\(\int \frac {x^{\frac {4}{\log (x^2)}} (8 \log (x)-4 \log (x^2))+(4 x-e^x x-2 x^2) \log ^2(x^2)}{x \log ^2(x^2)} \, dx\) [6707]

Optimal result

Integrand size = 55, antiderivative size = 26 \[ \int \frac {x^{\frac {4}{\log \left (x^2\right )}} \left (8 \log (x)-4 \log \left (x^2\right )\right )+\left (4 x-e^x x-2 x^2\right ) \log ^2\left (x^2\right )}{x \log ^2\left (x^2\right )} \, dx=2-e^x+x-(-3+x) x-x^{\frac {4}{\log \left (x^2\right )}} \]

[Out]

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

Rubi [F]

\[ \int \frac {x^{\frac {4}{\log \left (x^2\right )}} \left (8 \log (x)-4 \log \left (x^2\right )\right )+\left (4 x-e^x x-2 x^2\right ) \log ^2\left (x^2\right )}{x \log ^2\left (x^2\right )} \, dx=\int \frac {x^{\frac {4}{\log \left (x^2\right )}} \left (8 \log (x)-4 \log \left (x^2\right )\right )+\left (4 x-e^x x-2 x^2\right ) \log ^2\left (x^2\right )}{x \log ^2\left (x^2\right )} \, dx \]

[In]

Int[(x^(4/Log[x^2])*(8*Log[x] - 4*Log[x^2]) + (4*x - E^x*x - 2*x^2)*Log[x^2]^2)/(x*Log[x^2]^2),x]

[Out]

-E^x + 4*x - x^2 + 8*Defer[Int][(x^(-1 + 4/Log[x^2])*Log[x])/Log[x^2]^2, x] - 4*Defer[Int][x^(-1 + 4/Log[x^2])
/Log[x^2], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (4-e^x-2 x-\frac {4 x^{-1+\frac {4}{\log \left (x^2\right )}} \left (-2 \log (x)+\log \left (x^2\right )\right )}{\log ^2\left (x^2\right )}\right ) \, dx \\ & = 4 x-x^2-4 \int \frac {x^{-1+\frac {4}{\log \left (x^2\right )}} \left (-2 \log (x)+\log \left (x^2\right )\right )}{\log ^2\left (x^2\right )} \, dx-\int e^x \, dx \\ & = -e^x+4 x-x^2-4 \int \left (-\frac {2 x^{-1+\frac {4}{\log \left (x^2\right )}} \log (x)}{\log ^2\left (x^2\right )}+\frac {x^{-1+\frac {4}{\log \left (x^2\right )}}}{\log \left (x^2\right )}\right ) \, dx \\ & = -e^x+4 x-x^2-4 \int \frac {x^{-1+\frac {4}{\log \left (x^2\right )}}}{\log \left (x^2\right )} \, dx+8 \int \frac {x^{-1+\frac {4}{\log \left (x^2\right )}} \log (x)}{\log ^2\left (x^2\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x^{\frac {4}{\log \left (x^2\right )}} \left (8 \log (x)-4 \log \left (x^2\right )\right )+\left (4 x-e^x x-2 x^2\right ) \log ^2\left (x^2\right )}{x \log ^2\left (x^2\right )} \, dx=-e^x+4 x-x^2-x^{\frac {4}{\log \left (x^2\right )}} \]

[In]

Integrate[(x^(4/Log[x^2])*(8*Log[x] - 4*Log[x^2]) + (4*x - E^x*x - 2*x^2)*Log[x^2]^2)/(x*Log[x^2]^2),x]

[Out]

-E^x + 4*x - x^2 - x^(4/Log[x^2])

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00

method	result	size
default	\(4 x -x^{\frac {4}{\ln \left (x^{2}\right )}}-x^{2}-{\mathrm e}^{x}\)	\(26\)
parts	\(4 x -x^{\frac {4}{\ln \left (x^{2}\right )}}-x^{2}-{\mathrm e}^{x}\)	\(26\)
parallelrisch	\(-x^{2}+4 x -{\mathrm e}^{x}-{\mathrm e}^{\frac {4 \ln \left (x \right )}{\ln \left (x^{2}\right )}}\)	\(27\)
risch	\(-x^{2}+4 x -{\mathrm e}^{x}-x^{\frac {4}{2 \ln \left (x \right )-i \pi \,\operatorname {csgn}\left (i x^{2}\right )+i \pi \,\operatorname {csgn}\left (i x \right )}}\)	\(47\)

