\(\int \frac {-16 x+e^{e^{25 x^2}} (8-16 x+e^{25 x^2} (400 x^2-400 x^3))}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} (-25 x^3+75 x^4-75 x^5+25 x^6)+e^{2 e^{25 x^2}} (-75 x^2+150 x^3-150 x^5+75 x^6)+e^{e^{25 x^2}} (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6)} \, dx\) [5012]

Optimal result

Integrand size = 163, antiderivative size = 27 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4}{25 (1-x)^2 \left (1+x+e^{e^{25 x^2}} x\right )^2} \]

[Out]

4/5/(1-x)^2/(x+1+x*exp(exp(25*x^2)))/(5*x+5+5*x*exp(exp(25*x^2)))

Rubi [F]

\[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx \]

[In]

Int[(-16*x + E^E^(25*x^2)*(8 - 16*x + E^(25*x^2)*(400*x^2 - 400*x^3)))/(-25 + 75*x^2 - 75*x^4 + 25*x^6 + E^(3*
E^(25*x^2))*(-25*x^3 + 75*x^4 - 75*x^5 + 25*x^6) + E^(2*E^(25*x^2))*(-75*x^2 + 150*x^3 - 150*x^5 + 75*x^6) + E
^E^(25*x^2)*(-75*x + 75*x^2 + 150*x^3 - 150*x^4 - 75*x^5 + 75*x^6)),x]

[Out]

-16*Defer[Int][E^(E^(25*x^2) + 25*x^2)/(1 + x + E^E^(25*x^2)*x)^3, x] - (16*Defer[Int][1/((-1 + x)^3*(1 + x +
E^E^(25*x^2)*x)^3), x])/25 - (8*Defer[Int][E^E^(25*x^2)/((-1 + x)^3*(1 + x + E^E^(25*x^2)*x)^3), x])/25 - (16*
Defer[Int][1/((-1 + x)^2*(1 + x + E^E^(25*x^2)*x)^3), x])/25 - (16*Defer[Int][E^E^(25*x^2)/((-1 + x)^2*(1 + x
+ E^E^(25*x^2)*x)^3), x])/25 - 16*Defer[Int][E^(E^(25*x^2) + 25*x^2)/((-1 + x)^2*(1 + x + E^E^(25*x^2)*x)^3),
x] - 32*Defer[Int][E^(E^(25*x^2) + 25*x^2)/((-1 + x)*(1 + x + E^E^(25*x^2)*x)^3), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {8 \left (2 x+50 e^{e^{25 x^2}+25 x^2} (-1+x) x^2+e^{e^{25 x^2}} (-1+2 x)\right )}{25 (1-x)^3 \left (1+x+e^{e^{25 x^2}} x\right )^3} \, dx \\ & = \frac {8}{25} \int \frac {2 x+50 e^{e^{25 x^2}+25 x^2} (-1+x) x^2+e^{e^{25 x^2}} (-1+2 x)}{(1-x)^3 \left (1+x+e^{e^{25 x^2}} x\right )^3} \, dx \\ & = \frac {8}{25} \int \left (-\frac {2 x}{(-1+x)^3 \left (1+x+e^{e^{25 x^2}} x\right )^3}-\frac {50 e^{e^{25 x^2}+25 x^2} x^2}{(-1+x)^2 \left (1+x+e^{e^{25 x^2}} x\right )^3}-\frac {e^{e^{25 x^2}} (-1+2 x)}{(-1+x)^3 \left (1+x+e^{e^{25 x^2}} x\right )^3}\right ) \, dx \\ & = -\left (\frac {8}{25} \int \frac {e^{e^{25 x^2}} (-1+2 x)}{(-1+x)^3 \left (1+x+e^{e^{25 x^2}} x\right )^3} \, dx\right )-\frac {16}{25} \int \frac {x}{(-1+x)^3 \left (1+x+e^{e^{25 x^2}} x\right )^3} \, dx-16 \int \frac {e^{e^{25 x^2}+25 x^2} x^2}{(-1+x)^2 \left (1+x+e^{e^{25 x^2}} x\right )^3} \, dx \\ & = -\left (\frac {8}{25} \int \left (\frac {e^{e^{25 x^2}}}{(-1+x)^3 \left (1+x+e^{e^{25 x^2}} x\right )^3}+\frac {2 e^{e^{25 x^2}}}{(-1+x)^2 \left (1+x+e^{e^{25 x^2}} x\right )^3}\right ) \, dx\right )-\frac {16}{25} \int \left (\frac {1}{(-1+x)^3 \left (1+x+e^{e^{25 x^2}} x\right )^3}+\frac {1}{(-1+x)^2 \left (1+x+e^{e^{25 x^2}} x\right )^3}\right ) \, dx-16 \int \left (\frac {e^{e^{25 x^2}+25 x^2}}{\left (1+x+e^{e^{25 x^2}} x\right )^3}+\frac {e^{e^{25 x^2}+25 x^2}}{(-1+x)^2 \left (1+x+e^{e^{25 x^2}} x\right )^3}+\frac {2 e^{e^{25 x^2}+25 x^2}}{(-1+x) \left (1+x+e^{e^{25 x^2}} x\right )^3}\right ) \, dx \\ & = -\left (\frac {8}{25} \int \frac {e^{e^{25 x^2}}}{(-1+x)^3 \left (1+x+e^{e^{25 x^2}} x\right )^3} \, dx\right )-\frac {16}{25} \int \frac {1}{(-1+x)^3 \left (1+x+e^{e^{25 x^2}} x\right )^3} \, dx-\frac {16}{25} \int \frac {1}{(-1+x)^2 \left (1+x+e^{e^{25 x^2}} x\right )^3} \, dx-\frac {16}{25} \int \frac {e^{e^{25 x^2}}}{(-1+x)^2 \left (1+x+e^{e^{25 x^2}} x\right )^3} \, dx-16 \int \frac {e^{e^{25 x^2}+25 x^2}}{\left (1+x+e^{e^{25 x^2}} x\right )^3} \, dx-16 \int \frac {e^{e^{25 x^2}+25 x^2}}{(-1+x)^2 \left (1+x+e^{e^{25 x^2}} x\right )^3} \, dx-32 \int \frac {e^{e^{25 x^2}+25 x^2}}{(-1+x) \left (1+x+e^{e^{25 x^2}} x\right )^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4}{25 (-1+x)^2 \left (1+x+e^{e^{25 x^2}} x\right )^2} \]

