\(\int \frac {-4 x^3-8 e^{4 e^{2 x}+2 x} x^3-2 x^4+e^{2 e^{2 x}} (-2 x^3+e^{2 x} (-16 x^3-8 x^4))+(-32-x^3) \log ^2(2)}{x^3 \log ^2(2)} \, dx\) [7883]

Optimal result

Integrand size = 82, antiderivative size = 29 \[ \int \frac {-4 x^3-8 e^{4 e^{2 x}+2 x} x^3-2 x^4+e^{2 e^{2 x}} \left (-2 x^3+e^{2 x} \left (-16 x^3-8 x^4\right )\right )+\left (-32-x^3\right ) \log ^2(2)}{x^3 \log ^2(2)} \, dx=\frac {16}{x^2}-x-\frac {\left (2+e^{2 e^{2 x}}+x\right )^2}{\log ^2(2)} \]

[Out]

16/x^2-(x+exp(exp(x)^2)^2+2)^2/ln(2)^2-x

Rubi [F]

\[ \int \frac {-4 x^3-8 e^{4 e^{2 x}+2 x} x^3-2 x^4+e^{2 e^{2 x}} \left (-2 x^3+e^{2 x} \left (-16 x^3-8 x^4\right )\right )+\left (-32-x^3\right ) \log ^2(2)}{x^3 \log ^2(2)} \, dx=\int \frac {-4 x^3-8 e^{4 e^{2 x}+2 x} x^3-2 x^4+e^{2 e^{2 x}} \left (-2 x^3+e^{2 x} \left (-16 x^3-8 x^4\right )\right )+\left (-32-x^3\right ) \log ^2(2)}{x^3 \log ^2(2)} \, dx \]

[In]

Int[(-4*x^3 - 8*E^(4*E^(2*x) + 2*x)*x^3 - 2*x^4 + E^(2*E^(2*x))*(-2*x^3 + E^(2*x)*(-16*x^3 - 8*x^4)) + (-32 -
x^3)*Log[2]^2)/(x^3*Log[2]^2),x]

[Out]

16/x^2 - x*(1 + 4/Log[2]^2) - x^2/Log[2]^2 - ExpIntegralEi[2*E^(2*x)]/Log[2]^2 - (16*Defer[Int][E^(2*(E^(2*x)
+ x)), x])/Log[2]^2 - (8*Defer[Int][E^(2*(2*E^(2*x) + x)), x])/Log[2]^2 - (8*Defer[Int][E^(2*(E^(2*x) + x))*x,
x])/Log[2]^2

