3.28.84 \(\int \frac {e^{10} (1+4 x^2)+e^{5+x^2} (2 x^2+4 x^3-4 x^4)+e^{2 x^2} (-x^2+4 x^3-3 x^4)+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} (2 x^2-2 x^3)+e^{2 x^2} (x+x^3-2 x^4+x^5)+e^{2 x^2} x \log (x)} \, dx\)

Optimal. Leaf size=29 \[ \log \left (\frac {x}{1+\left (-e^{5-x^2}-x+x^2\right )^2+\log (x)}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 54.36, 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^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^10*(1 + 4*x^2) + E^(5 + x^2)*(2*x^2 + 4*x^3 - 4*x^4) + E^(2*x^2)*(-x^2 + 4*x^3 - 3*x^4) + E^(2*x^2)*Log
[x])/(E^10*x + E^(5 + x^2)*(2*x^2 - 2*x^3) + E^(2*x^2)*(x + x^3 - 2*x^4 + x^5) + E^(2*x^2)*x*Log[x]),x]

[Out]

Log[x] - Log[1 + x^2 - 2*x^3 + x^4 + Log[x]] + E^10*Defer[Int][1/(x*(1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2
*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4) + E^(2*x^2)*Log[x])), x] + 6*E^10*Defer[Int][x/((1
+ x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4) + E^(2*x^2
)*Log[x])), x] + 2*E^5*Defer[Int][(E^x^2*x)/((1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2*E^(5 + x^2)*(-1 + x)*x
+ E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4) + E^(2*x^2)*Log[x])), x] - 6*E^10*Defer[Int][x^2/((1 + x^2 - 2*x^3 + x^4
+ Log[x])*(E^10 - 2*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4) + E^(2*x^2)*Log[x])), x] + 6*E^
5*Defer[Int][(E^x^2*x^2)/((1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x
^2 - 2*x^3 + x^4) + E^(2*x^2)*Log[x])), x] + 8*E^10*Defer[Int][x^3/((1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2
*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4) + E^(2*x^2)*Log[x])), x] - 12*E^5*Defer[Int][(E^x^
2*x^3)/((1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4)
+ E^(2*x^2)*Log[x])), x] - 8*E^10*Defer[Int][x^4/((1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2*E^(5 + x^2)*(-1 +
x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4) + E^(2*x^2)*Log[x])), x] + 14*E^5*Defer[Int][(E^x^2*x^4)/((1 + x^2 -
2*x^3 + x^4 + Log[x])*(E^10 - 2*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4) + E^(2*x^2)*Log[x]
)), x] + 4*E^10*Defer[Int][x^5/((1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*
(1 + x^2 - 2*x^3 + x^4) + E^(2*x^2)*Log[x])), x] - 16*E^5*Defer[Int][(E^x^2*x^5)/((1 + x^2 - 2*x^3 + x^4 + Log
[x])*(E^10 - 2*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4) + E^(2*x^2)*Log[x])), x] + 12*E^5*De
fer[Int][(E^x^2*x^6)/((1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 -
2*x^3 + x^4) + E^(2*x^2)*Log[x])), x] - 4*E^5*Defer[Int][(E^x^2*x^7)/((1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10
- 2*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4) + E^(2*x^2)*Log[x])), x] - 2*E^5*Defer[Int][(E^
x^2*Log[x])/((1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 +