[In]

Integrate[(-16*x + E^E^(25*x^2)*(8 - 16*x + E^(25*x^2)*(400*x^2 - 400*x^3)))/(-25 + 75*x^2 - 75*x^4 + 25*x^6 +
E^(3*E^(25*x^2))*(-25*x^3 + 75*x^4 - 75*x^5 + 25*x^6) + E^(2*E^(25*x^2))*(-75*x^2 + 150*x^3 - 150*x^5 + 75*x^
6) + E^E^(25*x^2)*(-75*x + 75*x^2 + 150*x^3 - 150*x^4 - 75*x^5 + 75*x^6)),x]

[Out]

4/(25*(-1 + x)^2*(1 + x + E^E^(25*x^2)*x)^2)

Maple [A] (verified)

Time = 1.77 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {4}{25 \left (x^{2}-2 x +1\right ) \left (x +1+x \,{\mathrm e}^{{\mathrm e}^{25 x^{2}}}\right )^{2}}\)	\(27\)
parallelrisch	\(\frac {4}{25 \left ({\mathrm e}^{2 \,{\mathrm e}^{25 x^{2}}} x^{4}+2 \,{\mathrm e}^{{\mathrm e}^{25 x^{2}}} x^{4}-2 \,{\mathrm e}^{2 \,{\mathrm e}^{25 x^{2}}} x^{3}+x^{4}-2 \,{\mathrm e}^{{\mathrm e}^{25 x^{2}}} x^{3}+{\mathrm e}^{2 \,{\mathrm e}^{25 x^{2}}} x^{2}-2 \,{\mathrm e}^{{\mathrm e}^{25 x^{2}}} x^{2}-2 x^{2}+2 x \,{\mathrm e}^{{\mathrm e}^{25 x^{2}}}+1\right )}\)	\(101\)