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-4 x^3-8 e^{4 e^{2 x}+2 x} x^3-2 x^4+e^{2 e^{2 x}} \left (-2 x^3+e^{2 x} \left (-16 x^3-8 x^4\right )\right )+\left (-32-x^3\right ) \log ^2(2)}{x^3} \, dx}{\log ^2(2)} \\ & = \frac {\int \left (8 e^{2 \left (e^{2 x}+x\right )} \left (-2-e^{2 e^{2 x}}-x\right )+\frac {-2 e^{2 e^{2 x}} x^3-2 x^4-32 \log ^2(2)-4 x^3 \left (1+\frac {\log ^2(2)}{4}\right )}{x^3}\right ) \, dx}{\log ^2(2)} \\ & = \frac {\int \frac {-2 e^{2 e^{2 x}} x^3-2 x^4-32 \log ^2(2)-4 x^3 \left (1+\frac {\log ^2(2)}{4}\right )}{x^3} \, dx}{\log ^2(2)}+\frac {8 \int e^{2 \left (e^{2 x}+x\right )} \left (-2-e^{2 e^{2 x}}-x\right ) \, dx}{\log ^2(2)} \\ & = \frac {\int \left (-2 e^{2 e^{2 x}}+\frac {-2 x^4-32 \log ^2(2)-x^3 \left (4+\log ^2(2)\right )}{x^3}\right ) \, dx}{\log ^2(2)}+\frac {8 \int \left (-2 e^{2 \left (e^{2 x}+x\right )}-e^{2 e^{2 x}+2 \left (e^{2 x}+x\right )}-e^{2 \left (e^{2 x}+x\right )} x\right ) \, dx}{\log ^2(2)} \\ & = \frac {\int \frac {-2 x^4-32 \log ^2(2)-x^3 \left (4+\log ^2(2)\right )}{x^3} \, dx}{\log ^2(2)}-\frac {2 \int e^{2 e^{2 x}} \, dx}{\log ^2(2)}-\frac {8 \int e^{2 e^{2 x}+2 \left (e^{2 x}+x\right )} \, dx}{\log ^2(2)}-\frac {8 \int e^{2 \left (e^{2 x}+x\right )} x \, dx}{\log ^2(2)}-\frac {16 \int e^{2 \left (e^{2 x}+x\right )} \, dx}{\log ^2(2)} \\ & = \frac {\int \left (-2 x-\frac {32 \log ^2(2)}{x^3}-4 \left (1+\frac {\log ^2(2)}{4}\right )\right ) \, dx}{\log ^2(2)}-\frac {\text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,e^{2 x}\right )}{\log ^2(2)}-\frac {8 \int e^{2 \left (2 e^{2 x}+x\right )} \, dx}{\log ^2(2)}-\frac {8 \int e^{2 \left (e^{2 x}+x\right )} x \, dx}{\log ^2(2)}-\frac {16 \int e^{2 \left (e^{2 x}+x\right )} \, dx}{\log ^2(2)} \\ & = \frac {16}{x^2}-x \left (1+\frac {4}{\log ^2(2)}\right )-\frac {x^2}{\log ^2(2)}-\frac {\operatorname {ExpIntegralEi}\left (2 e^{2 x}\right )}{\log ^2(2)}-\frac {8 \int e^{2 \left (2 e^{2 x}+x\right )} \, dx}{\log ^2(2)}-\frac {8 \int e^{2 \left (e^{2 x}+x\right )} x \, dx}{\log ^2(2)}-\frac {16 \int e^{2 \left (e^{2 x}+x\right )} \, dx}{\log ^2(2)} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(29)=58\).

Time = 0.23 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \frac {-4 x^3-8 e^{4 e^{2 x}+2 x} x^3-2 x^4+e^{2 e^{2 x}} \left (-2 x^3+e^{2 x} \left (-16 x^3-8 x^4\right )\right )+\left (-32-x^3\right ) \log ^2(2)}{x^3 \log ^2(2)} \, dx=-\frac {e^{4 e^{2 x}} x^2+x^4+2 e^{2 e^{2 x}} x^2 (2+x)-16 \log ^2(2)+x^3 \left (4+\log ^2(2)\right )}{x^2 \log ^2(2)} \]

[In]

Integrate[(-4*x^3 - 8*E^(4*E^(2*x) + 2*x)*x^3 - 2*x^4 + E^(2*E^(2*x))*(-2*x^3 + E^(2*x)*(-16*x^3 - 8*x^4)) + (
-32 - x^3)*Log[2]^2)/(x^3*Log[2]^2),x]

[Out]

-((E^(4*E^(2*x))*x^2 + x^4 + 2*E^(2*E^(2*x))*x^2*(2 + x) - 16*Log[2]^2 + x^3*(4 + Log[2]^2))/(x^2*Log[2]^2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(27)=54\).

Time = 1.00 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93

method	result	size
risch	\(-x -\frac {x^{2}}{\ln \left (2\right )^{2}}-\frac {4 x}{\ln \left (2\right )^{2}}+\frac {16}{x^{2}}-\frac {{\mathrm e}^{4 \,{\mathrm e}^{2 x}}}{\ln \left (2\right )^{2}}+\frac {\left (-2 x -4\right ) {\mathrm e}^{2 \,{\mathrm e}^{2 x}}}{\ln \left (2\right )^{2}}\)	\(56\)
parallelrisch	\(-\frac {x^{3} \ln \left (2\right )^{2}+2 \,{\mathrm e}^{2 \,{\mathrm e}^{2 x}} x^{3}+x^{4}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{2 x}} x^{2}+{\mathrm e}^{4 \,{\mathrm e}^{2 x}} x^{2}+4 x^{3}-16 \ln \left (2\right )^{2}}{\ln \left (2\right )^{2} x^{2}}\)	\(68\)

