\(\int \frac {71+e^{18+12 x+2 x^2}+e^{9+6 x+x^2} (17-6 x-2 x^2)+(17+2 e^{9+6 x+x^2}) \log (x)+\log ^2(x)}{64+16 e^{9+6 x+x^2}+e^{18+12 x+2 x^2}+(16+2 e^{9+6 x+x^2}) \log (x)+\log ^2(x)} \, dx\) [10225]

Optimal result

Integrand size = 106, antiderivative size = 17 \[ \int \frac {71+e^{18+12 x+2 x^2}+e^{9+6 x+x^2} \left (17-6 x-2 x^2\right )+\left (17+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)}{64+16 e^{9+6 x+x^2}+e^{18+12 x+2 x^2}+\left (16+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)} \, dx=x+\frac {x}{8+e^{(3+x)^2}+\log (x)} \]

[Out]

x+x/(ln(x)+8+exp((3+x)^2))

Rubi [F]

\[ \int \frac {71+e^{18+12 x+2 x^2}+e^{9+6 x+x^2} \left (17-6 x-2 x^2\right )+\left (17+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)}{64+16 e^{9+6 x+x^2}+e^{18+12 x+2 x^2}+\left (16+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)} \, dx=\int \frac {71+e^{18+12 x+2 x^2}+e^{9+6 x+x^2} \left (17-6 x-2 x^2\right )+\left (17+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)}{64+16 e^{9+6 x+x^2}+e^{18+12 x+2 x^2}+\left (16+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)} \, dx \]

[In]

Int[(71 + E^(18 + 12*x + 2*x^2) + E^(9 + 6*x + x^2)*(17 - 6*x - 2*x^2) + (17 + 2*E^(9 + 6*x + x^2))*Log[x] + L
og[x]^2)/(64 + 16*E^(9 + 6*x + x^2) + E^(18 + 12*x + 2*x^2) + (16 + 2*E^(9 + 6*x + x^2))*Log[x] + Log[x]^2),x]

[Out]

x - Defer[Int][(8 + E^(3 + x)^2 + Log[x])^(-2), x] + 48*Defer[Int][x/(8 + E^(3 + x)^2 + Log[x])^2, x] + 16*Def
er[Int][x^2/(8 + E^(3 + x)^2 + Log[x])^2, x] + 6*Defer[Int][(x*Log[x])/(8 + E^(3 + x)^2 + Log[x])^2, x] + 2*De
fer[Int][(x^2*Log[x])/(8 + E^(3 + x)^2 + Log[x])^2, x] + Defer[Int][(8 + E^(3 + x)^2 + Log[x])^(-1), x] - 6*De
fer[Int][x/(8 + E^(3 + x)^2 + Log[x]), x] - 2*Defer[Int][x^2/(8 + E^(3 + x)^2 + Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {71+e^{2 (3+x)^2}+e^{(3+x)^2} \left (17-6 x-2 x^2\right )+\left (17+2 e^{(3+x)^2}\right ) \log (x)+\log ^2(x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx \\ & = \int \left (1-\frac {-1+6 x+2 x^2}{8+e^{(3+x)^2}+\log (x)}+\frac {-1+48 x+16 x^2+6 x \log (x)+2 x^2 \log (x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2}\right ) \, dx \\ & = x-\int \frac {-1+6 x+2 x^2}{8+e^{(3+x)^2}+\log (x)} \, dx+\int \frac {-1+48 x+16 x^2+6 x \log (x)+2 x^2 \log (x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx \\ & = x+\int \left (-\frac {1}{\left (8+e^{(3+x)^2}+\log (x)\right )^2}+\frac {48 x}{\left (8+e^{(3+x)^2}+\log (x)\right )^2}+\frac {16 x^2}{\left (8+e^{(3+x)^2}+\log (x)\right )^2}+\frac {6 x \log (x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2}+\frac {2 x^2 \log (x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2}\right ) \, dx-\int \left (-\frac {1}{8+e^{(3+x)^2}+\log (x)}+\frac {6 x}{8+e^{(3+x)^2}+\log (x)}+\frac {2 x^2}{8+e^{(3+x)^2}+\log (x)}\right ) \, dx \\ & = x+2 \int \frac {x^2 \log (x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx-2 \int \frac {x^2}{8+e^{(3+x)^2}+\log (x)} \, dx+6 \int \frac {x \log (x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx-6 \int \frac {x}{8+e^{(3+x)^2}+\log (x)} \, dx+16 \int \frac {x^2}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx+48 \int \frac {x}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx-\int \frac {1}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx+\int \frac {1}{8+e^{(3+x)^2}+\log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {71+e^{18+12 x+2 x^2}+e^{9+6 x+x^2} \left (17-6 x-2 x^2\right )+\left (17+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)}{64+16 e^{9+6 x+x^2}+e^{18+12 x+2 x^2}+\left (16+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)} \, dx=x \left (1+\frac {1}{8+e^{(3+x)^2}+\log (x)}\right ) \]

[In]

Integrate[(71 + E^(18 + 12*x + 2*x^2) + E^(9 + 6*x + x^2)*(17 - 6*x - 2*x^2) + (17 + 2*E^(9 + 6*x + x^2))*Log[
x] + Log[x]^2)/(64 + 16*E^(9 + 6*x + x^2) + E^(18 + 12*x + 2*x^2) + (16 + 2*E^(9 + 6*x + x^2))*Log[x] + Log[x]
^2),x]

[Out]

x*(1 + (8 + E^(3 + x)^2 + Log[x])^(-1))

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00

method	result	size
risch	\(x +\frac {x}{\ln \left (x \right )+8+{\mathrm e}^{\left (3+x \right )^{2}}}\)	\(17\)
parallelrisch	\(\frac {x \ln \left (x \right )+{\mathrm e}^{x^{2}+6 x +9} x +9 x}{8+\ln \left (x \right )+{\mathrm e}^{x^{2}+6 x +9}}\)	\(36\)

[In]

int((ln(x)^2+(2*exp(x^2+6*x+9)+17)*ln(x)+exp(x^2+6*x+9)^2+(-2*x^2-6*x+17)*exp(x^2+6*x+9)+71)/(ln(x)^2+(2*exp(x
^2+6*x+9)+16)*ln(x)+exp(x^2+6*x+9)^2+16*exp(x^2+6*x+9)+64),x,method=_RETURNVERBOSE)

[Out]

x+x/(ln(x)+8+exp((3+x)^2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (16) = 32\).

