\(\int \frac {(6 x+2 x^2) \log (x)+(-12-10 x-2 x^2+(4 x+2 x^2) \log (x)) \log (2+x)+(-18 x-21 x^2-8 x^3-x^4+e^x (36 x+42 x^2+16 x^3+2 x^4)) \log ^2(2+x)}{(36 x+42 x^2+16 x^3+2 x^4) \log ^2(2+x)} \, dx\) [8756]

Optimal result

Integrand size = 117, antiderivative size = 25 \[ \int \frac {\left (6 x+2 x^2\right ) \log (x)+\left (-12-10 x-2 x^2+\left (4 x+2 x^2\right ) \log (x)\right ) \log (2+x)+\left (-18 x-21 x^2-8 x^3-x^4+e^x \left (36 x+42 x^2+16 x^3+2 x^4\right )\right ) \log ^2(2+x)}{\left (36 x+42 x^2+16 x^3+2 x^4\right ) \log ^2(2+x)} \, dx=-4+e^x-\frac {x}{2}-\frac {\log (x)}{(3+x) \log (2+x)} \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (6 x+2 x^2\right ) \log (x)+\left (-12-10 x-2 x^2+\left (4 x+2 x^2\right ) \log (x)\right ) \log (2+x)+\left (-18 x-21 x^2-8 x^3-x^4+e^x \left (36 x+42 x^2+16 x^3+2 x^4\right )\right ) \log ^2(2+x)}{\left (36 x+42 x^2+16 x^3+2 x^4\right ) \log ^2(2+x)} \, dx=\int \frac {\left (6 x+2 x^2\right ) \log (x)+\left (-12-10 x-2 x^2+\left (4 x+2 x^2\right ) \log (x)\right ) \log (2+x)+\left (-18 x-21 x^2-8 x^3-x^4+e^x \left (36 x+42 x^2+16 x^3+2 x^4\right )\right ) \log ^2(2+x)}{\left (36 x+42 x^2+16 x^3+2 x^4\right ) \log ^2(2+x)} \, dx \]

[In]

Int[((6*x + 2*x^2)*Log[x] + (-12 - 10*x - 2*x^2 + (4*x + 2*x^2)*Log[x])*Log[2 + x] + (-18*x - 21*x^2 - 8*x^3 -
x^4 + E^x*(36*x + 42*x^2 + 16*x^3 + 2*x^4))*Log[2 + x]^2)/((36*x + 42*x^2 + 16*x^3 + 2*x^4)*Log[2 + x]^2),x]

[Out]

