3.60.46 \(\int \frac {e^x (-2-2 x)-x-2 x^2-x^3-e^x x^2 \log (x^2)}{45 x+120 x^2+110 x^3+40 x^4+5 x^5+e^x (30 x+40 x^2+10 x^3) \log (x^2)+5 e^{2 x} x \log ^2(x^2)} \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{5 \left (3+x+\frac {e^x x \log \left (x^2\right )}{x+x^2}\right )} \]

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Rubi [F]  time = 5.53, 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^x (-2-2 x)-x-2 x^2-x^3-e^x x^2 \log \left (x^2\right )}{45 x+120 x^2+110 x^3+40 x^4+5 x^5+e^x \left (30 x+40 x^2+10 x^3\right ) \log \left (x^2\right )+5 e^{2 x} x \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(-2 - 2*x) - x - 2*x^2 - x^3 - E^x*x^2*Log[x^2])/(45*x + 120*x^2 + 110*x^3 + 40*x^4 + 5*x^5 + E^x*(30
*x + 40*x^2 + 10*x^3)*Log[x^2] + 5*E^(2*x)*x*Log[x^2]^2),x]

[Out]

-1/5*Defer[Int][(3 + 4*x + x^2 + E^x*Log[x^2])^(-2), x] + Defer[Int][x/(3 + 4*x + x^2 + E^x*Log[x^2])^2, x]/5
+ (3*Defer[Int][x^2/(3 + 4*x + x^2 + E^x*Log[x^2])^2, x])/5 + Defer[Int][x^3/(3 + 4*x + x^2 + E^x*Log[x^2])^2,
x]/5 + (14*Defer[Int][1/(Log[x^2]*(3 + 4*x + x^2 + E^x*Log[x^2])^2), x])/5 + (6*Defer[Int][1/(x*Log[x^2]*(3 +
4*x + x^2 + E^x*Log[x^2])^2), x])/5 + 2*Defer[Int][x/(Log[x^2]*(3 + 4*x + x^2 + E^x*Log[x^2])^2), x] + (2*Def
er[Int][x^2/(Log[x^2]*(3 + 4*x + x^2 + E^x*Log[x^2])^2), x])/5 - Defer[Int][x/(3 + 4*x + x^2 + E^x*Log[x^2]),
x]/5 - (2*Defer[Int][1/(Log[x^2]*(3 + 4*x + x^2 + E^x*Log[x^2])), x])/5 - (2*Defer[Int][1/(x*Log[x^2]*(3 + 4*x
+ x^2 + E^x*Log[x^2])), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\left ((1+x) \left (2 e^x+x+x^2\right )\right )-e^x x^2 \log \left (x^2\right )}{5 x \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {-\left ((1+x) \left (2 e^x+x+x^2\right )\right )-e^x x^2 \log \left (x^2\right )}{x \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {2+2 x+x^2 \log \left (x^2\right )}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )}+\frac {(1+x) \left (6+8 x+2 x^2-x \log \left (x^2\right )+2 x^2 \log \left (x^2\right )+x^3 \log \left (x^2\right )\right )}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {2+2 x+x^2 \log \left (x^2\right )}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )} \, dx\right )+\frac {1}{5} \int \frac {(1+x) \left (6+8 x+2 x^2-x \log \left (x^2\right )+2 x^2 \log \left (x^2\right )+x^3 \log \left (x^2\right )\right )}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {(1+x) \left (2 \left (3+4 x+x^2\right )+x \left (-1+2 x+x^2\right ) \log \left (x^2\right )\right )}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx-\frac {1}{5} \int \left (\frac {x}{3+4 x+x^2+e^x \log \left (x^2\right )}+\frac {2}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )}+\frac {2}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {x}{3+4 x+x^2+e^x \log \left (x^2\right )} \, dx\right )+\frac {1}{5} \int \left (\frac {6+8 x+2 x^2-x \log \left (x^2\right )+2 x^2 \log \left (x^2\right )+x^3 \log \left (x^2\right )}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}+\frac {6+8 x+2 x^2-x \log \left (x^2\right )+2 x^2 \log \left (x^2\right )+x^3 \log \left (x^2\right )}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}\right ) \, dx-\frac {2}{5} \int \frac {1}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )} \, dx-\frac {2}{5} \int \frac {1}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )} \, dx\\ &=-\left (\frac {1}{5} \int \frac {x}{3+4 x+x^2+e^x \log \left (x^2\right )} \, dx\right )+\frac {1}{5} \int \frac {6+8 x+2 x^2-x \log \left (x^2\right )+2 x^2 \log \left (x^2\right )+x^3 \log \left (x^2\right )}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx+\frac {1}{5} \int \frac {6+8 x+2 x^2-x \log \left (x^2\right )+2 x^2 \log \left (x^2\right )+x^3 \log \left (x^2\right )}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx-\frac {2}{5} \int \frac {1}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )} \, dx-\frac {2}{5} \int \frac {1}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )} \, dx\\ &=-\left (\frac {1}{5} \int \frac {x}{3+4 x+x^2+e^x \log \left (x^2\right )} \, dx\right )+\frac {1}{5} \int \frac {2 \left (3+4 x+x^2\right )+x \left (-1+2 x+x^2\right ) \log \left (x^2\right )}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx+\frac {1}{5} \int \frac {2 \left (3+4 x+x^2\right )+x \left (-1+2 x+x^2\right ) \log \left (x^2\right )}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx-\frac {2}{5} \int \frac {1}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )} \, dx-\frac {2}{5} \int \frac {1}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )} \, dx\\ &=-\left (\frac {1}{5} \int \frac {x}{3+4 x+x^2+e^x \log \left (x^2\right )} \, dx\right )+\frac {1}{5} \int \left (-\frac {1}{\left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}+\frac {2 x}{\left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}+\frac {x^2}{\left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}+\frac {8}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}+\frac {6}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}+\frac {2 x}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}\right ) \, dx+\frac {1}{5} \int \left (-\frac {x}{\left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}+\frac {2 x^2}{\left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}+\frac {x^3}{\left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}+\frac {6}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}+\frac {8 x}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}+\frac {2 x^2}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2}\right ) \, dx-\frac {2}{5} \int \frac {1}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )} \, dx-\frac {2}{5} \int \frac {1}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )} \, dx\\ &=-\left (\frac {1}{5} \int \frac {1}{\left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx\right )-\frac {1}{5} \int \frac {x}{\left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx+\frac {1}{5} \int \frac {x^2}{\left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx+\frac {1}{5} \int \frac {x^3}{\left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx-\frac {1}{5} \int \frac {x}{3+4 x+x^2+e^x \log \left (x^2\right )} \, dx+\frac {2}{5} \int \frac {x}{\left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx+\frac {2}{5} \int \frac {x^2}{\left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx+\frac {2}{5} \int \frac {x}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx+\frac {2}{5} \int \frac {x^2}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx-\frac {2}{5} \int \frac {1}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )} \, dx-\frac {2}{5} \int \frac {1}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )} \, dx+\frac {6}{5} \int \frac {1}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx+\frac {6}{5} \int \frac {1}{x \log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx+\frac {8}{5} \int \frac {1}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx+\frac {8}{5} \int \frac {x}{\log \left (x^2\right ) \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.66, size = 27, normalized size = 1.08 \begin {gather*} -\frac {-1-x}{5 \left (3+4 x+x^2+e^x \log \left (x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-2 - 2*x) - x - 2*x^2 - x^3 - E^x*x^2*Log[x^2])/(45*x + 120*x^2 + 110*x^3 + 40*x^4 + 5*x^5 + E
^x*(30*x + 40*x^2 + 10*x^3)*Log[x^2] + 5*E^(2*x)*x*Log[x^2]^2),x]

[Out]

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

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fricas [A]  time = 0.68, size = 22, normalized size = 0.88 \begin {gather*} \frac {x + 1}{5 \, {\left (x^{2} + e^{x} \log \left (x^{2}\right ) + 4 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(x)*log(x^2)+(-2*x-2)*exp(x)-x^3-2*x^2-x)/(5*x*exp(x)^2*log(x^2)^2+(10*x^3+40*x^2+30*x)*exp
(x)*log(x^2)+5*x^5+40*x^4+110*x^3+120*x^2+45*x),x, algorithm="fricas")

[Out]

1/5*(x + 1)/(x^2 + e^x*log(x^2) + 4*x + 3)