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.06 \[ \int \frac {71+e^{18+12 x+2 x^2}+e^{9+6 x+x^2} \left (17-6 x-2 x^2\right )+\left (17+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)}{64+16 e^{9+6 x+x^2}+e^{18+12 x+2 x^2}+\left (16+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)} \, dx=\frac {x e^{\left (x^{2} + 6 \, x + 9\right )} + x \log \left (x\right ) + 9 \, x}{e^{\left (x^{2} + 6 \, x + 9\right )} + \log \left (x\right ) + 8} \]

[In]

integrate((log(x)^2+(2*exp(x^2+6*x+9)+17)*log(x)+exp(x^2+6*x+9)^2+(-2*x^2-6*x+17)*exp(x^2+6*x+9)+71)/(log(x)^2
+(2*exp(x^2+6*x+9)+16)*log(x)+exp(x^2+6*x+9)^2+16*exp(x^2+6*x+9)+64),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {71+e^{18+12 x+2 x^2}+e^{9+6 x+x^2} \left (17-6 x-2 x^2\right )+\left (17+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)}{64+16 e^{9+6 x+x^2}+e^{18+12 x+2 x^2}+\left (16+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)} \, dx=x + \frac {x}{e^{x^{2} + 6 x + 9} + \log {\left (x \right )} + 8} \]

[In]

integrate((ln(x)**2+(2*exp(x**2+6*x+9)+17)*ln(x)+exp(x**2+6*x+9)**2+(-2*x**2-6*x+17)*exp(x**2+6*x+9)+71)/(ln(x
)**2+(2*exp(x**2+6*x+9)+16)*ln(x)+exp(x**2+6*x+9)**2+16*exp(x**2+6*x+9)+64),x)

[Out]

x + x/(exp(x**2 + 6*x + 9) + log(x) + 8)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (16) = 32\).

Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.06 \[ \int \frac {71+e^{18+12 x+2 x^2}+e^{9+6 x+x^2} \left (17-6 x-2 x^2\right )+\left (17+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)}{64+16 e^{9+6 x+x^2}+e^{18+12 x+2 x^2}+\left (16+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)} \, dx=\frac {x e^{\left (x^{2} + 6 \, x + 9\right )} + x \log \left (x\right ) + 9 \, x}{e^{\left (x^{2} + 6 \, x + 9\right )} + \log \left (x\right ) + 8} \]

[In]

integrate((log(x)^2+(2*exp(x^2+6*x+9)+17)*log(x)+exp(x^2+6*x+9)^2+(-2*x^2-6*x+17)*exp(x^2+6*x+9)+71)/(log(x)^2
+(2*exp(x^2+6*x+9)+16)*log(x)+exp(x^2+6*x+9)^2+16*exp(x^2+6*x+9)+64),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (16) = 32\).

Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.06 \[ \int \frac {71+e^{18+12 x+2 x^2}+e^{9+6 x+x^2} \left (17-6 x-2 x^2\right )+\left (17+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)}{64+16 e^{9+6 x+x^2}+e^{18+12 x+2 x^2}+\left (16+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)} \, dx=\frac {x e^{\left (x^{2} + 6 \, x + 9\right )} + x \log \left (x\right ) + 9 \, x}{e^{\left (x^{2} + 6 \, x + 9\right )} + \log \left (x\right ) + 8} \]

[In]

integrate((log(x)^2+(2*exp(x^2+6*x+9)+17)*log(x)+exp(x^2+6*x+9)^2+(-2*x^2-6*x+17)*exp(x^2+6*x+9)+71)/(log(x)^2
+(2*exp(x^2+6*x+9)+16)*log(x)+exp(x^2+6*x+9)^2+16*exp(x^2+6*x+9)+64),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 16.99 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76 \[ \int \frac {71+e^{18+12 x+2 x^2}+e^{9+6 x+x^2} \left (17-6 x-2 x^2\right )+\left (17+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)}{64+16 e^{9+6 x+x^2}+e^{18+12 x+2 x^2}+\left (16+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)} \, dx=\frac {x\,\left ({\mathrm {e}}^{x^2+6\,x+9}+\ln \left (x\right )+9\right )}{{\mathrm {e}}^{x^2+6\,x+9}+\ln \left (x\right )+8} \]

[In]

int((exp(12*x + 2*x^2 + 18) - exp(6*x + x^2 + 9)*(6*x + 2*x^2 - 17) + log(x)^2 + log(x)*(2*exp(6*x + x^2 + 9)
+ 17) + 71)/(16*exp(6*x + x^2 + 9) + exp(12*x + 2*x^2 + 18) + log(x)^2 + log(x)*(2*exp(6*x + x^2 + 9) + 16) +
64),x)

[Out]

(x*(exp(6*x + x^2 + 9) + log(x) + 9))/(exp(6*x + x^2 + 9) + log(x) + 8)