\(\int (3+(-8+4 x^2) \log ^3(\frac {e^{\frac {x^2}{2}}}{x^2})+\log ^4(\frac {e^{\frac {x^2}{2}}}{x^2})) \, dx\) [277]

Optimal result

Integrand size = 42, antiderivative size = 25 \[ \int \left (3+\left (-8+4 x^2\right ) \log ^3\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )+\log ^4\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )\right ) \, dx=\log \left (2 e^{x \left (3+\log ^4\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )\right )}\right ) \]

[Out]

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

Rubi [F]

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

[In]

Int[3 + (-8 + 4*x^2)*Log[E^(x^2/2)/x^2]^3 + Log[E^(x^2/2)/x^2]^4,x]

[Out]

3*x - 8*Defer[Int][Log[E^(x^2/2)/x^2]^3, x] + 4*Defer[Int][x^2*Log[E^(x^2/2)/x^2]^3, x] + Defer[Int][Log[E^(x^
2/2)/x^2]^4, x]

Rubi steps \begin{align*} \text {integral}& = 3 x+\int \left (-8+4 x^2\right ) \log ^3\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right ) \, dx+\int \log ^4\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right ) \, dx \\ & = 3 x+\int \log ^4\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right ) \, dx+\int \left (-8 \log ^3\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )+4 x^2 \log ^3\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )\right ) \, dx \\ & = 3 x+4 \int x^2 \log ^3\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right ) \, dx-8 \int \log ^3\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right ) \, dx+\int \log ^4\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \left (3+\left (-8+4 x^2\right ) \log ^3\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )+\log ^4\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )\right ) \, dx=x \left (3+\log ^4\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )\right ) \]

[In]

Integrate[3 + (-8 + 4*x^2)*Log[E^(x^2/2)/x^2]^3 + Log[E^(x^2/2)/x^2]^4,x]

[Out]

x*(3 + Log[E^(x^2/2)/x^2]^4)

Maple [A] (verified)

Time = 1.99 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
parallelrisch	\(\ln \left (\frac {{\mathrm e}^{\frac {x^{2}}{2}}}{x^{2}}\right )^{4} x +3 x\)	\(20\)
default	\(\text {Expression too large to display}\)	\(740\)
parts	\(\text {Expression too large to display}\)	\(740\)
risch	\(\text {Expression too large to display}\)	\(17452\)

[In]

int(ln(exp(1/2*x^2)/x^2)^4+(4*x^2-8)*ln(exp(1/2*x^2)/x^2)^3+3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (3+\left (-8+4 x^2\right ) \log ^3\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )+\log ^4\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )\right ) \, dx=x \log \left (\frac {e^{\left (\frac {1}{2} \, x^{2}\right )}}{x^{2}}\right )^{4} + 3 \, x \]

[In]

integrate(log(exp(1/2*x^2)/x^2)^4+(4*x^2-8)*log(exp(1/2*x^2)/x^2)^3+3,x, algorithm="fricas")

[Out]

x*log(e^(1/2*x^2)/x^2)^4 + 3*x

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \left (3+\left (-8+4 x^2\right ) \log ^3\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )+\log ^4\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )\right ) \, dx=x \log {\left (\frac {e^{\frac {x^{2}}{2}}}{x^{2}} \right )}^{4} + 3 x \]

[In]

integrate(ln(exp(1/2*x**2)/x**2)**4+(4*x**2-8)*ln(exp(1/2*x**2)/x**2)**3+3,x)

[Out]

x*log(exp(x**2/2)/x**2)**4 + 3*x

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (3+\left (-8+4 x^2\right ) \log ^3\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )+\log ^4\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )\right ) \, dx=x \log \left (\frac {e^{\left (\frac {1}{2} \, x^{2}\right )}}{x^{2}}\right )^{4} + 3 \, x \]

[In]

integrate(log(exp(1/2*x^2)/x^2)^4+(4*x^2-8)*log(exp(1/2*x^2)/x^2)^3+3,x, algorithm="maxima")

[Out]

x*log(e^(1/2*x^2)/x^2)^4 + 3*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (21) = 42\).