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giac [A]  time = 0.28, size = 22, normalized size = 0.88 \begin {gather*} \frac {x + 1}{5 \, {\left (x^{2} + e^{x} \log \left (x^{2}\right ) + 4 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(x)*log(x^2)+(-2*x-2)*exp(x)-x^3-2*x^2-x)/(5*x*exp(x)^2*log(x^2)^2+(10*x^3+40*x^2+30*x)*exp
(x)*log(x^2)+5*x^5+40*x^4+110*x^3+120*x^2+45*x),x, algorithm="giac")

[Out]

1/5*(x + 1)/(x^2 + e^x*log(x^2) + 4*x + 3)

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maple [C]  time = 0.15, size = 79, normalized size = 3.16

method	result	size
risch	\(\frac {\frac {2 x}{5}+\frac {2}{5}}{-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) {\mathrm e}^{x}+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}+2 x^{2}+4 \,{\mathrm e}^{x} \ln \relax (x )+8 x +6}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2*exp(x)*ln(x^2)+(-2*x-2)*exp(x)-x^3-2*x^2-x)/(5*x*exp(x)^2*ln(x^2)^2+(10*x^3+40*x^2+30*x)*exp(x)*ln(x
^2)+5*x^5+40*x^4+110*x^3+120*x^2+45*x),x,method=_RETURNVERBOSE)

[Out]

2/5*(x+1)/(-I*Pi*csgn(I*x)^2*csgn(I*x^2)*exp(x)+2*I*Pi*csgn(I*x)*csgn(I*x^2)^2*exp(x)-I*Pi*csgn(I*x^2)^3*exp(x
)+2*x^2+4*exp(x)*ln(x)+8*x+6)

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maxima [A]  time = 0.40, size = 21, normalized size = 0.84 \begin {gather*} \frac {x + 1}{5 \, {\left (x^{2} + 2 \, e^{x} \log \relax (x) + 4 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(x)*log(x^2)+(-2*x-2)*exp(x)-x^3-2*x^2-x)/(5*x*exp(x)^2*log(x^2)^2+(10*x^3+40*x^2+30*x)*exp
(x)*log(x^2)+5*x^5+40*x^4+110*x^3+120*x^2+45*x),x, algorithm="maxima")

[Out]

1/5*(x + 1)/(x^2 + 2*e^x*log(x) + 4*x + 3)

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mupad [B]  time = 4.58, size = 90, normalized size = 3.60 \begin {gather*} \frac {\frac {2\,x\,{\mathrm {e}}^{2\,x}}{5}+\frac {2\,x^2\,{\mathrm {e}}^{2\,x}}{5}-{\mathrm {e}}^x\,\left (\frac {x^5}{5}+\frac {3\,x^4}{5}+\frac {x^3}{5}-\frac {x^2}{5}\right )}{\left (4\,x+\ln \left (x^2\right )\,{\mathrm {e}}^x+x^2+3\right )\,\left (2\,x\,{\mathrm {e}}^{2\,x}+x^2\,{\mathrm {e}}^x-2\,x^3\,{\mathrm {e}}^x-x^4\,{\mathrm {e}}^x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + exp(x)*(2*x + 2) + 2*x^2 + x^3 + x^2*log(x^2)*exp(x))/(45*x + 120*x^2 + 110*x^3 + 40*x^4 + 5*x^5 + l
og(x^2)*exp(x)*(30*x + 40*x^2 + 10*x^3) + 5*x*log(x^2)^2*exp(2*x)),x)

[Out]

((2*x*exp(2*x))/5 + (2*x^2*exp(2*x))/5 - exp(x)*(x^3/5 - x^2/5 + (3*x^4)/5 + x^5/5))/((4*x + log(x^2)*exp(x) +
x^2 + 3)*(2*x*exp(2*x) + x^2*exp(x) - 2*x^3*exp(x) - x^4*exp(x)))

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sympy [A]  time = 0.33, size = 22, normalized size = 0.88 \begin {gather*} \frac {x + 1}{5 x^{2} + 20 x + 5 e^{x} \log {\left (x^{2} \right )} + 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2*exp(x)*ln(x**2)+(-2*x-2)*exp(x)-x**3-2*x**2-x)/(5*x*exp(x)**2*ln(x**2)**2+(10*x**3+40*x**2+30
*x)*exp(x)*ln(x**2)+5*x**5+40*x**4+110*x**3+120*x**2+45*x),x)

[Out]

(x + 1)/(5*x**2 + 20*x + 5*exp(x)*log(x**2) + 15)

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