\(\int (1+32 x+48 x^2+16 x^3+e^{2 e^4} (32 x+e^5 (-64-48 x^2)+e^{10} (64 x+16 x^3))+e^{e^4} (-64 x-48 x^2+e^5 (64+64 x+48 x^2+32 x^3))) \, dx\) [3319]

Optimal result

Integrand size = 85, antiderivative size = 31 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=x+4 \left (x (2+x)+e^{e^4} \left (-2 x+e^5 \left (4+x^2\right )\right )\right )^2 \]

[Out]

x+2*(x*(2+x)+(exp(5)*(x^2+4)-2*x)*exp(exp(4)))*(2*x*(2+x)+2*(exp(5)*(x^2+4)-2*x)*exp(exp(4)))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(149\) vs. \(2(31)=62\).

Time = 0.03 (sec) , antiderivative size = 149, normalized size of antiderivative = 4.81, number of steps used = 6, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=4 e^{2 \left (5+e^4\right )} x^4+8 e^{5+e^4} x^4+4 x^4-16 e^{5+2 e^4} x^3+16 e^{5+e^4} x^3-16 e^{e^4} x^3+16 x^3+32 e^{2 \left (5+e^4\right )} x^2+32 e^{5+e^4} x^2+16 e^{2 e^4} x^2-32 e^{e^4} x^2+16 x^2-64 e^{5+2 e^4} x+64 e^{5+e^4} x+x \]

[In]

Int[1 + 32*x + 48*x^2 + 16*x^3 + E^(2*E^4)*(32*x + E^5*(-64 - 48*x^2) + E^10*(64*x + 16*x^3)) + E^E^4*(-64*x -
48*x^2 + E^5*(64 + 64*x + 48*x^2 + 32*x^3)),x]

[Out]

x + 64*E^(5 + E^4)*x - 64*E^(5 + 2*E^4)*x + 16*x^2 - 32*E^E^4*x^2 + 16*E^(2*E^4)*x^2 + 32*E^(5 + E^4)*x^2 + 32
*E^(2*(5 + E^4))*x^2 + 16*x^3 - 16*E^E^4*x^3 + 16*E^(5 + E^4)*x^3 - 16*E^(5 + 2*E^4)*x^3 + 4*x^4 + 8*E^(5 + E^
4)*x^4 + 4*E^(2*(5 + E^4))*x^4

Rubi steps \begin{align*} \text {integral}& = x+16 x^2+16 x^3+4 x^4+e^{e^4} \int \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right ) \, dx+e^{2 e^4} \int \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right ) \, dx \\ & = x+16 x^2-32 e^{e^4} x^2+16 e^{2 e^4} x^2+16 x^3-16 e^{e^4} x^3+4 x^4+e^{5+e^4} \int \left (64+64 x+48 x^2+32 x^3\right ) \, dx+e^{2 \left (5+e^4\right )} \int \left (64 x+16 x^3\right ) \, dx+e^{5+2 e^4} \int \left (-64-48 x^2\right ) \, dx \\ & = x+64 e^{5+e^4} x-64 e^{5+2 e^4} x+16 x^2-32 e^{e^4} x^2+16 e^{2 e^4} x^2+32 e^{5+e^4} x^2+32 e^{2 \left (5+e^4\right )} x^2+16 x^3-16 e^{e^4} x^3+16 e^{5+e^4} x^3-16 e^{5+2 e^4} x^3+4 x^4+8 e^{5+e^4} x^4+4 e^{2 \left (5+e^4\right )} x^4 \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(31)=62\).

Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.00 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=x \left (1+16 x+16 e^{2 e^4} x+16 x^2+4 x^3-16 e^{e^4} x (2+x)-16 e^{5+2 e^4} \left (4+x^2\right )+4 e^{2 \left (5+e^4\right )} x \left (8+x^2\right )+8 e^{5+e^4} \left (8+4 x+2 x^2+x^3\right )\right ) \]

[In]

Integrate[1 + 32*x + 48*x^2 + 16*x^3 + E^(2*E^4)*(32*x + E^5*(-64 - 48*x^2) + E^10*(64*x + 16*x^3)) + E^E^4*(-
64*x - 48*x^2 + E^5*(64 + 64*x + 48*x^2 + 32*x^3)),x]

[Out]

x*(1 + 16*x + 16*E^(2*E^4)*x + 16*x^2 + 4*x^3 - 16*E^E^4*x*(2 + x) - 16*E^(5 + 2*E^4)*(4 + x^2) + 4*E^(2*(5 +
E^4))*x*(8 + x^2) + 8*E^(5 + E^4)*(8 + 4*x + 2*x^2 + x^3))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(50)=100\).