[In]

int((((-400*x^3+400*x^2)*exp(25*x^2)-16*x+8)*exp(exp(25*x^2))-16*x)/((25*x^6-75*x^5+75*x^4-25*x^3)*exp(exp(25*
x^2))^3+(75*x^6-150*x^5+150*x^3-75*x^2)*exp(exp(25*x^2))^2+(75*x^6-75*x^5-150*x^4+150*x^3+75*x^2-75*x)*exp(exp
(25*x^2))+25*x^6-75*x^4+75*x^2-25),x,method=_RETURNVERBOSE)

[Out]

4/25/(x^2-2*x+1)/(x+1+x*exp(exp(25*x^2)))^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (21) = 42\).

Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4}{25 \, {\left (x^{4} - 2 \, x^{2} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (2 \, e^{\left (25 \, x^{2}\right )}\right )} + 2 \, {\left (x^{4} - x^{3} - x^{2} + x\right )} e^{\left (e^{\left (25 \, x^{2}\right )}\right )} + 1\right )}} \]

[In]

integrate((((-400*x^3+400*x^2)*exp(25*x^2)-16*x+8)*exp(exp(25*x^2))-16*x)/((25*x^6-75*x^5+75*x^4-25*x^3)*exp(e
xp(25*x^2))^3+(75*x^6-150*x^5+150*x^3-75*x^2)*exp(exp(25*x^2))^2+(75*x^6-75*x^5-150*x^4+150*x^3+75*x^2-75*x)*e
xp(exp(25*x^2))+25*x^6-75*x^4+75*x^2-25),x, algorithm="fricas")

[Out]

4/25/(x^4 - 2*x^2 + (x^4 - 2*x^3 + x^2)*e^(2*e^(25*x^2)) + 2*(x^4 - x^3 - x^2 + x)*e^(e^(25*x^2)) + 1)

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4}{25 x^{4} - 50 x^{2} + \left (25 x^{4} - 50 x^{3} + 25 x^{2}\right ) e^{2 e^{25 x^{2}}} + \left (50 x^{4} - 50 x^{3} - 50 x^{2} + 50 x\right ) e^{e^{25 x^{2}}} + 25} \]

[In]

integrate((((-400*x**3+400*x**2)*exp(25*x**2)-16*x+8)*exp(exp(25*x**2))-16*x)/((25*x**6-75*x**5+75*x**4-25*x**
3)*exp(exp(25*x**2))**3+(75*x**6-150*x**5+150*x**3-75*x**2)*exp(exp(25*x**2))**2+(75*x**6-75*x**5-150*x**4+150
*x**3+75*x**2-75*x)*exp(exp(25*x**2))+25*x**6-75*x**4+75*x**2-25),x)

[Out]

4/(25*x**4 - 50*x**2 + (25*x**4 - 50*x**3 + 25*x**2)*exp(2*exp(25*x**2)) + (50*x**4 - 50*x**3 - 50*x**2 + 50*x
)*exp(exp(25*x**2)) + 25)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (21) = 42\).

Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4}{25 \, {\left (x^{4} - 2 \, x^{2} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (2 \, e^{\left (25 \, x^{2}\right )}\right )} + 2 \, {\left (x^{4} - x^{3} - x^{2} + x\right )} e^{\left (e^{\left (25 \, x^{2}\right )}\right )} + 1\right )}} \]

[In]

integrate((((-400*x^3+400*x^2)*exp(25*x^2)-16*x+8)*exp(exp(25*x^2))-16*x)/((25*x^6-75*x^5+75*x^4-25*x^3)*exp(e
xp(25*x^2))^3+(75*x^6-150*x^5+150*x^3-75*x^2)*exp(exp(25*x^2))^2+(75*x^6-75*x^5-150*x^4+150*x^3+75*x^2-75*x)*e
xp(exp(25*x^2))+25*x^6-75*x^4+75*x^2-25),x, algorithm="maxima")

[Out]

4/25/(x^4 - 2*x^2 + (x^4 - 2*x^3 + x^2)*e^(2*e^(25*x^2)) + 2*(x^4 - x^3 - x^2 + x)*e^(e^(25*x^2)) + 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (21) = 42\).