[In]

int((-8*x^3*exp(x)^2*exp(exp(x)^2)^4+((-8*x^4-16*x^3)*exp(x)^2-2*x^3)*exp(exp(x)^2)^2+(-x^3-32)*ln(2)^2-2*x^4-
4*x^3)/x^3/ln(2)^2,x,method=_RETURNVERBOSE)

[Out]

-x-x^2/ln(2)^2-4*x/ln(2)^2+16/x^2-1/ln(2)^2*exp(4*exp(2*x))+1/ln(2)^2*(-2*x-4)*exp(2*exp(2*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (27) = 54\).

Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97 \[ \int \frac {-4 x^3-8 e^{4 e^{2 x}+2 x} x^3-2 x^4+e^{2 e^{2 x}} \left (-2 x^3+e^{2 x} \left (-16 x^3-8 x^4\right )\right )+\left (-32-x^3\right ) \log ^2(2)}{x^3 \log ^2(2)} \, dx=-\frac {x^{4} + 4 \, x^{3} + x^{2} e^{\left (4 \, e^{\left (2 \, x\right )}\right )} + {\left (x^{3} - 16\right )} \log \left (2\right )^{2} + 2 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{\left (2 \, e^{\left (2 \, x\right )}\right )}}{x^{2} \log \left (2\right )^{2}} \]

[In]

integrate((-8*x^3*exp(x)^2*exp(exp(x)^2)^4+((-8*x^4-16*x^3)*exp(x)^2-2*x^3)*exp(exp(x)^2)^2+(-x^3-32)*log(2)^2
-2*x^4-4*x^3)/x^3/log(2)^2,x, algorithm="fricas")

[Out]

-(x^4 + 4*x^3 + x^2*e^(4*e^(2*x)) + (x^3 - 16)*log(2)^2 + 2*(x^3 + 2*x^2)*e^(2*e^(2*x)))/(x^2*log(2)^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (24) = 48\).

Time = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.41 \[ \int \frac {-4 x^3-8 e^{4 e^{2 x}+2 x} x^3-2 x^4+e^{2 e^{2 x}} \left (-2 x^3+e^{2 x} \left (-16 x^3-8 x^4\right )\right )+\left (-32-x^3\right ) \log ^2(2)}{x^3 \log ^2(2)} \, dx=\frac {\left (- 2 x \log {\left (2 \right )}^{2} - 4 \log {\left (2 \right )}^{2}\right ) e^{2 e^{2 x}} - e^{4 e^{2 x}} \log {\left (2 \right )}^{2}}{\log {\left (2 \right )}^{4}} + \frac {- x^{2} - x \left (\log {\left (2 \right )}^{2} + 4\right ) + \frac {16 \log {\left (2 \right )}^{2}}{x^{2}}}{\log {\left (2 \right )}^{2}} \]

[In]

integrate((-8*x**3*exp(x)**2*exp(exp(x)**2)**4+((-8*x**4-16*x**3)*exp(x)**2-2*x**3)*exp(exp(x)**2)**2+(-x**3-3
2)*ln(2)**2-2*x**4-4*x**3)/x**3/ln(2)**2,x)

[Out]

((-2*x*log(2)**2 - 4*log(2)**2)*exp(2*exp(2*x)) - exp(4*exp(2*x))*log(2)**2)/log(2)**4 + (-x**2 - x*(log(2)**2
+ 4) + 16*log(2)**2/x**2)/log(2)**2

Maxima [F]

\[ \int \frac {-4 x^3-8 e^{4 e^{2 x}+2 x} x^3-2 x^4+e^{2 e^{2 x}} \left (-2 x^3+e^{2 x} \left (-16 x^3-8 x^4\right )\right )+\left (-32-x^3\right ) \log ^2(2)}{x^3 \log ^2(2)} \, dx=\int { -\frac {2 \, x^{4} + 8 \, x^{3} e^{\left (2 \, x + 4 \, e^{\left (2 \, x\right )}\right )} + 4 \, x^{3} + {\left (x^{3} + 32\right )} \log \left (2\right )^{2} + 2 \, {\left (x^{3} + 4 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (2 \, e^{\left (2 \, x\right )}\right )}}{x^{3} \log \left (2\right )^{2}} \,d x } \]