x^4) + E^(2*x^2)*Log[x])), x] + 4*E^10*Defer[Int][(x*Log[x])/((1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2*E^(5
+ x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4) + E^(2*x^2)*Log[x])), x] + 4*E^5*Defer[Int][(E^x^2*x*Log
[x])/((1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4) +
E^(2*x^2)*Log[x])), x] + 4*E^5*Defer[Int][(E^x^2*x^2*Log[x])/((1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2*E^(5
+ x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4) + E^(2*x^2)*Log[x])), x] - 4*E^5*Defer[Int][(E^x^2*x^3*L
og[x])/((1 + x^2 - 2*x^3 + x^4 + Log[x])*(E^10 - 2*E^(5 + x^2)*(-1 + x)*x + E^(2*x^2)*(1 + x^2 - 2*x^3 + x^4)
+ E^(2*x^2)*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-x^2+4 x^3-3 x^4+\log (x)}{x \left (1+x^2-2 x^3+x^4+\log (x)\right )}+\frac {e^5 \left (e^5+6 e^5 x^2+2 e^{x^2} x^2-6 e^5 x^3+6 e^{x^2} x^3+8 e^5 x^4-12 e^{x^2} x^4-8 e^5 x^5+14 e^{x^2} x^5+4 e^5 x^6-16 e^{x^2} x^6+12 e^{x^2} x^7-4 e^{x^2} x^8-2 e^{x^2} x \log (x)+4 e^5 x^2 \log (x)+4 e^{x^2} x^2 \log (x)+4 e^{x^2} x^3 \log (x)-4 e^{x^2} x^4 \log (x)\right )}{x \left (1+x^2-2 x^3+x^4+\log (x)\right ) \left (e^{10}+e^{2 x^2}+2 e^{5+x^2} x+e^{2 x^2} x^2-2 e^{5+x^2} x^2-2 e^{2 x^2} x^3+e^{2 x^2} x^4+e^{2 x^2} \log (x)\right )}\right ) \, dx\\ &=e^5 \int \frac {e^5+6 e^5 x^2+2 e^{x^2} x^2-6 e^5 x^3+6 e^{x^2} x^3+8 e^5 x^4-12 e^{x^2} x^4-8 e^5 x^5+14 e^{x^2} x^5+4 e^5 x^6-16 e^{x^2} x^6+12 e^{x^2} x^7-4 e^{x^2} x^8-2 e^{x^2} x \log (x)+4 e^5 x^2 \log (x)+4 e^{x^2} x^2 \log (x)+4 e^{x^2} x^3 \log (x)-4 e^{x^2} x^4 \log (x)}{x \left (1+x^2-2 x^3+x^4+\log (x)\right ) \left (e^{10}+e^{2 x^2}+2 e^{5+x^2} x+e^{2 x^2} x^2-2 e^{5+x^2} x^2-2 e^{2 x^2} x^3+e^{2 x^2} x^4+e^{2 x^2} \log (x)\right )} \, dx+\int \frac {-x^2+4 x^3-3 x^4+\log (x)}{x \left (1+x^2-2 x^3+x^4+\log (x)\right )} \, dx\\ &=e^5 \int \frac {-2 e^{x^2} x^2 \left (-1-3 x+6 x^2-7 x^3+8 x^4-6 x^5+2 x^6\right )+e^5 \left (1+6 x^2-6 x^3+8 x^4-8 x^5+4 x^6\right )-2 x \left (-2 e^5 x+e^{x^2} \left (1-2 x-2 x^2+2 x^3\right )\right ) \log (x)}{x \left (1+x^2-2 x^3+x^4+\log (x)\right ) \left (e^{10}-2 e^{5+x^2} (-1+x) x+e^{2 x^2} \left (1+x^2-2 x^3+x^4\right )+e^{2 x^2} \log (x)\right )} \, dx+\int \left (\frac {1}{x}+\frac {-1-2 x^2+6 x^3-4 x^4}{x \left (1+x^2-2 x^3+x^4+\log (x)\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.16, size = 88, normalized size = 3.03 \begin {gather*} 2 x^2+\log (x)-\log \left (e^{10}+e^{2 x^2}+2 e^{5+x^2} x+e^{2 x^2} x^2-2 e^{5+x^2} x^2-2 e^{2 x^2} x^3+e^{2 x^2} x^4+e^{2 x^2} \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^10*(1 + 4*x^2) + E^(5 + x^2)*(2*x^2 + 4*x^3 - 4*x^4) + E^(2*x^2)*(-x^2 + 4*x^3 - 3*x^4) + E^(2*x^
2)*Log[x])/(E^10*x + E^(5 + x^2)*(2*x^2 - 2*x^3) + E^(2*x^2)*(x + x^3 - 2*x^4 + x^5) + E^(2*x^2)*x*Log[x]),x]

[Out]

2*x^2 + Log[x] - Log[E^10 + E^(2*x^2) + 2*E^(5 + x^2)*x + E^(2*x^2)*x^2 - 2*E^(5 + x^2)*x^2 - 2*E^(2*x^2)*x^3
+ E^(2*x^2)*x^4 + E^(2*x^2)*Log[x]]