Time = 0.36 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.70 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4}{25 \, {\left (x^{4} e^{\left (2 \, e^{\left (25 \, x^{2}\right )}\right )} + 2 \, x^{4} e^{\left (e^{\left (25 \, x^{2}\right )}\right )} + x^{4} - 2 \, x^{3} e^{\left (2 \, e^{\left (25 \, x^{2}\right )}\right )} - 2 \, x^{3} e^{\left (e^{\left (25 \, x^{2}\right )}\right )} + x^{2} e^{\left (2 \, e^{\left (25 \, x^{2}\right )}\right )} - 2 \, x^{2} e^{\left (e^{\left (25 \, x^{2}\right )}\right )} - 2 \, x^{2} + 2 \, x e^{\left (e^{\left (25 \, x^{2}\right )}\right )} + 1\right )}} \]

[In]

integrate((((-400*x^3+400*x^2)*exp(25*x^2)-16*x+8)*exp(exp(25*x^2))-16*x)/((25*x^6-75*x^5+75*x^4-25*x^3)*exp(e
xp(25*x^2))^3+(75*x^6-150*x^5+150*x^3-75*x^2)*exp(exp(25*x^2))^2+(75*x^6-75*x^5-150*x^4+150*x^3+75*x^2-75*x)*e
xp(exp(25*x^2))+25*x^6-75*x^4+75*x^2-25),x, algorithm="giac")

[Out]

4/25/(x^4*e^(2*e^(25*x^2)) + 2*x^4*e^(e^(25*x^2)) + x^4 - 2*x^3*e^(2*e^(25*x^2)) - 2*x^3*e^(e^(25*x^2)) + x^2*
e^(2*e^(25*x^2)) - 2*x^2*e^(e^(25*x^2)) - 2*x^2 + 2*x*e^(e^(25*x^2)) + 1)

Mupad [B] (verification not implemented)

Time = 11.52 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.41 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4\,\left (x-50\,x^2\,{\mathrm {e}}^{25\,x^2}+50\,x^4\,{\mathrm {e}}^{25\,x^2}-1\right )}{25\,{\left (x-1\right )}^3\,\left (50\,x^2\,{\mathrm {e}}^{25\,x^2}+50\,x^3\,{\mathrm {e}}^{25\,x^2}+1\right )\,\left ({\left (x+1\right )}^2+x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^{25\,x^2}}+2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{25\,x^2}}\,\left (x+1\right )\right )} \]

[In]

int((16*x - exp(exp(25*x^2))*(exp(25*x^2)*(400*x^2 - 400*x^3) - 16*x + 8))/(exp(exp(25*x^2))*(75*x - 75*x^2 -
150*x^3 + 150*x^4 + 75*x^5 - 75*x^6) - 75*x^2 + 75*x^4 - 25*x^6 + exp(3*exp(25*x^2))*(25*x^3 - 75*x^4 + 75*x^5
- 25*x^6) + exp(2*exp(25*x^2))*(75*x^2 - 150*x^3 + 150*x^5 - 75*x^6) + 25),x)

[Out]

(4*(x - 50*x^2*exp(25*x^2) + 50*x^4*exp(25*x^2) - 1))/(25*(x - 1)^3*(50*x^2*exp(25*x^2) + 50*x^3*exp(25*x^2) +
1)*((x + 1)^2 + x^2*exp(2*exp(25*x^2)) + 2*x*exp(exp(25*x^2))*(x + 1)))