3.9.87 \(\int \frac {e^{4+2 x+x^2} (-3-6 x-6 x^2)+e^{2 x} (6+24 x+12 x^2)}{(4+e^{4+2 x+x^2} x+e^{2 x} (-2 x-2 x^2)) \log ^2(4+e^{4+2 x+x^2} x+e^{2 x} (-2 x-2 x^2))} \, dx\)

Optimal. Leaf size=27 \[ \frac {3}{\log \left (4+e^{2 x} x \left (e^{4+x^2}-2 (1+x)\right )\right )} \]

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Rubi [F]  time = 5.15, 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^{4+2 x+x^2} \left (-3-6 x-6 x^2\right )+e^{2 x} \left (6+24 x+12 x^2\right )}{\left (4+e^{4+2 x+x^2} x+e^{2 x} \left (-2 x-2 x^2\right )\right ) \log ^2\left (4+e^{4+2 x+x^2} x+e^{2 x} \left (-2 x-2 x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(4 + 2*x + x^2)*(-3 - 6*x - 6*x^2) + E^(2*x)*(6 + 24*x + 12*x^2))/((4 + E^(4 + 2*x + x^2)*x + E^(2*x)*(
-2*x - 2*x^2))*Log[4 + E^(4 + 2*x + x^2)*x + E^(2*x)*(-2*x - 2*x^2)]^2),x]

[Out]

-6*Defer[Int][Log[4 + E^(4 + 2*x + x^2)*x - 2*E^(2*x)*x*(1 + x)]^(-2), x] - 3*Defer[Int][1/(x*Log[4 + E^(4 + 2
*x + x^2)*x - 2*E^(2*x)*x*(1 + x)]^2), x] - 6*Defer[Int][x/Log[4 + E^(4 + 2*x + x^2)*x - 2*E^(2*x)*x*(1 + x)]^
2, x] - 24*Defer[Int][1/((-4 + 2*E^(2*x)*x - E^(4 + 2*x + x^2)*x + 2*E^(2*x)*x^2)*Log[4 + E^(4 + 2*x + x^2)*x
- 2*E^(2*x)*x*(1 + x)]^2), x] - 12*Defer[Int][1/(x*(-4 + 2*E^(2*x)*x - E^(4 + 2*x + x^2)*x + 2*E^(2*x)*x^2)*Lo
g[4 + E^(4 + 2*x + x^2)*x - 2*E^(2*x)*x*(1 + x)]^2), x] - 24*Defer[Int][x/((-4 + 2*E^(2*x)*x - E^(4 + 2*x + x^
2)*x + 2*E^(2*x)*x^2)*Log[4 + E^(4 + 2*x + x^2)*x - 2*E^(2*x)*x*(1 + x)]^2), x] - 6*Defer[Int][(E^(2*x)*x)/((-
4 + 2*E^(2*x)*x - E^(4 + 2*x + x^2)*x + 2*E^(2*x)*x^2)*Log[4 + E^(4 + 2*x + x^2)*x - 2*E^(2*x)*x*(1 + x)]^2),
x] + 12*Defer[Int][(E^(2*x)*x^2)/((-4 + 2*E^(2*x)*x - E^(4 + 2*x + x^2)*x + 2*E^(2*x)*x^2)*Log[4 + E^(4 + 2*x
+ x^2)*x - 2*E^(2*x)*x*(1 + x)]^2), x] + 12*Defer[Int][(E^(2*x)*x^3)/((-4 + 2*E^(2*x)*x - E^(4 + 2*x + x^2)*x
+ 2*E^(2*x)*x^2)*Log[4 + E^(4 + 2*x + x^2)*x - 2*E^(2*x)*x*(1 + x)]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {3 \left (1+2 x+2 x^2\right )}{x \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )}+\frac {6 \left (-2-4 x-4 x^2-e^{2 x} x^2+2 e^{2 x} x^3+2 e^{2 x} x^4\right )}{x \left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )}\right ) \, dx\\ &=-\left (3 \int \frac {1+2 x+2 x^2}{x \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )} \, dx\right )+6 \int \frac {-2-4 x-4 x^2-e^{2 x} x^2+2 e^{2 x} x^3+2 e^{2 x} x^4}{x \left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )} \, dx\\ &=-\left (3 \int \left (\frac {2}{\log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )}+\frac {1}{x \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )}+\frac {2 x}{\log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )}\right ) \, dx\right )+6 \int \left (-\frac {4}{\left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )}-\frac {2}{x \left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )}-\frac {4 x}{\left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )}-\frac {e^{2 x} x}{\left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )}+\frac {2 e^{2 x} x^2}{\left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )}+\frac {2 e^{2 x} x^3}{\left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )}\right ) \, dx\\ &=-\left (3 \int \frac {1}{x \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )} \, dx\right )-6 \int \frac {1}{\log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )} \, dx-6 \int \frac {x}{\log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )} \, dx-6 \int \frac {e^{2 x} x}{\left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )} \, dx-12 \int \frac {1}{x \left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )} \, dx+12 \int \frac {e^{2 x} x^2}{\left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )} \, dx+12 \int \frac {e^{2 x} x^3}{\left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )} \, dx-24 \int \frac {1}{\left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )} \, dx-24 \int \frac {x}{\left (-4+2 e^{2 x} x-e^{4+2 x+x^2} x+2 e^{2 x} x^2\right ) \log ^2\left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 30, normalized size = 1.11 \begin {gather*} \frac {3}{\log \left (4+e^{4+2 x+x^2} x-2 e^{2 x} x (1+x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4 + 2*x + x^2)*(-3 - 6*x - 6*x^2) + E^(2*x)*(6 + 24*x + 12*x^2))/((4 + E^(4 + 2*x + x^2)*x + E^(
2*x)*(-2*x - 2*x^2))*Log[4 + E^(4 + 2*x + x^2)*x + E^(2*x)*(-2*x - 2*x^2)]^2),x]

