3.45.87 \(\int \frac {e^{5-2 e^4+6 x} (-1+6 x)+e^2 \log (2)}{e^{6-4 e^4+12 x}+x^2+e^{3-2 e^4+6 x} (-2 x-2 \log (2))+2 x \log (2)+\log ^2(2)} \, dx\)

Optimal. Leaf size=25 \[ \frac {e^2 x}{-e^{3-2 e^4+6 x}+x+\log (2)} \]

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Rubi [F]  time = 2.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{5-2 e^4+6 x} (-1+6 x)+e^2 \log (2)}{e^{6-4 e^4+12 x}+x^2+e^{3-2 e^4+6 x} (-2 x-2 \log (2))+2 x \log (2)+\log ^2(2)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(5 - 2*E^4 + 6*x)*(-1 + 6*x) + E^2*Log[2])/(E^(6 - 4*E^4 + 12*x) + x^2 + E^(3 - 2*E^4 + 6*x)*(-2*x - 2*
Log[2]) + 2*x*Log[2] + Log[2]^2),x]

[Out]

-6*E^(2 + 2*E^4)*Defer[Int][x/(-E^(3 + 6*x) + E^(2*E^4)*x + E^(2*E^4)*Log[2]), x] - E^(2 + 4*E^4)*(1 - Log[64]
)*Defer[Int][x/(E^(3 + 6*x) - E^(2*E^4)*(x + Log[2]))^2, x] + 6*E^(2 + 4*E^4)*Defer[Int][x^2/(E^(3 + 6*x) - E^
(2*E^4)*(x + Log[2]))^2, x] - E^(2 + 2*E^4)*Defer[Int][(E^(3 + 6*x) - E^(2*E^4)*(x + Log[2]))^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{5+2 e^4+6 x} (-1+6 x)+e^{2+4 e^4} \log (2)}{\left (e^{3+6 x}-e^{2 e^4} (x+\log (2))\right )^2} \, dx\\ &=\int \left (-\frac {e^{2+2 e^4} (-1+6 x)}{-e^{3+6 x}+e^{2 e^4} x+e^{2 e^4} \log (2)}+\frac {e^{2+4 e^4} x (-1+6 x+\log (64))}{\left (-e^{3+6 x}+e^{2 e^4} x+e^{2 e^4} \log (2)\right )^2}\right ) \, dx\\ &=-\left (e^{2+2 e^4} \int \frac {-1+6 x}{-e^{3+6 x}+e^{2 e^4} x+e^{2 e^4} \log (2)} \, dx\right )+e^{2+4 e^4} \int \frac {x (-1+6 x+\log (64))}{\left (-e^{3+6 x}+e^{2 e^4} x+e^{2 e^4} \log (2)\right )^2} \, dx\\ &=-\left (e^{2+2 e^4} \int \left (\frac {1}{e^{3+6 x}-e^{2 e^4} x-e^{2 e^4} \log (2)}+\frac {6 x}{-e^{3+6 x}+e^{2 e^4} x+e^{2 e^4} \log (2)}\right ) \, dx\right )+e^{2+4 e^4} \int \frac {x (-1+6 x+\log (64))}{\left (e^{3+6 x}-e^{2 e^4} (x+\log (2))\right )^2} \, dx\\ &=-\left (e^{2+2 e^4} \int \frac {1}{e^{3+6 x}-e^{2 e^4} x-e^{2 e^4} \log (2)} \, dx\right )-\left (6 e^{2+2 e^4}\right ) \int \frac {x}{-e^{3+6 x}+e^{2 e^4} x+e^{2 e^4} \log (2)} \, dx+e^{2+4 e^4} \int \left (\frac {6 x^2}{\left (-e^{3+6 x}+e^{2 e^4} x+e^{2 e^4} \log (2)\right )^2}+\frac {x (-1+\log (64))}{\left (-e^{3+6 x}+e^{2 e^4} x+e^{2 e^4} \log (2)\right )^2}\right ) \, dx\\ &=-\left (e^{2+2 e^4} \int \frac {1}{e^{3+6 x}-e^{2 e^4} (x+\log (2))} \, dx\right )-\left (6 e^{2+2 e^4}\right ) \int \frac {x}{-e^{3+6 x}+e^{2 e^4} x+e^{2 e^4} \log (2)} \, dx+\left (6 e^{2+4 e^4}\right ) \int \frac {x^2}{\left (-e^{3+6 x}+e^{2 e^4} x+e^{2 e^4} \log (2)\right )^2} \, dx-\left (e^{2+4 e^4} (1-\log (64))\right ) \int \frac {x}{\left (-e^{3+6 x}+e^{2 e^4} x+e^{2 e^4} \log (2)\right )^2} \, dx\\ &=-\left (e^{2+2 e^4} \int \frac {1}{e^{3+6 x}-e^{2 e^4} (x+\log (2))} \, dx\right )-\left (6 e^{2+2 e^4}\right ) \int \frac {x}{-e^{3+6 x}+e^{2 e^4} x+e^{2 e^4} \log (2)} \, dx+\left (6 e^{2+4 e^4}\right ) \int \frac {x^2}{\left (e^{3+6 x}-e^{2 e^4} (x+\log (2))\right )^2} \, dx-\left (e^{2+4 e^4} (1-\log (64))\right ) \int \frac {x}{\left (e^{3+6 x}-e^{2 e^4} (x+\log (2))\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.47, size = 35, normalized size = 1.40 \begin {gather*} \frac {e^{2+2 e^4} x}{-e^{3+6 x}+e^{2 e^4} (x+\log (2))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(5 - 2*E^4 + 6*x)*(-1 + 6*x) + E^2*Log[2])/(E^(6 - 4*E^4 + 12*x) + x^2 + E^(3 - 2*E^4 + 6*x)*(-2*
x - 2*Log[2]) + 2*x*Log[2] + Log[2]^2),x]

