∫ −10x2+180e−x2 x3+(−10x+45e−x2 x+5x2) log(−2+9e−x2 +x)___________________________________________

3.72.39 \(\int \frac {-10 x^2+180 e^{-x^2} x^3+(-10 x+45 e^{-x^2} x+5 x^2) \log (-2+9 e^{-x^2}+x)}{(-2+9 e^{-x^2}+x) \log ^5(-2+9 e^{-x^2}+x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {5 x^2}{2 \log ^4\left (-2+9 e^{-x^2}+x\right )} \]

________________________________________________________________________________________

Rubi [F] time = 4.41, 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 {-10 x^2+180 e^{-x^2} x^3+\left (-10 x+45 e^{-x^2} x+5 x^2\right ) \log \left (-2+9 e^{-x^2}+x\right )}{\left (-2+9 e^{-x^2}+x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(-10*x^2 + (180*x^3)/E^x^2 + (-10*x + (45*x)/E^x^2 + 5*x^2)*Log[-2 + 9/E^x^2 + x])/((-2 + 9/E^x^2 + x)*Log

[-2 + 9/E^x^2 + x]^5),x]

[Out]

-20*Defer[Int][Log[-2 + 9/E^x^2 + x]^(-5), x] + 180*Defer[Int][1/((9 + E^x^2*(-2 + x))*Log[-2 + 9/E^x^2 + x]^5

), x] - 40*Defer[Int][1/((-2 + x)*Log[-2 + 9/E^x^2 + x]^5), x] - 10*Defer[Int][x/Log[-2 + 9/E^x^2 + x]^5, x] +

90*Defer[Int][x/((9 + E^x^2*(-2 + x))*Log[-2 + 9/E^x^2 + x]^5), x] + 180*Defer[Int][x^3/((9 + E^x^2*(-2 + x))

*Log[-2 + 9/E^x^2 + x]^5), x] + 360*Defer[Int][1/((-2 + x)*(9 - 2*E^x^2 + E^x^2*x)*Log[-2 + 9/E^x^2 + x]^5), x

] + 5*Defer[Int][x/Log[-2 + 9/E^x^2 + x]^4, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 x \left (-\frac {2 \left (e^{x^2}-18 x\right ) x}{9+e^{x^2} (-2+x)}+\log \left (-2+9 e^{-x^2}+x\right )\right )}{\log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx\\ &=5 \int \frac {x \left (-\frac {2 \left (e^{x^2}-18 x\right ) x}{9+e^{x^2} (-2+x)}+\log \left (-2+9 e^{-x^2}+x\right )\right )}{\log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx\\ &=5 \int \left (\frac {18 x^2 \left (1-4 x+2 x^2\right )}{(-2+x) \left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {x \left (-2 x-2 \log \left (-2+9 e^{-x^2}+x\right )+x \log \left (-2+9 e^{-x^2}+x\right )\right )}{(-2+x) \log ^5\left (-2+9 e^{-x^2}+x\right )}\right ) \, dx\\ &=5 \int \frac {x \left (-2 x-2 \log \left (-2+9 e^{-x^2}+x\right )+x \log \left (-2+9 e^{-x^2}+x\right )\right )}{(-2+x) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+90 \int \frac {x^2 \left (1-4 x+2 x^2\right )}{(-2+x) \left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx\\ &=5 \int \left (-\frac {2 x^2}{(-2+x) \log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {x}{\log ^4\left (-2+9 e^{-x^2}+x\right )}\right ) \, dx+90 \int \left (\frac {2}{\left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {4}{(-2+x) \left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {x}{\left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {2 x^3}{\left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )}\right ) \, dx\\ &=5 \int \frac {x}{\log ^4\left (-2+9 e^{-x^2}+x\right )} \, dx-10 \int \frac {x^2}{(-2+x) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+90 \int \frac {x}{\left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+180 \int \frac {1}{\left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+180 \int \frac {x^3}{\left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+360 \int \frac {1}{(-2+x) \left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx\\ &=5 \int \frac {x}{\log ^4\left (-2+9 e^{-x^2}+x\right )} \, dx-10 \int \left (\frac {2}{\log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {4}{(-2+x) \log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {x}{\log ^5\left (-2+9 e^{-x^2}+x\right )}\right ) \, dx+90 \int \frac {x}{\left (9+e^{x^2} (-2+x)\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+180 \int \frac {1}{\left (9+e^{x^2} (-2+x)\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+180 \int \frac {x^3}{\left (9+e^{x^2} (-2+x)\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+360 \int \frac {1}{(-2+x) \left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx\\ &=5 \int \frac {x}{\log ^4\left (-2+9 e^{-x^2}+x\right )} \, dx-10 \int \frac {x}{\log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx-20 \int \frac {1}{\log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx-40 \int \frac {1}{(-2+x) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+90 \int \frac {x}{\left (9+e^{x^2} (-2+x)\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+180 \int \frac {1}{\left (9+e^{x^2} (-2+x)\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+180 \int \frac {x^3}{\left (9+e^{x^2} (-2+x)\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+360 \int \frac {1}{(-2+x) \left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A] time = 0.57, size = 22, normalized size = 1.00 \begin {gather*} \frac {5 x^2}{2 \log ^4\left (-2+9 e^{-x^2}+x\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-10*x^2 + (180*x^3)/E^x^2 + (-10*x + (45*x)/E^x^2 + 5*x^2)*Log[-2 + 9/E^x^2 + x])/((-2 + 9/E^x^2 +

