\(\int \frac {-4-x+2^x e^{-2+2^x x^{2+x}} x^x (6 x+5 x^2+x^3+(3 x^2+x^3) \log (2 x))}{-3 x-x^2+e^{-2+2^x x^{2+x}} (3+x)+(-3-x) \log (3+x)} \, dx\) [606]

Optimal result

Integrand size = 90, antiderivative size = 22 \[ \int \frac {-4-x+2^x e^{-2+2^x x^{2+x}} x^x \left (6 x+5 x^2+x^3+\left (3 x^2+x^3\right ) \log (2 x)\right )}{-3 x-x^2+e^{-2+2^x x^{2+x}} (3+x)+(-3-x) \log (3+x)} \, dx=\log \left (-e^{-2+2^x x^{2+x}}+x+\log (3+x)\right ) \]

[Out]

ln(x-exp(x^2*exp(x*ln(2*x))-2)+ln(3+x))

Rubi [F]

\[ \int \frac {-4-x+2^x e^{-2+2^x x^{2+x}} x^x \left (6 x+5 x^2+x^3+\left (3 x^2+x^3\right ) \log (2 x)\right )}{-3 x-x^2+e^{-2+2^x x^{2+x}} (3+x)+(-3-x) \log (3+x)} \, dx=\int \frac {-4-x+2^x e^{-2+2^x x^{2+x}} x^x \left (6 x+5 x^2+x^3+\left (3 x^2+x^3\right ) \log (2 x)\right )}{-3 x-x^2+e^{-2+2^x x^{2+x}} (3+x)+(-3-x) \log (3+x)} \, dx \]

[In]

Int[(-4 - x + 2^x*E^(-2 + 2^x*x^(2 + x))*x^x*(6*x + 5*x^2 + x^3 + (3*x^2 + x^3)*Log[2*x]))/(-3*x - x^2 + E^(-2
+ 2^x*x^(2 + x))*(3 + x) + (-3 - x)*Log[3 + x]),x]

[Out]

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

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

Mathematica [F]

\[ \int \frac {-4-x+2^x e^{-2+2^x x^{2+x}} x^x \left (6 x+5 x^2+x^3+\left (3 x^2+x^3\right ) \log (2 x)\right )}{-3 x-x^2+e^{-2+2^x x^{2+x}} (3+x)+(-3-x) \log (3+x)} \, dx=\int \frac {-4-x+2^x e^{-2+2^x x^{2+x}} x^x \left (6 x+5 x^2+x^3+\left (3 x^2+x^3\right ) \log (2 x)\right )}{-3 x-x^2+e^{-2+2^x x^{2+x}} (3+x)+(-3-x) \log (3+x)} \, dx \]

[In]

Integrate[(-4 - x + 2^x*E^(-2 + 2^x*x^(2 + x))*x^x*(6*x + 5*x^2 + x^3 + (3*x^2 + x^3)*Log[2*x]))/(-3*x - x^2 +
E^(-2 + 2^x*x^(2 + x))*(3 + x) + (-3 - x)*Log[3 + x]),x]

[Out]

Integrate[(-4 - x + 2^x*E^(-2 + 2^x*x^(2 + x))*x^x*(6*x + 5*x^2 + x^3 + (3*x^2 + x^3)*Log[2*x]))/(-3*x - x^2 +
E^(-2 + 2^x*x^(2 + x))*(3 + x) + (-3 - x)*Log[3 + x]), x]

Maple [A] (verified)

Time = 62.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09

method	result	size
parallelrisch	\(\ln \left (x -{\mathrm e}^{x^{2} {\mathrm e}^{x \ln \left (2 x \right )}-2}+\ln \left (3+x \right )\right )\)	\(24\)
risch	\(2+\ln \left (-x +{\mathrm e}^{x^{2} \left (2 x \right )^{x}-2}-\ln \left (3+x \right )\right )\)	\(26\)

[In]

int((((x^3+3*x^2)*ln(2*x)+x^3+5*x^2+6*x)*exp(x*ln(2*x))*exp(x^2*exp(x*ln(2*x))-2)-x-4)/((3+x)*exp(x^2*exp(x*ln
(2*x))-2)+(-3-x)*ln(3+x)-x^2-3*x),x,method=_RETURNVERBOSE)

[Out]

ln(x-exp(x^2*exp(x*ln(2*x))-2)+ln(3+x))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-4-x+2^x e^{-2+2^x x^{2+x}} x^x \left (6 x+5 x^2+x^3+\left (3 x^2+x^3\right ) \log (2 x)\right )}{-3 x-x^2+e^{-2+2^x x^{2+x}} (3+x)+(-3-x) \log (3+x)} \, dx=\log \left (-x + e^{\left (\left (2 \, x\right )^{x} x^{2} - 2\right )} - \log \left (x + 3\right )\right ) \]