[Out]

(E^(2 + 2*E^4)*x)/(-E^(3 + 6*x) + E^(2*E^4)*(x + Log[2]))

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fricas [A]  time = 0.55, size = 28, normalized size = 1.12 \begin {gather*} \frac {x e^{4}}{x e^{2} + e^{2} \log \relax (2) - e^{\left (6 \, x - 2 \, e^{4} + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x-1)*exp(2)*exp(-2*exp(4)+6*x+3)+exp(2)*log(2))/(exp(-2*exp(4)+6*x+3)^2+(-2*log(2)-2*x)*exp(-2*e
xp(4)+6*x+3)+log(2)^2+2*x*log(2)+x^2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.19, size = 267, normalized size = 10.68 \begin {gather*} \frac {6 \, x^{3} e^{\left (2 \, e^{4} + 2\right )} + 12 \, x^{2} e^{\left (2 \, e^{4} + 2\right )} \log \relax (2) + 6 \, x e^{\left (2 \, e^{4} + 2\right )} \log \relax (2)^{2} - 6 \, x^{2} e^{\left (6 \, x + 5\right )} - x^{2} e^{\left (2 \, e^{4} + 2\right )} - 6 \, x e^{\left (6 \, x + 5\right )} \log \relax (2) - x e^{\left (2 \, e^{4} + 2\right )} \log \relax (2) + x e^{\left (6 \, x + 5\right )}}{6 \, x^{3} e^{\left (2 \, e^{4}\right )} + 18 \, x^{2} e^{\left (2 \, e^{4}\right )} \log \relax (2) + 18 \, x e^{\left (2 \, e^{4}\right )} \log \relax (2)^{2} + 6 \, e^{\left (2 \, e^{4}\right )} \log \relax (2)^{3} - 12 \, x^{2} e^{\left (6 \, x + 3\right )} - x^{2} e^{\left (2 \, e^{4}\right )} - 24 \, x e^{\left (6 \, x + 3\right )} \log \relax (2) - 2 \, x e^{\left (2 \, e^{4}\right )} \log \relax (2) - 12 \, e^{\left (6 \, x + 3\right )} \log \relax (2)^{2} - e^{\left (2 \, e^{4}\right )} \log \relax (2)^{2} + 6 \, x e^{\left (12 \, x - 2 \, e^{4} + 6\right )} + 2 \, x e^{\left (6 \, x + 3\right )} + 6 \, e^{\left (12 \, x - 2 \, e^{4} + 6\right )} \log \relax (2) + 2 \, e^{\left (6 \, x + 3\right )} \log \relax (2) - e^{\left (12 \, x - 2 \, e^{4} + 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x-1)*exp(2)*exp(-2*exp(4)+6*x+3)+exp(2)*log(2))/(exp(-2*exp(4)+6*x+3)^2+(-2*log(2)-2*x)*exp(-2*e
xp(4)+6*x+3)+log(2)^2+2*x*log(2)+x^2),x, algorithm="giac")