x)*Log[-2 + 9/E^x^2 + x]^5),x]

[Out]

(5*x^2)/(2*Log[-2 + 9/E^x^2 + x]^4)

________________________________________________________________________________________

fricas [A] time = 0.67, size = 22, normalized size = 1.00 \begin {gather*} \frac {5 \, x^{2}}{2 \, \log \left (x + e^{\left (-x^{2} + 2 \, \log \relax (3)\right )} - 2\right )^{4}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x*exp(2*log(3)-x^2)+5*x^2-10*x)*log(exp(2*log(3)-x^2)+x-2)+20*x^3*exp(2*log(3)-x^2)-10*x^2)/(exp

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

[Out]

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

________________________________________________________________________________________

giac [A] time = 0.59, size = 19, normalized size = 0.86 \begin {gather*} \frac {5 \, x^{2}}{2 \, \log \left (x + 9 \, e^{\left (-x^{2}\right )} - 2\right )^{4}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x*exp(2*log(3)-x^2)+5*x^2-10*x)*log(exp(2*log(3)-x^2)+x-2)+20*x^3*exp(2*log(3)-x^2)-10*x^2)/(exp

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

[Out]

5/2*x^2/log(x + 9*e^(-x^2) - 2)^4

________________________________________________________________________________________

maple [A] time = 0.03, size = 20, normalized size = 0.91


method	result	size

risch	\(\frac {5 x^{2}}{2 \ln \left (9 \,{\mathrm e}^{-x^{2}}+x -2\right )^{4}}\)	\(20\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x*exp(2*ln(3)-x^2)+5*x^2-10*x)*ln(exp(2*ln(3)-x^2)+x-2)+20*x^3*exp(2*ln(3)-x^2)-10*x^2)/(exp(2*ln(3)-x

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

[Out]

5/2/ln(9*exp(-x^2)+x-2)^4*x^2

________________________________________________________________________________________

maxima [B] time = 0.46, size = 76, normalized size = 3.45 \begin {gather*} \frac {5 \, x^{2}}{2 \, {\left (x^{8} - 4 \, x^{6} \log \left ({\left (x - 2\right )} e^{\left (x^{2}\right )} + 9\right ) + 6 \, x^{4} \log \left ({\left (x - 2\right )} e^{\left (x^{2}\right )} + 9\right )^{2} - 4 \, x^{2} \log \left ({\left (x - 2\right )} e^{\left (x^{2}\right )} + 9\right )^{3} + \log \left ({\left (x - 2\right )} e^{\left (x^{2}\right )} + 9\right )^{4}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x*exp(2*log(3)-x^2)+5*x^2-10*x)*log(exp(2*log(3)-x^2)+x-2)+20*x^3*exp(2*log(3)-x^2)-10*x^2)/(exp

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

[Out]

5/2*x^2/(x^8 - 4*x^6*log((x - 2)*e^(x^2) + 9) + 6*x^4*log((x - 2)*e^(x^2) + 9)^2 - 4*x^2*log((x - 2)*e^(x^2) +

9)^3 + log((x - 2)*e^(x^2) + 9)^4)

________________________________________________________________________________________

mupad [B] time = 5.66, size = 2169, normalized size = 98.59 result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + exp(2*log(3) - x^2) - 2)*(5*x*exp(2*log(3) - x^2) - 10*x + 5*x^2) - 10*x^2 + 20*x^3*exp(2*log(3)

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

[Out]

((5*x^2)/2 + (5*x*log(x + 9*exp(-x^2) - 2)*(x + 9*exp(-x^2) - 2))/(4*(18*x*exp(-x^2) - 1)))/log(x + 9*exp(-x^2

) - 2)^4 - ((5*(x + 9*exp(-x^2) - 2)*(4*x + 243*exp(-x^2) - 1944*exp(-2*x^2) + 4374*exp(-3*x^2) - 990*x^2*exp(