E^x - x/2 - Defer[Int][Log[x]/((3 + x)*Log[2 + x]^2), x] - Defer[Int][1/(x*Log[2 + x]), x]/3 + Defer[Int][1/((
3 + x)*Log[2 + x]), x]/3 + Defer[Int][Log[x]/((3 + x)^2*Log[2 + x]), x] + Defer[Subst][Defer[Int][Log[-2 + x]/
(x*Log[x]^2), x], x, 2 + x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x \log (x) (3+x+(2+x) \log (2+x))+\left (6+5 x+x^2\right ) \log (2+x) \left (-2+\left (-1+2 e^x\right ) x (3+x) \log (2+x)\right )}{2 x (2+x) (3+x)^2 \log ^2(2+x)} \, dx \\ & = \frac {1}{2} \int \frac {2 x \log (x) (3+x+(2+x) \log (2+x))+\left (6+5 x+x^2\right ) \log (2+x) \left (-2+\left (-1+2 e^x\right ) x (3+x) \log (2+x)\right )}{x (2+x) (3+x)^2 \log ^2(2+x)} \, dx \\ & = \frac {1}{2} \int \left (2 e^x+\frac {6 x \log (x)+2 x^2 \log (x)-12 \log (2+x)-10 x \log (2+x)-2 x^2 \log (2+x)+4 x \log (x) \log (2+x)+2 x^2 \log (x) \log (2+x)-18 x \log ^2(2+x)-21 x^2 \log ^2(2+x)-8 x^3 \log ^2(2+x)-x^4 \log ^2(2+x)}{x (2+x) (3+x)^2 \log ^2(2+x)}\right ) \, dx \\ & = \frac {1}{2} \int \frac {6 x \log (x)+2 x^2 \log (x)-12 \log (2+x)-10 x \log (2+x)-2 x^2 \log (2+x)+4 x \log (x) \log (2+x)+2 x^2 \log (x) \log (2+x)-18 x \log ^2(2+x)-21 x^2 \log ^2(2+x)-8 x^3 \log ^2(2+x)-x^4 \log ^2(2+x)}{x (2+x) (3+x)^2 \log ^2(2+x)} \, dx+\int e^x \, dx \\ & = e^x+\frac {1}{2} \int \frac {2 x \log (x) (3+x+(2+x) \log (2+x))-\left (6+5 x+x^2\right ) \log (2+x) (2+x (3+x) \log (2+x))}{x (2+x) (3+x)^2 \log ^2(2+x)} \, dx \\ & = e^x+\frac {1}{2} \int \left (-1+\frac {2 \log (x)}{(2+x) (3+x) \log ^2(2+x)}+\frac {2 (-3-x+x \log (x))}{x (3+x)^2 \log (2+x)}\right ) \, dx \\ & = e^x-\frac {x}{2}+\int \frac {\log (x)}{(2+x) (3+x) \log ^2(2+x)} \, dx+\int \frac {-3-x+x \log (x)}{x (3+x)^2 \log (2+x)} \, dx \\ & = e^x-\frac {x}{2}+\int \left (\frac {\log (x)}{(2+x) \log ^2(2+x)}-\frac {\log (x)}{(3+x) \log ^2(2+x)}\right ) \, dx+\int \left (\frac {3+x-x \log (x)}{3 (3+x)^2 \log (2+x)}+\frac {3+x-x \log (x)}{9 (3+x) \log (2+x)}+\frac {-3-x+x \log (x)}{9 x \log (2+x)}\right ) \, dx \\ & = e^x-\frac {x}{2}+\frac {1}{9} \int \frac {3+x-x \log (x)}{(3+x) \log (2+x)} \, dx+\frac {1}{9} \int \frac {-3-x+x \log (x)}{x \log (2+x)} \, dx+\frac {1}{3} \int \frac {3+x-x \log (x)}{(3+x)^2 \log (2+x)} \, dx+\int \frac {\log (x)}{(2+x) \log ^2(2+x)} \, dx-\int \frac {\log (x)}{(3+x) \log ^2(2+x)} \, dx \\ & = e^x-\frac {x}{2}+\frac {1}{9} \int \left (-\frac {1}{\log (2+x)}-\frac {3}{x \log (2+x)}+\frac {\log (x)}{\log (2+x)}\right ) \, dx+\frac {1}{9} \int \left (\frac {3}{(3+x) \log (2+x)}+\frac {x}{(3+x) \log (2+x)}-\frac {x \log (x)}{(3+x) \log (2+x)}\right ) \, dx+\frac {1}{3} \int \left (\frac {3}{(3+x)^2 \log (2+x)}+\frac {x}{(3+x)^2 \log (2+x)}-\frac {x \log (x)}{(3+x)^2 \log (2+x)}\right ) \, dx-\int \frac {\log (x)}{(3+x) \log ^2(2+x)} \, dx+\text {Subst}\left (\int \frac {\log (-2+x)}{x \log ^2(x)} \, dx,x,2+x\right ) \\ & = e^x-\frac {x}{2}-\frac {1}{9} \int \frac {1}{\log (2+x)} \, dx+\frac {1}{9} \int \frac {x}{(3+x) \log (2+x)} \, dx+\frac {1}{9} \int \frac {\log (x)}{\log (2+x)} \, dx-\frac {1}{9} \int \frac {x \log (x)}{(3+x) \log (2+x)} \, dx-\frac {1}{3} \int \frac {1}{x \log (2+x)} \, dx+\frac {1}{3} \int \frac {x}{(3+x)^2 \log (2+x)} \, dx+\frac {1}{3} \int \frac {1}{(3+x) \log (2+x)} \, dx-\frac {1}{3} \int \frac {x \log (x)}{(3+x)^2 \log (2+x)} \, dx-\int \frac {\log (x)}{(3+x) \log ^2(2+x)} \, dx+\int \frac {1}{(3+x)^2 \log (2+x)} \, dx+\text {Subst}\left (\int \frac {\log (-2+x)}{x \log ^2(x)} \, dx,x,2+x\right ) \\ & = e^x-\frac {x}{2}+\frac {1}{9} \int \left (\frac {1}{\log (2+x)}-\frac {3}{(3+x) \log (2+x)}\right ) \, dx-\frac {1}{9} \int \left (\frac {\log (x)}{\log (2+x)}-\frac {3 \log (x)}{(3+x) \log (2+x)}\right ) \, dx+\frac {1}{9} \int \frac {\log (x)}{\log (2+x)} \, dx-\frac {1}{9} \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,2+x\right )+\frac {1}{3} \int \left (-\frac {3}{(3+x)^2 \log (2+x)}+\frac {1}{(3+x) \log (2+x)}\right ) \, dx-\frac {1}{3} \int \left (-\frac {3 \log (x)}{(3+x)^2 \log (2+x)}+\frac {\log (x)}{(3+x) \log (2+x)}\right ) \, dx-\frac {1}{3} \int \frac {1}{x \log (2+x)} \, dx+\frac {1}{3} \int \frac {1}{(3+x) \log (2+x)} \, dx-\int \frac {\log (x)}{(3+x) \log ^2(2+x)} \, dx+\int \frac {1}{(3+x)^2 \log (2+x)} \, dx+\text {Subst}\left (\int \frac {\log (-2+x)}{x \log ^2(x)} \, dx,x,2+x\right ) \\ & = e^x-\frac {x}{2}-\frac {\operatorname {LogIntegral}(2+x)}{9}+\frac {1}{9} \int \frac {1}{\log (2+x)} \, dx-\frac {1}{3} \int \frac {1}{x \log (2+x)} \, dx+\frac {1}{3} \int \frac {1}{(3+x) \log (2+x)} \, dx-\int \frac {\log (x)}{(3+x) \log ^2(2+x)} \, dx+\int \frac {\log (x)}{(3+x)^2 \log (2+x)} \, dx+\text {Subst}\left (\int \frac {\log (-2+x)}{x \log ^2(x)} \, dx,x,2+x\right ) \\ & = e^x-\frac {x}{2}-\frac {\operatorname {LogIntegral}(2+x)}{9}+\frac {1}{9} \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,2+x\right )-\frac {1}{3} \int \frac {1}{x \log (2+x)} \, dx+\frac {1}{3} \int \frac {1}{(3+x) \log (2+x)} \, dx-\int \frac {\log (x)}{(3+x) \log ^2(2+x)} \, dx+\int \frac {\log (x)}{(3+x)^2 \log (2+x)} \, dx+\text {Subst}\left (\int \frac {\log (-2+x)}{x \log ^2(x)} \, dx,x,2+x\right ) \\ & = e^x-\frac {x}{2}-\frac {1}{3} \int \frac {1}{x \log (2+x)} \, dx+\frac {1}{3} \int \frac {1}{(3+x) \log (2+x)} \, dx-\int \frac {\log (x)}{(3+x) \log ^2(2+x)} \, dx+\int \frac {\log (x)}{(3+x)^2 \log (2+x)} \, dx+\text {Subst}\left (\int \frac {\log (-2+x)}{x \log ^2(x)} \, dx,x,2+x\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {\left (6 x+2 x^2\right ) \log (x)+\left (-12-10 x-2 x^2+\left (4 x+2 x^2\right ) \log (x)\right ) \log (2+x)+\left (-18 x-21 x^2-8 x^3-x^4+e^x \left (36 x+42 x^2+16 x^3+2 x^4\right )\right ) \log ^2(2+x)}{\left (36 x+42 x^2+16 x^3+2 x^4\right ) \log ^2(2+x)} \, dx=\frac {1}{2} \left (2 e^x-x-\frac {2 \log (x)}{(3+x) \log (2+x)}\right ) \]