[Out]

(6*x^3*e^(2*e^4 + 2) + 12*x^2*e^(2*e^4 + 2)*log(2) + 6*x*e^(2*e^4 + 2)*log(2)^2 - 6*x^2*e^(6*x + 5) - x^2*e^(2
*e^4 + 2) - 6*x*e^(6*x + 5)*log(2) - x*e^(2*e^4 + 2)*log(2) + x*e^(6*x + 5))/(6*x^3*e^(2*e^4) + 18*x^2*e^(2*e^
4)*log(2) + 18*x*e^(2*e^4)*log(2)^2 + 6*e^(2*e^4)*log(2)^3 - 12*x^2*e^(6*x + 3) - x^2*e^(2*e^4) - 24*x*e^(6*x
+ 3)*log(2) - 2*x*e^(2*e^4)*log(2) - 12*e^(6*x + 3)*log(2)^2 - e^(2*e^4)*log(2)^2 + 6*x*e^(12*x - 2*e^4 + 6) +
2*x*e^(6*x + 3) + 6*e^(12*x - 2*e^4 + 6)*log(2) + 2*e^(6*x + 3)*log(2) - e^(12*x - 2*e^4 + 6))

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maple [A]  time = 0.21, size = 23, normalized size = 0.92

method	result	size
risch	\(\frac {x \,{\mathrm e}^{2}}{x +\ln \relax (2)-{\mathrm e}^{-2 \,{\mathrm e}^{4}+6 x +3}}\)	\(23\)
norman	\(\frac {{\mathrm e}^{2} {\mathrm e}^{-2 \,{\mathrm e}^{4}+6 x +3}-{\mathrm e}^{2} \ln \relax (2)}{x +\ln \relax (2)-{\mathrm e}^{-2 \,{\mathrm e}^{4}+6 x +3}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x-1)*exp(2)*exp(-2*exp(4)+6*x+3)+exp(2)*ln(2))/(exp(-2*exp(4)+6*x+3)^2+(-2*ln(2)-2*x)*exp(-2*exp(4)+6*
x+3)+ln(2)^2+2*x*ln(2)+x^2),x,method=_RETURNVERBOSE)

[Out]

x*exp(2)/(x+ln(2)-exp(-2*exp(4)+6*x+3))

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maxima [A]  time = 0.50, size = 35, normalized size = 1.40 \begin {gather*} \frac {x e^{\left (2 \, e^{4} + 2\right )}}{x e^{\left (2 \, e^{4}\right )} + e^{\left (2 \, e^{4}\right )} \log \relax (2) - e^{\left (6 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x-1)*exp(2)*exp(-2*exp(4)+6*x+3)+exp(2)*log(2))/(exp(-2*exp(4)+6*x+3)^2+(-2*log(2)-2*x)*exp(-2*e
xp(4)+6*x+3)+log(2)^2+2*x*log(2)+x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.40, size = 23, normalized size = 0.92 \begin {gather*} \frac {x\,{\mathrm {e}}^2}{x+\ln \relax (2)-{\mathrm {e}}^{-2\,{\mathrm {e}}^4}\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2)*log(2) + exp(6*x - 2*exp(4) + 3)*exp(2)*(6*x - 1))/(exp(12*x - 4*exp(4) + 6) + 2*x*log(2) - exp(6*
x - 2*exp(4) + 3)*(2*x + 2*log(2)) + log(2)^2 + x^2),x)

[Out]

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

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sympy [A]  time = 0.13, size = 22, normalized size = 0.88 \begin {gather*} - \frac {x e^{2}}{- x + e^{6 x - 2 e^{4} + 3} - \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x-1)*exp(2)*exp(-2*exp(4)+6*x+3)+exp(2)*ln(2))/(exp(-2*exp(4)+6*x+3)**2+(-2*ln(2)-2*x)*exp(-2*ex
p(4)+6*x+3)+ln(2)**2+2*x*ln(2)+x**2),x)

[Out]

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

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