Time = 0.51 (sec) , antiderivative size = 231, normalized size of antiderivative = 9.24 \[ \int \left (3+\left (-8+4 x^2\right ) \log ^3\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )+\log ^4\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )\right ) \, dx=-\frac {17}{210} \, x^{9} + \frac {8}{35} \, x^{7} \log \left (\frac {e^{\left (\frac {1}{2} \, x^{2}\right )}}{x^{2}}\right ) + \frac {59}{75} \, x^{7} - \frac {4}{5} \, x^{5} \log \left (\frac {e^{\left (\frac {1}{2} \, x^{2}\right )}}{x^{2}}\right )^{2} - \frac {368}{75} \, x^{5} \log \left (\frac {e^{\left (\frac {1}{2} \, x^{2}\right )}}{x^{2}}\right ) + \frac {44}{9} \, x^{5} + \frac {32}{3} \, x^{3} \log \left (\frac {e^{\left (\frac {1}{2} \, x^{2}\right )}}{x^{2}}\right )^{2} + x \log \left (\frac {e^{\left (\frac {1}{2} \, x^{2}\right )}}{x^{2}}\right )^{4} + \frac {4}{3} \, {\left (x^{3} - 6 \, x\right )} \log \left (x^{2}\right )^{3} + \frac {128}{9} \, x^{3} \log \left (\frac {e^{\left (\frac {1}{2} \, x^{2}\right )}}{x^{2}}\right ) + \frac {4}{3} \, {\left (x^{3} - 6 \, x\right )} \log \left (\frac {e^{\left (\frac {1}{2} \, x^{2}\right )}}{x^{2}}\right )^{3} - \frac {2}{15} \, {\left (9 \, x^{5} - 10 \, x^{3} - 360 \, x\right )} \log \left (x^{2}\right )^{2} - 48 \, x \log \left (\frac {e^{\left (\frac {1}{2} \, x^{2}\right )}}{x^{2}}\right )^{2} + \frac {1}{1575} \, {\left (675 \, x^{7} - 378 \, x^{5} - 2800 \, x^{3} - 302400 \, x\right )} \log \left (x^{2}\right ) - 32 \, {\left (x^{3} - 6 \, x\right )} \log \left (x^{2}\right ) + 3 \, x \]

[In]

integrate(log(exp(1/2*x^2)/x^2)^4+(4*x^2-8)*log(exp(1/2*x^2)/x^2)^3+3,x, algorithm="giac")

[Out]

-17/210*x^9 + 8/35*x^7*log(e^(1/2*x^2)/x^2) + 59/75*x^7 - 4/5*x^5*log(e^(1/2*x^2)/x^2)^2 - 368/75*x^5*log(e^(1
/2*x^2)/x^2) + 44/9*x^5 + 32/3*x^3*log(e^(1/2*x^2)/x^2)^2 + x*log(e^(1/2*x^2)/x^2)^4 + 4/3*(x^3 - 6*x)*log(x^2
)^3 + 128/9*x^3*log(e^(1/2*x^2)/x^2) + 4/3*(x^3 - 6*x)*log(e^(1/2*x^2)/x^2)^3 - 2/15*(9*x^5 - 10*x^3 - 360*x)*
log(x^2)^2 - 48*x*log(e^(1/2*x^2)/x^2)^2 + 1/1575*(675*x^7 - 378*x^5 - 2800*x^3 - 302400*x)*log(x^2) - 32*(x^3
- 6*x)*log(x^2) + 3*x

Mupad [B] (verification not implemented)

Time = 8.59 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \left (3+\left (-8+4 x^2\right ) \log ^3\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )+\log ^4\left (\frac {e^{\frac {x^2}{2}}}{x^2}\right )\right ) \, dx=x\,\left ({\ln \left (\frac {{\mathrm {e}}^{\frac {x^2}{2}}}{x^2}\right )}^4+3\right ) \]

[In]

int(log(exp(x^2/2)/x^2)^3*(4*x^2 - 8) + log(exp(x^2/2)/x^2)^4 + 3,x)

[Out]

x*(log(exp(x^2/2)/x^2)^4 + 3)