[In]

Integrate[((6*x + 2*x^2)*Log[x] + (-12 - 10*x - 2*x^2 + (4*x + 2*x^2)*Log[x])*Log[2 + x] + (-18*x - 21*x^2 - 8
*x^3 - x^4 + E^x*(36*x + 42*x^2 + 16*x^3 + 2*x^4))*Log[2 + x]^2)/((36*x + 42*x^2 + 16*x^3 + 2*x^4)*Log[2 + x]^
2),x]

[Out]

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

Maple [A] (verified)

Time = 11.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88

method	result	size
risch	\(-\frac {x}{2}+{\mathrm e}^{x}-\frac {\ln \left (x \right )}{\ln \left (2+x \right ) \left (3+x \right )}\)	\(22\)
parallelrisch	\(-\frac {7 \ln \left (2+x \right ) x^{2}-14 x \,{\mathrm e}^{x} \ln \left (2+x \right )+5 x \ln \left (2+x \right )-42 \,{\mathrm e}^{x} \ln \left (2+x \right )+14 \ln \left (x \right )-48 \ln \left (2+x \right )}{14 \left (3+x \right ) \ln \left (2+x \right )}\)	\(58\)

[In]

int((((2*x^4+16*x^3+42*x^2+36*x)*exp(x)-x^4-8*x^3-21*x^2-18*x)*ln(2+x)^2+((2*x^2+4*x)*ln(x)-2*x^2-10*x-12)*ln(
2+x)+(2*x^2+6*x)*ln(x))/(2*x^4+16*x^3+42*x^2+36*x)/ln(2+x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/2*x+exp(x)-1/ln(2+x)*ln(x)/(3+x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {\left (6 x+2 x^2\right ) \log (x)+\left (-12-10 x-2 x^2+\left (4 x+2 x^2\right ) \log (x)\right ) \log (2+x)+\left (-18 x-21 x^2-8 x^3-x^4+e^x \left (36 x+42 x^2+16 x^3+2 x^4\right )\right ) \log ^2(2+x)}{\left (36 x+42 x^2+16 x^3+2 x^4\right ) \log ^2(2+x)} \, dx=-\frac {{\left (x^{2} - 2 \, {\left (x + 3\right )} e^{x} + 3 \, x\right )} \log \left (x + 2\right ) + 2 \, \log \left (x\right )}{2 \, {\left (x + 3\right )} \log \left (x + 2\right )} \]

[In]

integrate((((2*x^4+16*x^3+42*x^2+36*x)*exp(x)-x^4-8*x^3-21*x^2-18*x)*log(2+x)^2+((2*x^2+4*x)*log(x)-2*x^2-10*x
-12)*log(2+x)+(2*x^2+6*x)*log(x))/(2*x^4+16*x^3+42*x^2+36*x)/log(2+x)^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {\left (6 x+2 x^2\right ) \log (x)+\left (-12-10 x-2 x^2+\left (4 x+2 x^2\right ) \log (x)\right ) \log (2+x)+\left (-18 x-21 x^2-8 x^3-x^4+e^x \left (36 x+42 x^2+16 x^3+2 x^4\right )\right ) \log ^2(2+x)}{\left (36 x+42 x^2+16 x^3+2 x^4\right ) \log ^2(2+x)} \, dx=- \frac {x}{2} + e^{x} - \frac {\log {\left (x \right )}}{\left (x + 3\right ) \log {\left (x + 2 \right )}} \]

[In]

integrate((((2*x**4+16*x**3+42*x**2+36*x)*exp(x)-x**4-8*x**3-21*x**2-18*x)*ln(2+x)**2+((2*x**2+4*x)*ln(x)-2*x*
*2-10*x-12)*ln(2+x)+(2*x**2+6*x)*ln(x))/(2*x**4+16*x**3+42*x**2+36*x)/ln(2+x)**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {\left (6 x+2 x^2\right ) \log (x)+\left (-12-10 x-2 x^2+\left (4 x+2 x^2\right ) \log (x)\right ) \log (2+x)+\left (-18 x-21 x^2-8 x^3-x^4+e^x \left (36 x+42 x^2+16 x^3+2 x^4\right )\right ) \log ^2(2+x)}{\left (36 x+42 x^2+16 x^3+2 x^4\right ) \log ^2(2+x)} \, dx=-\frac {{\left (x^{2} - 2 \, {\left (x + 3\right )} e^{x} + 3 \, x\right )} \log \left (x + 2\right ) + 2 \, \log \left (x\right )}{2 \, {\left (x + 3\right )} \log \left (x + 2\right )} \]