Time = 0.74 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.52

method	result	size
norman	\(\left (4 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}}+8 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+4\right ) x^{4}+\left (-16 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}}+16 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-16 \,{\mathrm e}^{{\mathrm e}^{4}}+16\right ) x^{3}+\left (32 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}}+32 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}-32 \,{\mathrm e}^{{\mathrm e}^{4}}+16\right ) x^{2}+\left (-64 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}}+64 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+1\right ) x\)	\(109\)
gosper	\(x \left (4 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{3}+32 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x -16 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}+8 x^{3} {\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+16 x^{2} {\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-64 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}}+32 x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x -16 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+4 x^{3}+64 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-32 x \,{\mathrm e}^{{\mathrm e}^{4}}+16 x^{2}+16 x +1\right )\)	\(122\)
default	\(4 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{4}+32 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{3}+8 x^{4} {\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+16 x^{3} {\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-64 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x +32 x^{2} {\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 x^{3} {\mathrm e}^{{\mathrm e}^{4}}+4 x^{4}+64 x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-32 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+16 x^{3}+16 x^{2}+x\)	\(132\)
parallelrisch	\(4 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{4}+32 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{3}+8 x^{4} {\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+16 x^{3} {\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-64 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x +32 x^{2} {\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 x^{3} {\mathrm e}^{{\mathrm e}^{4}}+4 x^{4}+64 x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-32 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+16 x^{3}+16 x^{2}+x\)	\(132\)
parts	\(4 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{4}+32 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{3}+8 x^{4} {\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+16 x^{3} {\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-64 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x +32 x^{2} {\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 x^{3} {\mathrm e}^{{\mathrm e}^{4}}+4 x^{4}+64 x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-32 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+16 x^{3}+16 x^{2}+x\)	\(132\)
risch	\(4 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{4}+32 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{3}+8 x^{4} {\mathrm e}^{5+{\mathrm e}^{4}}+16 x^{3} {\mathrm e}^{5+{\mathrm e}^{4}}-16 x^{3} {\mathrm e}^{{\mathrm e}^{4}}+4 x^{4}+64 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}}-64 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x +16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}+32 x^{2} {\mathrm e}^{5+{\mathrm e}^{4}}-32 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+16 x^{3}+64 x \,{\mathrm e}^{5+{\mathrm e}^{4}}+16 x^{2}+x\)	\(143\)

[In]

int(((16*x^3+64*x)*exp(5)^2+(-48*x^2-64)*exp(5)+32*x)*exp(exp(4))^2+((32*x^3+48*x^2+64*x+64)*exp(5)-48*x^2-64*
x)*exp(exp(4))+16*x^3+48*x^2+32*x+1,x,method=_RETURNVERBOSE)

[Out]

(4*exp(5)^2*exp(exp(4))^2+8*exp(5)*exp(exp(4))+4)*x^4+(-16*exp(5)*exp(exp(4))^2+16*exp(5)*exp(exp(4))-16*exp(e
xp(4))+16)*x^3+(32*exp(5)^2*exp(exp(4))^2+32*exp(5)*exp(exp(4))+16*exp(exp(4))^2-32*exp(exp(4))+16)*x^2+(-64*e
xp(5)*exp(exp(4))^2+64*exp(5)*exp(exp(4))+1)*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (28) = 56\).

Time = 0.23 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.90 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=4 \, x^{4} + 16 \, x^{3} + 16 \, x^{2} + 4 \, {\left (4 \, x^{2} + {\left (x^{4} + 8 \, x^{2}\right )} e^{10} - 4 \, {\left (x^{3} + 4 \, x\right )} e^{5}\right )} e^{\left (2 \, e^{4}\right )} - 8 \, {\left (2 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 2 \, x^{3} + 4 \, x^{2} + 8 \, x\right )} e^{5}\right )} e^{\left (e^{4}\right )} + x \]

[In]

integrate(((16*x^3+64*x)*exp(5)^2+(-48*x^2-64)*exp(5)+32*x)*exp(exp(4))^2+((32*x^3+48*x^2+64*x+64)*exp(5)-48*x
^2-64*x)*exp(exp(4))+16*x^3+48*x^2+32*x+1,x, algorithm="fricas")

[Out]

4*x^4 + 16*x^3 + 16*x^2 + 4*(4*x^2 + (x^4 + 8*x^2)*e^10 - 4*(x^3 + 4*x)*e^5)*e^(2*e^4) - 8*(2*x^3 + 4*x^2 - (x
^4 + 2*x^3 + 4*x^2 + 8*x)*e^5)*e^(e^4) + x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (51) = 102\).

Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.13 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=x^{4} \cdot \left (4 + 8 e^{5} e^{e^{4}} + 4 e^{10} e^{2 e^{4}}\right ) + x^{3} \left (- 16 e^{5} e^{2 e^{4}} - 16 e^{e^{4}} + 16 + 16 e^{5} e^{e^{4}}\right ) + x^{2} \left (- 32 e^{e^{4}} + 16 + 32 e^{5} e^{e^{4}} + 16 e^{2 e^{4}} + 32 e^{10} e^{2 e^{4}}\right ) + x \left (- 64 e^{5} e^{2 e^{4}} + 1 + 64 e^{5} e^{e^{4}}\right ) \]