[In]

integrate((-8*x^3*exp(x)^2*exp(exp(x)^2)^4+((-8*x^4-16*x^3)*exp(x)^2-2*x^3)*exp(exp(x)^2)^2+(-x^3-32)*log(2)^2
-2*x^4-4*x^3)/x^3/log(2)^2,x, algorithm="maxima")

[Out]

-(x*log(2)^2 + x^2 + 2*x*e^(2*e^(2*x)) + 4*x - 16*log(2)^2/x^2 + Ei(2*e^(2*x)) + e^(4*e^(2*x)) + 4*e^(2*e^(2*x
)) - 2*integrate(e^(2*e^(2*x)), x))/log(2)^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (27) = 54\).

Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.45 \[ \int \frac {-4 x^3-8 e^{4 e^{2 x}+2 x} x^3-2 x^4+e^{2 e^{2 x}} \left (-2 x^3+e^{2 x} \left (-16 x^3-8 x^4\right )\right )+\left (-32-x^3\right ) \log ^2(2)}{x^3 \log ^2(2)} \, dx=-\frac {{\left (x^{3} e^{\left (2 \, x\right )} \log \left (2\right )^{2} + x^{4} e^{\left (2 \, x\right )} + 4 \, x^{3} e^{\left (2 \, x\right )} + 2 \, x^{3} e^{\left (2 \, x + 2 \, e^{\left (2 \, x\right )}\right )} + x^{2} e^{\left (2 \, x + 4 \, e^{\left (2 \, x\right )}\right )} + 4 \, x^{2} e^{\left (2 \, x + 2 \, e^{\left (2 \, x\right )}\right )} - 16 \, e^{\left (2 \, x\right )} \log \left (2\right )^{2}\right )} e^{\left (-2 \, x\right )}}{x^{2} \log \left (2\right )^{2}} \]

[In]

integrate((-8*x^3*exp(x)^2*exp(exp(x)^2)^4+((-8*x^4-16*x^3)*exp(x)^2-2*x^3)*exp(exp(x)^2)^2+(-x^3-32)*log(2)^2
-2*x^4-4*x^3)/x^3/log(2)^2,x, algorithm="giac")

[Out]

-(x^3*e^(2*x)*log(2)^2 + x^4*e^(2*x) + 4*x^3*e^(2*x) + 2*x^3*e^(2*x + 2*e^(2*x)) + x^2*e^(2*x + 4*e^(2*x)) + 4
*x^2*e^(2*x + 2*e^(2*x)) - 16*e^(2*x)*log(2)^2)*e^(-2*x)/(x^2*log(2)^2)

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.21 \[ \int \frac {-4 x^3-8 e^{4 e^{2 x}+2 x} x^3-2 x^4+e^{2 e^{2 x}} \left (-2 x^3+e^{2 x} \left (-16 x^3-8 x^4\right )\right )+\left (-32-x^3\right ) \log ^2(2)}{x^3 \log ^2(2)} \, dx=\frac {16}{x^2}-\frac {{\mathrm {e}}^{4\,{\mathrm {e}}^{2\,x}}}{{\ln \left (2\right )}^2}-{\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}}\,\left (\frac {2\,x}{{\ln \left (2\right )}^2}+\frac {4}{{\ln \left (2\right )}^2}\right )-\frac {x^2}{{\ln \left (2\right )}^2}-\frac {x\,\left ({\ln \left (2\right )}^2+4\right )}{{\ln \left (2\right )}^2} \]

[In]

int(-(exp(2*exp(2*x))*(exp(2*x)*(16*x^3 + 8*x^4) + 2*x^3) + log(2)^2*(x^3 + 32) + 4*x^3 + 2*x^4 + 8*x^3*exp(4*
exp(2*x))*exp(2*x))/(x^3*log(2)^2),x)

[Out]

16/x^2 - exp(4*exp(2*x))/log(2)^2 - exp(2*exp(2*x))*((2*x)/log(2)^2 + 4/log(2)^2) - x^2/log(2)^2 - (x*(log(2)^
2 + 4))/log(2)^2