________________________________________________________________________________________

fricas [B]  time = 0.52, size = 66, normalized size = 2.28 \begin {gather*} -\log \left ({\left ({\left (x^{4} - 2 \, x^{3} + x^{2} + 1\right )} e^{\left (2 \, x^{2} + 10\right )} - 2 \, {\left (x^{2} - x\right )} e^{\left (x^{2} + 15\right )} + e^{\left (2 \, x^{2} + 10\right )} \log \relax (x) + e^{20}\right )} e^{\left (-2 \, x^{2} - 10\right )}\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x^2)^2*log(x)+(-3*x^4+4*x^3-x^2)*exp(x^2)^2+(-4*x^4+4*x^3+2*x^2)*exp(5)*exp(x^2)+(4*x^2+1)*exp(
5)^2)/(x*exp(x^2)^2*log(x)+(x^5-2*x^4+x^3+x)*exp(x^2)^2+(-2*x^3+2*x^2)*exp(5)*exp(x^2)+x*exp(5)^2),x, algorith
m="fricas")

[Out]

-log(((x^4 - 2*x^3 + x^2 + 1)*e^(2*x^2 + 10) - 2*(x^2 - x)*e^(x^2 + 15) + e^(2*x^2 + 10)*log(x) + e^20)*e^(-2*
x^2 - 10)) + log(x)

________________________________________________________________________________________

giac [B]  time = 1.00, size = 120, normalized size = 4.14 \begin {gather*} 2 \, x^{2} - \log \left (x^{4} e^{\left (2 \, x^{2}\right )} - 2 \, x^{3} e^{\left (2 \, x^{2}\right )} + x^{2} e^{\left (2 \, x^{2}\right )} - 2 \, x^{2} e^{\left (x^{2} + 5\right )} + 2 \, x e^{\left (x^{2} + 5\right )} + e^{\left (2 \, x^{2}\right )} \log \relax (x) + e^{10} + e^{\left (2 \, x^{2}\right )}\right ) + \log \left (x^{4} - 2 \, x^{3} + x^{2} + \log \relax (x) + 1\right ) - \log \left (-x^{4} + 2 \, x^{3} - x^{2} - \log \relax (x) - 1\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x^2)^2*log(x)+(-3*x^4+4*x^3-x^2)*exp(x^2)^2+(-4*x^4+4*x^3+2*x^2)*exp(5)*exp(x^2)+(4*x^2+1)*exp(
5)^2)/(x*exp(x^2)^2*log(x)+(x^5-2*x^4+x^3+x)*exp(x^2)^2+(-2*x^3+2*x^2)*exp(5)*exp(x^2)+x*exp(5)^2),x, algorith
m="giac")

[Out]

2*x^2 - log(x^4*e^(2*x^2) - 2*x^3*e^(2*x^2) + x^2*e^(2*x^2) - 2*x^2*e^(x^2 + 5) + 2*x*e^(x^2 + 5) + e^(2*x^2)*
log(x) + e^10 + e^(2*x^2)) + log(x^4 - 2*x^3 + x^2 + log(x) + 1) - log(-x^4 + 2*x^3 - x^2 - log(x) - 1) + log(
x)

________________________________________________________________________________________

maple [B]  time = 0.04, size = 77, normalized size = 2.66

method	result	size
risch	\(\ln \relax (x )-\ln \left (\ln \relax (x )+\left (x^{4} {\mathrm e}^{2 x^{2}}-2 \,{\mathrm e}^{2 x^{2}} x^{3}-2 \,{\mathrm e}^{x^{2}+5} x^{2}+{\mathrm e}^{2 x^{2}} x^{2}+2 \,{\mathrm e}^{x^{2}+5} x +{\mathrm e}^{10}+{\mathrm e}^{2 x^{2}}\right ) {\mathrm e}^{-2 x^{2}}\right )\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2)^2*ln(x)+(-3*x^4+4*x^3-x^2)*exp(x^2)^2+(-4*x^4+4*x^3+2*x^2)*exp(5)*exp(x^2)+(4*x^2+1)*exp(5)^2)/(
x*exp(x^2)^2*ln(x)+(x^5-2*x^4+x^3+x)*exp(x^2)^2+(-2*x^3+2*x^2)*exp(5)*exp(x^2)+x*exp(5)^2),x,method=_RETURNVER
BOSE)

[Out]

ln(x)-ln(ln(x)+(x^4*exp(2*x^2)-2*exp(2*x^2)*x^3-2*exp(x^2+5)*x^2+exp(2*x^2)*x^2+2*exp(x^2+5)*x+exp(10)+exp(2*x
^2))*exp(-2*x^2))