-x^2) + 5832*x^2*exp(-2*x^2) + 1080*x^3*exp(-x^2) - 8748*x^2*exp(-3*x^2) - 1944*x^3*exp(-2*x^2) - 36*x^4*exp(-

x^2) - 10368*x^4*exp(-2*x^2) - 288*x^5*exp(-x^2) + 17496*x^4*exp(-3*x^2) + 15552*x^5*exp(-2*x^2) + 72*x^6*exp(

-x^2) - 23328*x^5*exp(-3*x^2) - 10368*x^6*exp(-2*x^2) + 11664*x^6*exp(-3*x^2) + 2592*x^7*exp(-2*x^2) - 6))/(24

*(18*x*exp(-x^2) - 1)^5) + (5*log(x + 9*exp(-x^2) - 2)*(x + 9*exp(-x^2) - 2)*(50544*exp(-2*x^2) - 1359*exp(-x^

2) - 8*x - 551124*exp(-3*x^2) + 2361960*exp(-4*x^2) - 3542940*exp(-5*x^2) - 4032*x*exp(-x^2) + 45360*x*exp(-2*

x^2) - 157464*x*exp(-3*x^2) + 157464*x*exp(-4*x^2) + 11952*x^2*exp(-x^2) - 250776*x^2*exp(-2*x^2) - 3168*x^3*e

xp(-x^2) + 1872072*x^2*exp(-3*x^2) - 2592*x^3*exp(-2*x^2) - 8280*x^4*exp(-x^2) - 5668704*x^2*exp(-4*x^2) + 233

28*x^3*exp(-3*x^2) + 562464*x^4*exp(-2*x^2) + 4896*x^5*exp(-x^2) + 5668704*x^2*exp(-5*x^2) - 3732480*x^4*exp(-

3*x^2) - 601344*x^5*exp(-2*x^2) + 576*x^6*exp(-x^2) + 8398080*x^4*exp(-4*x^2) + 2169504*x^5*exp(-3*x^2) + 9072

0*x^6*exp(-2*x^2) - 864*x^7*exp(-x^2) - 5668704*x^4*exp(-5*x^2) - 1259712*x^5*exp(-4*x^2) + 3639168*x^6*exp(-3

*x^2) + 207360*x^7*exp(-2*x^2) + 144*x^8*exp(-x^2) - 13436928*x^6*exp(-4*x^2) - 6158592*x^7*exp(-3*x^2) - 1244

16*x^8*exp(-2*x^2) + 7558272*x^6*exp(-5*x^2) + 16796160*x^7*exp(-4*x^2) + 4339008*x^8*exp(-3*x^2) + 20736*x^9*

exp(-2*x^2) - 7558272*x^7*exp(-5*x^2) - 10077696*x^8*exp(-4*x^2) - 1679616*x^9*exp(-3*x^2) + 3779136*x^8*exp(-

5*x^2) + 2519424*x^9*exp(-4*x^2) + 279936*x^10*exp(-3*x^2) + 14))/(24*(18*x*exp(-x^2) - 1)^7))/log(x + 9*exp(-

x^2) - 2) - ((5*(x + 9*exp(-x^2) - 2)*(2*x + 9*exp(-x^2) - 36*x^2*exp(-x^2) + 72*x^3*exp(-x^2) - 36*x^4*exp(-x

^2) - 2))/(12*(18*x*exp(-x^2) - 1)^3) - (5*log(x + 9*exp(-x^2) - 2)*(x + 9*exp(-x^2) - 2)*(4*x + 243*exp(-x^2)

- 1944*exp(-2*x^2) + 4374*exp(-3*x^2) - 990*x^2*exp(-x^2) + 5832*x^2*exp(-2*x^2) + 1080*x^3*exp(-x^2) - 8748*

x^2*exp(-3*x^2) - 1944*x^3*exp(-2*x^2) - 36*x^4*exp(-x^2) - 10368*x^4*exp(-2*x^2) - 288*x^5*exp(-x^2) + 17496*

x^4*exp(-3*x^2) + 15552*x^5*exp(-2*x^2) + 72*x^6*exp(-x^2) - 23328*x^5*exp(-3*x^2) - 10368*x^6*exp(-2*x^2) + 1

1664*x^6*exp(-3*x^2) + 2592*x^7*exp(-2*x^2) - 6))/(24*(18*x*exp(-x^2) - 1)^5))/log(x + 9*exp(-x^2) - 2)^2 - ((

5*x*(x + 9*exp(-x^2) - 2))/(4*(18*x*exp(-x^2) - 1)) - (5*log(x + 9*exp(-x^2) - 2)*(x + 9*exp(-x^2) - 2)*(2*x +

9*exp(-x^2) - 36*x^2*exp(-x^2) + 72*x^3*exp(-x^2) - 36*x^4*exp(-x^2) - 2))/(12*(18*x*exp(-x^2) - 1)^3))/log(x