[In]

integrate((((2*x^4+16*x^3+42*x^2+36*x)*exp(x)-x^4-8*x^3-21*x^2-18*x)*log(2+x)^2+((2*x^2+4*x)*log(x)-2*x^2-10*x
-12)*log(2+x)+(2*x^2+6*x)*log(x))/(2*x^4+16*x^3+42*x^2+36*x)/log(2+x)^2,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).

Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \frac {\left (6 x+2 x^2\right ) \log (x)+\left (-12-10 x-2 x^2+\left (4 x+2 x^2\right ) \log (x)\right ) \log (2+x)+\left (-18 x-21 x^2-8 x^3-x^4+e^x \left (36 x+42 x^2+16 x^3+2 x^4\right )\right ) \log ^2(2+x)}{\left (36 x+42 x^2+16 x^3+2 x^4\right ) \log ^2(2+x)} \, dx=-\frac {x^{2} \log \left (x + 2\right ) - 2 \, x e^{x} \log \left (x + 2\right ) + 3 \, x \log \left (x + 2\right ) - 6 \, e^{x} \log \left (x + 2\right ) + 2 \, \log \left (x\right )}{2 \, {\left (x \log \left (x + 2\right ) + 3 \, \log \left (x + 2\right )\right )}} \]

[In]

integrate((((2*x^4+16*x^3+42*x^2+36*x)*exp(x)-x^4-8*x^3-21*x^2-18*x)*log(2+x)^2+((2*x^2+4*x)*log(x)-2*x^2-10*x
-12)*log(2+x)+(2*x^2+6*x)*log(x))/(2*x^4+16*x^3+42*x^2+36*x)/log(2+x)^2,x, algorithm="giac")

[Out]

-1/2*(x^2*log(x + 2) - 2*x*e^x*log(x + 2) + 3*x*log(x + 2) - 6*e^x*log(x + 2) + 2*log(x))/(x*log(x + 2) + 3*lo
g(x + 2))

Mupad [B] (verification not implemented)

Time = 14.37 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.12 \[ \int \frac {\left (6 x+2 x^2\right ) \log (x)+\left (-12-10 x-2 x^2+\left (4 x+2 x^2\right ) \log (x)\right ) \log (2+x)+\left (-18 x-21 x^2-8 x^3-x^4+e^x \left (36 x+42 x^2+16 x^3+2 x^4\right )\right ) \log ^2(2+x)}{\left (36 x+42 x^2+16 x^3+2 x^4\right ) \log ^2(2+x)} \, dx={\mathrm {e}}^x-\frac {x}{2}-\frac {x+2}{x^2+3\,x}-\frac {\frac {\ln \left (x\right )}{x+3}-\frac {\ln \left (x+2\right )\,\left (x+2\right )\,\left (x-x\,\ln \left (x\right )+3\right )}{x\,{\left (x+3\right )}^2}}{\ln \left (x+2\right )}+\frac {\ln \left (x\right )\,\left (x+2\right )}{x^2+6\,x+9} \]

[In]

int(-(log(x + 2)^2*(18*x - exp(x)*(36*x + 42*x^2 + 16*x^3 + 2*x^4) + 21*x^2 + 8*x^3 + x^4) - log(x)*(6*x + 2*x
^2) + log(x + 2)*(10*x - log(x)*(4*x + 2*x^2) + 2*x^2 + 12))/(log(x + 2)^2*(36*x + 42*x^2 + 16*x^3 + 2*x^4)),x
)

[Out]

exp(x) - x/2 - (x + 2)/(3*x + x^2) - (log(x)/(x + 3) - (log(x + 2)*(x + 2)*(x - x*log(x) + 3))/(x*(x + 3)^2))/
log(x + 2) + (log(x)*(x + 2))/(6*x + x^2 + 9)