[In]

integrate((((x^3+3*x^2)*log(2*x)+x^3+5*x^2+6*x)*exp(x*log(2*x))*exp(x^2*exp(x*log(2*x))-2)-x-4)/((3+x)*exp(x^2
*exp(x*log(2*x))-2)+(-3-x)*log(3+x)-x^2-3*x),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-4-x+2^x e^{-2+2^x x^{2+x}} x^x \left (6 x+5 x^2+x^3+\left (3 x^2+x^3\right ) \log (2 x)\right )}{-3 x-x^2+e^{-2+2^x x^{2+x}} (3+x)+(-3-x) \log (3+x)} \, dx=\log {\left (- x + e^{x^{2} e^{x \log {\left (2 x \right )}} - 2} - \log {\left (x + 3 \right )} \right )} \]

[In]

integrate((((x**3+3*x**2)*ln(2*x)+x**3+5*x**2+6*x)*exp(x*ln(2*x))*exp(x**2*exp(x*ln(2*x))-2)-x-4)/((3+x)*exp(x
**2*exp(x*ln(2*x))-2)+(-3-x)*ln(3+x)-x**2-3*x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {-4-x+2^x e^{-2+2^x x^{2+x}} x^x \left (6 x+5 x^2+x^3+\left (3 x^2+x^3\right ) \log (2 x)\right )}{-3 x-x^2+e^{-2+2^x x^{2+x}} (3+x)+(-3-x) \log (3+x)} \, dx=\log \left (-x e^{2} - e^{2} \log \left (x + 3\right ) + e^{\left (x^{2} e^{\left (x \log \left (2\right ) + x \log \left (x\right )\right )}\right )}\right ) \]

[In]

integrate((((x^3+3*x^2)*log(2*x)+x^3+5*x^2+6*x)*exp(x*log(2*x))*exp(x^2*exp(x*log(2*x))-2)-x-4)/((3+x)*exp(x^2
*exp(x*log(2*x))-2)+(-3-x)*log(3+x)-x^2-3*x),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (21) = 42\).

Time = 3.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.32 \[ \int \frac {-4-x+2^x e^{-2+2^x x^{2+x}} x^x \left (6 x+5 x^2+x^3+\left (3 x^2+x^3\right ) \log (2 x)\right )}{-3 x-x^2+e^{-2+2^x x^{2+x}} (3+x)+(-3-x) \log (3+x)} \, dx=-\left (2 \, x\right )^{x} x^{3} \log \left (2 \, x\right ) - 2 \, \left (2 \, x\right )^{x} x^{2} \log \left (2 \, x\right ) + \left (2 \, x\right )^{x} x^{2} + \log \left (-2 \, x e^{2} + 2 \, e^{2} \log \left (2\right ) - 2 \, e^{2} \log \left (2 \, x + 6\right ) + 2 \, e^{\left (\left (2 \, x\right )^{x} x^{2}\right )}\right ) \]

[In]

integrate((((x^3+3*x^2)*log(2*x)+x^3+5*x^2+6*x)*exp(x*log(2*x))*exp(x^2*exp(x*log(2*x))-2)-x-4)/((3+x)*exp(x^2
*exp(x*log(2*x))-2)+(-3-x)*log(3+x)-x^2-3*x),x, algorithm="giac")

[Out]

-(2*x)^x*x^3*log(2*x) - 2*(2*x)^x*x^2*log(2*x) + (2*x)^x*x^2 + log(-2*x*e^2 + 2*e^2*log(2) - 2*e^2*log(2*x + 6
) + 2*e^((2*x)^x*x^2))

Mupad [B] (verification not implemented)

Time = 9.93 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-4-x+2^x e^{-2+2^x x^{2+x}} x^x \left (6 x+5 x^2+x^3+\left (3 x^2+x^3\right ) \log (2 x)\right )}{-3 x-x^2+e^{-2+2^x x^{2+x}} (3+x)+(-3-x) \log (3+x)} \, dx=\ln \left (x+\ln \left (x+3\right )-{\mathrm {e}}^{2^x\,x^{x+2}-2}\right ) \]

[In]

int((x - exp(x^2*exp(x*log(2*x)) - 2)*exp(x*log(2*x))*(6*x + log(2*x)*(3*x^2 + x^3) + 5*x^2 + x^3) + 4)/(3*x +
log(x + 3)*(x + 3) - exp(x^2*exp(x*log(2*x)) - 2)*(x + 3) + x^2),x)

[Out]

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