4 x+4 x^2}{\log (4)}} \log (4)+(-2 x-2 \log (2)) \log (4)} \, dx\)

Optimal. Leaf size=30 \[ e^3-\frac {1}{2} \log \left (e^{\frac {4 \left (x+x^2\right )}{\log (4)}}-x-\log (2)\right ) \]

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Rubi [A]  time = 0.13, antiderivative size = 32, normalized size of antiderivative = 1.07, number of steps used = 1, number of rules used = 1, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6684} \begin {gather*} -\frac {1}{2} \log \left (e^{\frac {4 x^2}{\log (4)}+\frac {4 x}{\log (4)}}-x-\log (2)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((4*x + 4*x^2)/Log[4])*(-4 - 8*x) + Log[4])/(2*E^((4*x + 4*x^2)/Log[4])*Log[4] + (-2*x - 2*Log[2])*Log[
4]),x]

[Out]

-1/2*Log[E^((4*x)/Log[4] + (4*x^2)/Log[4]) - x - Log[2]]

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} &=-\frac {1}{2} \log \left (e^{\frac {4 x}{\log (4)}+\frac {4 x^2}{\log (4)}}-x-\log (2)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.27, size = 32, normalized size = 1.07 \begin {gather*} -\frac {1}{2} \log \left (e^{\frac {4 x}{\log (4)}+\frac {4 x^2}{\log (4)}}-x-\log (2)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4*x + 4*x^2)/Log[4])*(-4 - 8*x) + Log[4])/(2*E^((4*x + 4*x^2)/Log[4])*Log[4] + (-2*x - 2*Log[2]
)*Log[4]),x]

[Out]

-1/2*Log[E^((4*x)/Log[4] + (4*x^2)/Log[4]) - x - Log[2]]

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fricas [A]  time = 0.91, size = 23, normalized size = 0.77 \begin {gather*} -\frac {1}{2} \, \log \left (-x + e^{\left (\frac {2 \, {\left (x^{2} + x\right )}}{\log \relax (2)}\right )} - \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-4)*exp(1/2*(4*x^2+4*x)/log(2))+2*log(2))/(4*log(2)*exp(1/2*(4*x^2+4*x)/log(2))+2*(-2*log(2)-2
*x)*log(2)),x, algorithm="fricas")

[Out]

-1/2*log(-x + e^(2*(x^2 + x)/log(2)) - log(2))

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giac [A]  time = 0.67, size = 21, normalized size = 0.70 \begin {gather*} -\frac {1}{2} \, \log \left (x - e^{\left (\frac {2 \, {\left (x^{2} + x\right )}}{\log \relax (2)}\right )} + \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-4)*exp(1/2*(4*x^2+4*x)/log(2))+2*log(2))/(4*log(2)*exp(1/2*(4*x^2+4*x)/log(2))+2*(-2*log(2)-2
*x)*log(2)),x, algorithm="giac")

[Out]

-1/2*log(x - e^(2*(x^2 + x)/log(2)) + log(2))

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maple [A]  time = 0.12, size = 23, normalized size = 0.77

method	result	size
risch	\(-\frac {\ln \left ({\mathrm e}^{\frac {2 \left (x +1\right ) x}{\ln \relax (2)}}-\ln \relax (2)-x \right )}{2}\)	\(23\)
norman	\(-\frac {\ln \left (-{\mathrm e}^{\frac {4 x^{2}+4 x}{2 \ln \relax (2)}}+\ln \relax (2)+x \right )}{2}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x-4)*exp(1/2*(4*x^2+4*x)/ln(2))+2*ln(2))/(4*ln(2)*exp(1/2*(4*x^2+4*x)/ln(2))+2*(-2*ln(2)-2*x)*ln(2)),
x,method=_RETURNVERBOSE)

[Out]

-1/2*ln(exp(2*(x+1)*x/ln(2))-ln(2)-x)

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maxima [A]  time = 0.38, size = 27, normalized size = 0.90 \begin {gather*} -\frac {1}{2} \, \log \left ({\left (x + \log \relax (2)\right )} \log \relax (2) - e^{\left (\frac {2 \, {\left (x^{2} + x\right )}}{\log \relax (2)}\right )} \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-4)*exp(1/2*(4*x^2+4*x)/log(2))+2*log(2))/(4*log(2)*exp(1/2*(4*x^2+4*x)/log(2))+2*(-2*log(2)-2
*x)*log(2)),x, algorithm="maxima")

[Out]

-1/2*log((x + log(2))*log(2) - e^(2*(x^2 + x)/log(2))*log(2))

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mupad [B]  time = 0.29, size = 20, normalized size = 0.67 \begin {gather*} -\frac {\ln \left (x+\ln \relax (2)-{\mathrm {e}}^{\frac {2\,x\,\left (x+1\right )}{\ln \relax (2)}}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(2) - exp((2*x + 2*x^2)/log(2))*(8*x + 4))/(4*exp((2*x + 2*x^2)/log(2))*log(2) - 2*log(2)*(2*x + 2*l
og(2))),x)

[Out]

-log(x + log(2) - exp((2*x*(x + 1))/log(2)))/2

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sympy [A]  time = 0.18, size = 22, normalized size = 0.73 \begin {gather*} - \frac {\log {\left (- x + e^{\frac {2 x^{2} + 2 x}{\log {\relax (2 )}}} - \log {\relax (2 )} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-4)*exp(1/2*(4*x**2+4*x)/ln(2))+2*ln(2))/(4*ln(2)*exp(1/2*(4*x**2+4*x)/ln(2))+2*(-2*ln(2)-2*x)
*ln(2)),x)

[Out]

-log(-x + exp((2*x**2 + 2*x)/log(2)) - log(2))/2

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