________________________________________________________________________________________

maxima [B]  time = 0.72, size = 90, normalized size = 3.10 \begin {gather*} 2 \, x^{2} - \log \left (x^{4} - 2 \, x^{3} + x^{2} + \log \relax (x) + 1\right ) + \log \relax (x) - \log \left (\frac {{\left (x^{4} - 2 \, x^{3} + x^{2} + \log \relax (x) + 1\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} e^{5} - x e^{5}\right )} e^{\left (x^{2}\right )} + e^{10}}{x^{4} - 2 \, x^{3} + x^{2} + \log \relax (x) + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x^2)^2*log(x)+(-3*x^4+4*x^3-x^2)*exp(x^2)^2+(-4*x^4+4*x^3+2*x^2)*exp(5)*exp(x^2)+(4*x^2+1)*exp(
5)^2)/(x*exp(x^2)^2*log(x)+(x^5-2*x^4+x^3+x)*exp(x^2)^2+(-2*x^3+2*x^2)*exp(5)*exp(x^2)+x*exp(5)^2),x, algorith
m="maxima")

[Out]

2*x^2 - log(x^4 - 2*x^3 + x^2 + log(x) + 1) + log(x) - log(((x^4 - 2*x^3 + x^2 + log(x) + 1)*e^(2*x^2) - 2*(x^
2*e^5 - x*e^5)*e^(x^2) + e^10)/(x^4 - 2*x^3 + x^2 + log(x) + 1))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{10}\,\left (4\,x^2+1\right )-{\mathrm {e}}^{2\,x^2}\,\left (3\,x^4-4\,x^3+x^2\right )+{\mathrm {e}}^{2\,x^2}\,\ln \relax (x)+{\mathrm {e}}^{x^2+5}\,\left (-4\,x^4+4\,x^3+2\,x^2\right )}{x\,{\mathrm {e}}^{10}+{\mathrm {e}}^{2\,x^2}\,\left (x^5-2\,x^4+x^3+x\right )+{\mathrm {e}}^{x^2+5}\,\left (2\,x^2-2\,x^3\right )+x\,{\mathrm {e}}^{2\,x^2}\,\ln \relax (x)} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(10)*(4*x^2 + 1) - exp(2*x^2)*(x^2 - 4*x^3 + 3*x^4) + exp(2*x^2)*log(x) + exp(x^2)*exp(5)*(2*x^2 + 4*x
^3 - 4*x^4))/(x*exp(10) + exp(2*x^2)*(x + x^3 - 2*x^4 + x^5) + x*exp(2*x^2)*log(x) + exp(x^2)*exp(5)*(2*x^2 -
2*x^3)),x)

[Out]

int((exp(10)*(4*x^2 + 1) - exp(2*x^2)*(x^2 - 4*x^3 + 3*x^4) + exp(2*x^2)*log(x) + exp(x^2 + 5)*(2*x^2 + 4*x^3
- 4*x^4))/(x*exp(10) + exp(2*x^2)*(x + x^3 - 2*x^4 + x^5) + exp(x^2 + 5)*(2*x^2 - 2*x^3) + x*exp(2*x^2)*log(x)
), x)

________________________________________________________________________________________

sympy [B]  time = 9.63, size = 92, normalized size = 3.17 \begin {gather*} 2 x^{2} + \log {\relax (x )} - \log {\left (\frac {\left (- 2 x^{2} e^{5} + 2 x e^{5}\right ) e^{x^{2}}}{x^{4} - 2 x^{3} + x^{2} + \log {\relax (x )} + 1} + e^{2 x^{2}} + \frac {e^{10}}{x^{4} - 2 x^{3} + x^{2} + \log {\relax (x )} + 1} \right )} - \log {\left (x^{4} - 2 x^{3} + x^{2} + \log {\relax (x )} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x**2)**2*ln(x)+(-3*x**4+4*x**3-x**2)*exp(x**2)**2+(-4*x**4+4*x**3+2*x**2)*exp(5)*exp(x**2)+(4*x
**2+1)*exp(5)**2)/(x*exp(x**2)**2*ln(x)+(x**5-2*x**4+x**3+x)*exp(x**2)**2+(-2*x**3+2*x**2)*exp(5)*exp(x**2)+x*
exp(5)**2),x)

[Out]

2*x**2 + log(x) - log((-2*x**2*exp(5) + 2*x*exp(5))*exp(x**2)/(x**4 - 2*x**3 + x**2 + log(x) + 1) + exp(2*x**2
) + exp(10)/(x**4 - 2*x**3 + x**2 + log(x) + 1)) - log(x**4 - 2*x**3 + x**2 + log(x) + 1)

________________________________________________________________________________________