[In]

int(((-4*ln(x^2)+8*ln(x))*exp(4*ln(x)/ln(x^2))+(-exp(x)*x-2*x^2+4*x)*ln(x^2)^2)/x/ln(x^2)^2,x,method=_RETURNVE
RBOSE)

[Out]

4*x-(x^(1/ln(x^2)))^4-x^2-exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.50 \[ \int \frac {x^{\frac {4}{\log \left (x^2\right )}} \left (8 \log (x)-4 \log \left (x^2\right )\right )+\left (4 x-e^x x-2 x^2\right ) \log ^2\left (x^2\right )}{x \log ^2\left (x^2\right )} \, dx=-x^{2} + 4 \, x - e^{x} \]

[In]

integrate(((-4*log(x^2)+8*log(x))*exp(4*log(x)/log(x^2))+(-exp(x)*x-2*x^2+4*x)*log(x^2)^2)/x/log(x^2)^2,x, alg
orithm="fricas")

[Out]

-x^2 + 4*x - e^x

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.31 \[ \int \frac {x^{\frac {4}{\log \left (x^2\right )}} \left (8 \log (x)-4 \log \left (x^2\right )\right )+\left (4 x-e^x x-2 x^2\right ) \log ^2\left (x^2\right )}{x \log ^2\left (x^2\right )} \, dx=- x^{2} + 4 x - e^{x} \]

[In]

integrate(((-4*ln(x**2)+8*ln(x))*exp(4*ln(x)/ln(x**2))+(-exp(x)*x-2*x**2+4*x)*ln(x**2)**2)/x/ln(x**2)**2,x)

[Out]

-x**2 + 4*x - exp(x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int \frac {x^{\frac {4}{\log \left (x^2\right )}} \left (8 \log (x)-4 \log \left (x^2\right )\right )+\left (4 x-e^x x-2 x^2\right ) \log ^2\left (x^2\right )}{x \log ^2\left (x^2\right )} \, dx=-x^{2} + 4 \, x - 2 \, e^{2} - e^{x} \]

[In]

integrate(((-4*log(x^2)+8*log(x))*exp(4*log(x)/log(x^2))+(-exp(x)*x-2*x^2+4*x)*log(x^2)^2)/x/log(x^2)^2,x, alg
orithm="maxima")

[Out]

-x^2 + 4*x - 2*e^2 - e^x

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.50 \[ \int \frac {x^{\frac {4}{\log \left (x^2\right )}} \left (8 \log (x)-4 \log \left (x^2\right )\right )+\left (4 x-e^x x-2 x^2\right ) \log ^2\left (x^2\right )}{x \log ^2\left (x^2\right )} \, dx=-x^{2} + 4 \, x - e^{x} \]

[In]

integrate(((-4*log(x^2)+8*log(x))*exp(4*log(x)/log(x^2))+(-exp(x)*x-2*x^2+4*x)*log(x^2)^2)/x/log(x^2)^2,x, alg
orithm="giac")

[Out]

-x^2 + 4*x - e^x

Mupad [B] (verification not implemented)

Time = 12.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {x^{\frac {4}{\log \left (x^2\right )}} \left (8 \log (x)-4 \log \left (x^2\right )\right )+\left (4 x-e^x x-2 x^2\right ) \log ^2\left (x^2\right )}{x \log ^2\left (x^2\right )} \, dx=4\,x-{\mathrm {e}}^x-x^2-x^{\frac {4}{\ln \left (x^2\right )}} \]

[In]

int(-(log(x^2)^2*(x*exp(x) - 4*x + 2*x^2) + exp((4*log(x))/log(x^2))*(4*log(x^2) - 8*log(x)))/(x*log(x^2)^2),x
)

[Out]

4*x - exp(x) - x^2 - x^(4/log(x^2))