+ 9*exp(-x^2) - 2)^3 + (5*(756*x^2 - 1200*x + 4944*x^3 - 6612*x^4 - 7936*x^5 + 16320*x^6 + 7296*x^7 - 32336*x

^8 + 16896*x^9 + 20288*x^10 - 35584*x^11 + 24640*x^12 - 9216*x^13 + 1536*x^14 + 225))/(384*x^6*(2*x^2 - 1)*(54

*x*exp(-x^2) - 972*x^2*exp(-2*x^2) + 5832*x^3*exp(-3*x^2) - 1)) + (5*(8226*x^2 - 2400*x - 3312*x^3 - 39212*x^4

+ 68320*x^5 + 43608*x^6 - 209792*x^7 + 102224*x^8 + 259840*x^9 - 384032*x^10 + 78080*x^11 + 267968*x^12 - 324

096*x^13 + 180864*x^14 - 53248*x^15 + 6656*x^16 + 225))/(384*x^6*(2*x^2 - 1)*(90*x*exp(-x^2) - 3240*x^2*exp(-2

*x^2) + 58320*x^3*exp(-3*x^2) - 524880*x^4*exp(-4*x^2) + 1889568*x^5*exp(-5*x^2) - 1)) + (5*(72*x^4 - 54*x^2 -

80*x^6 + 32*x^7 + 48*x^8 - 64*x^9 + 32*x^10 + 15))/(384*x^6*(2*x^2 - 1)*(18*x*exp(-x^2) - 1)) + (5*(1251*x^2

- 600*x + 1356*x^3 - 7240*x^4 + 2904*x^5 + 15520*x^6 - 16640*x^7 - 16696*x^8 + 41216*x^9 - 17552*x^10 - 26048*

x^11 + 42560*x^12 - 29568*x^13 + 12160*x^14 - 3072*x^15 + 384*x^16 + 75))/(96*x^6*(2*x^2 - 1)*(1944*x^2*exp(-2

*x^2) - 72*x*exp(-x^2) - 23328*x^3*exp(-3*x^2) + 104976*x^4*exp(-4*x^2) + 1)) - (5*(120*x + 153*x^2 - 504*x^3

- 174*x^4 + 896*x^5 + 120*x^6 - 1408*x^7 + 736*x^8 + 896*x^9 - 1456*x^10 + 896*x^11 - 224*x^12 - 45))/(192*x^6

*(2*x^2 - 1)*(324*x^2*exp(-2*x^2) - 36*x*exp(-x^2) + 1)) + (25*(567*x^2 - 120*x - 816*x^3 - 1930*x^4 + 7648*x^

5 - 3892*x^6 - 17472*x^7 + 27864*x^8 + 3968*x^9 - 44208*x^10 + 35584*x^11 + 8736*x^12 - 32256*x^13 + 22592*x^1

4 - 7168*x^15 + 896*x^16 + 9))/(192*x^6*(2*x^2 - 1)*(4860*x^2*exp(-2*x^2) - 108*x*exp(-x^2) - 116640*x^3*exp(-

3*x^2) + 1574640*x^4*exp(-4*x^2) - 11337408*x^5*exp(-5*x^2) + 34012224*x^6*exp(-6*x^2) + 1)) - (25*(16*x - 96*

x^2 + 224*x^3 + 144*x^4 - 1728*x^5 + 2432*x^6 + 2432*x^7 - 9312*x^8 + 4864*x^9 + 9728*x^10 - 13824*x^11 + 2304

*x^12 + 7168*x^13 - 6144*x^14 + 2048*x^15 - 256*x^16 - 1))/(128*x^6*(2*x^2 - 1)*(126*x*exp(-x^2) - 6804*x^2*ex

p(-2*x^2) + 204120*x^3*exp(-3*x^2) - 3674160*x^4*exp(-4*x^2) + 39680928*x^5*exp(-5*x^2) - 238085568*x^6*exp(-6

*x^2) + 612220032*x^7*exp(-7*x^2) - 1))

________________________________________________________________________________________

sympy [A] time = 0.38, size = 19, normalized size = 0.86 \begin {gather*} \frac {5 x^{2}}{2 \log {\left (x - 2 + 9 e^{- x^{2}} \right )}^{4}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x*exp(2*ln(3)-x**2)+5*x**2-10*x)*ln(exp(2*ln(3)-x**2)+x-2)+20*x**3*exp(2*ln(3)-x**2)-10*x**2)/(e

xp(2*ln(3)-x**2)+x-2)/ln(exp(2*ln(3)-x**2)+x-2)**5,x)

[Out]

5*x**2/(2*log(x - 2 + 9*exp(-x**2))**4)

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]