[Out]

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

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fricas [A]  time = 0.95, size = 29, normalized size = 1.07 \begin {gather*} \frac {3}{\log \left (x e^{\left (x^{2} + 2 \, x + 4\right )} - 2 \, {\left (x^{2} + x\right )} e^{\left (2 \, x\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^2-6*x-3)*exp(x)^2*exp(x^2+4)+(12*x^2+24*x+6)*exp(x)^2)/(x*exp(x)^2*exp(x^2+4)+(-2*x^2-2*x)*ex
p(x)^2+4)/log(x*exp(x)^2*exp(x^2+4)+(-2*x^2-2*x)*exp(x)^2+4)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 2.44, size = 34, normalized size = 1.26 \begin {gather*} \frac {3}{\log \left (-2 \, x^{2} e^{\left (2 \, x\right )} + x e^{\left (x^{2} + 2 \, x + 4\right )} - 2 \, x e^{\left (2 \, x\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^2-6*x-3)*exp(x)^2*exp(x^2+4)+(12*x^2+24*x+6)*exp(x)^2)/(x*exp(x)^2*exp(x^2+4)+(-2*x^2-2*x)*ex
p(x)^2+4)/log(x*exp(x)^2*exp(x^2+4)+(-2*x^2-2*x)*exp(x)^2+4)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 33, normalized size = 1.22

method	result	size
risch	\(\frac {3}{\ln \left (x \,{\mathrm e}^{x^{2}+2 x +4}+\left (-2 x^{2}-2 x \right ) {\mathrm e}^{2 x}+4\right )}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^2-6*x-3)*exp(x)^2*exp(x^2+4)+(12*x^2+24*x+6)*exp(x)^2)/(x*exp(x)^2*exp(x^2+4)+(-2*x^2-2*x)*exp(x)^2
+4)/ln(x*exp(x)^2*exp(x^2+4)+(-2*x^2-2*x)*exp(x)^2+4)^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.89, size = 29, normalized size = 1.07 \begin {gather*} \frac {3}{\log \left (x e^{\left (x^{2} + 2 \, x + 4\right )} - 2 \, {\left (x^{2} + x\right )} e^{\left (2 \, x\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^2-6*x-3)*exp(x)^2*exp(x^2+4)+(12*x^2+24*x+6)*exp(x)^2)/(x*exp(x)^2*exp(x^2+4)+(-2*x^2-2*x)*ex
p(x)^2+4)/log(x*exp(x)^2*exp(x^2+4)+(-2*x^2-2*x)*exp(x)^2+4)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.22, size = 34, normalized size = 1.26 \begin {gather*} \frac {3}{\ln \left (x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^4-{\mathrm {e}}^{2\,x}\,\left (2\,x^2+2\,x\right )+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(24*x + 12*x^2 + 6) - exp(2*x)*exp(x^2 + 4)*(6*x + 6*x^2 + 3))/(log(x*exp(2*x)*exp(x^2 + 4) - ex
p(2*x)*(2*x + 2*x^2) + 4)^2*(x*exp(2*x)*exp(x^2 + 4) - exp(2*x)*(2*x + 2*x^2) + 4)),x)

[Out]

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

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sympy [A]  time = 0.87, size = 32, normalized size = 1.19 \begin {gather*} \frac {3}{\log {\left (x e^{2 x} e^{x^{2} + 4} + \left (- 2 x^{2} - 2 x\right ) e^{2 x} + 4 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**2-6*x-3)*exp(x)**2*exp(x**2+4)+(12*x**2+24*x+6)*exp(x)**2)/(x*exp(x)**2*exp(x**2+4)+(-2*x**2
-2*x)*exp(x)**2+4)/ln(x*exp(x)**2*exp(x**2+4)+(-2*x**2-2*x)*exp(x)**2+4)**2,x)

[Out]

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

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