[In]

integrate(((16*x**3+64*x)*exp(5)**2+(-48*x**2-64)*exp(5)+32*x)*exp(exp(4))**2+((32*x**3+48*x**2+64*x+64)*exp(5
)-48*x**2-64*x)*exp(exp(4))+16*x**3+48*x**2+32*x+1,x)

[Out]

x**4*(4 + 8*exp(5)*exp(exp(4)) + 4*exp(10)*exp(2*exp(4))) + x**3*(-16*exp(5)*exp(2*exp(4)) - 16*exp(exp(4)) +
16 + 16*exp(5)*exp(exp(4))) + x**2*(-32*exp(exp(4)) + 16 + 32*exp(5)*exp(exp(4)) + 16*exp(2*exp(4)) + 32*exp(1
0)*exp(2*exp(4))) + x*(-64*exp(5)*exp(2*exp(4)) + 1 + 64*exp(5)*exp(exp(4)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (28) = 56\).

Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.90 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=4 \, x^{4} + 16 \, x^{3} + 16 \, x^{2} + 4 \, {\left (4 \, x^{2} + {\left (x^{4} + 8 \, x^{2}\right )} e^{10} - 4 \, {\left (x^{3} + 4 \, x\right )} e^{5}\right )} e^{\left (2 \, e^{4}\right )} - 8 \, {\left (2 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 2 \, x^{3} + 4 \, x^{2} + 8 \, x\right )} e^{5}\right )} e^{\left (e^{4}\right )} + x \]

[In]

integrate(((16*x^3+64*x)*exp(5)^2+(-48*x^2-64)*exp(5)+32*x)*exp(exp(4))^2+((32*x^3+48*x^2+64*x+64)*exp(5)-48*x
^2-64*x)*exp(exp(4))+16*x^3+48*x^2+32*x+1,x, algorithm="maxima")

[Out]

4*x^4 + 16*x^3 + 16*x^2 + 4*(4*x^2 + (x^4 + 8*x^2)*e^10 - 4*(x^3 + 4*x)*e^5)*e^(2*e^4) - 8*(2*x^3 + 4*x^2 - (x
^4 + 2*x^3 + 4*x^2 + 8*x)*e^5)*e^(e^4) + x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (28) = 56\).

Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.90 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=4 \, x^{4} + 16 \, x^{3} + 16 \, x^{2} + 4 \, {\left (4 \, x^{2} + {\left (x^{4} + 8 \, x^{2}\right )} e^{10} - 4 \, {\left (x^{3} + 4 \, x\right )} e^{5}\right )} e^{\left (2 \, e^{4}\right )} - 8 \, {\left (2 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 2 \, x^{3} + 4 \, x^{2} + 8 \, x\right )} e^{5}\right )} e^{\left (e^{4}\right )} + x \]

[In]

integrate(((16*x^3+64*x)*exp(5)^2+(-48*x^2-64)*exp(5)+32*x)*exp(exp(4))^2+((32*x^3+48*x^2+64*x+64)*exp(5)-48*x
^2-64*x)*exp(exp(4))+16*x^3+48*x^2+32*x+1,x, algorithm="giac")

[Out]

4*x^4 + 16*x^3 + 16*x^2 + 4*(4*x^2 + (x^4 + 8*x^2)*e^10 - 4*(x^3 + 4*x)*e^5)*e^(2*e^4) - 8*(2*x^3 + 4*x^2 - (x
^4 + 2*x^3 + 4*x^2 + 8*x)*e^5)*e^(e^4) + x

Mupad [B] (verification not implemented)

Time = 8.55 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.19 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=\left (8\,{\mathrm {e}}^{{\mathrm {e}}^4+5}+4\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+10}+4\right )\,x^4+\left (\frac {{\mathrm {e}}^{{\mathrm {e}}^4}\,\left (48\,{\mathrm {e}}^5-48\right )}{3}-16\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+5}+16\right )\,x^3+\left (\frac {{\mathrm {e}}^{{\mathrm {e}}^4}\,\left (64\,{\mathrm {e}}^5-64\right )}{2}+\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^4}\,\left (64\,{\mathrm {e}}^{10}+32\right )}{2}+16\right )\,x^2+\left (64\,{\mathrm {e}}^{{\mathrm {e}}^4+5}-64\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+5}+1\right )\,x \]

[In]

int(32*x - exp(exp(4))*(64*x - exp(5)*(64*x + 48*x^2 + 32*x^3 + 64) + 48*x^2) + exp(2*exp(4))*(32*x + exp(10)*
(64*x + 16*x^3) - exp(5)*(48*x^2 + 64)) + 48*x^2 + 16*x^3 + 1,x)

[Out]

x^4*(8*exp(exp(4) + 5) + 4*exp(2*exp(4) + 10) + 4) + x^2*((exp(exp(4))*(64*exp(5) - 64))/2 + (exp(2*exp(4))*(6
4*exp(10) + 32))/2 + 16) + x*(64*exp(exp(4) + 5) - 64*exp(2*exp(4) + 5) + 1) + x^3*((exp(exp(4))*(48*exp(5) -
48))/3 - 16*exp(2*exp(4) + 5) + 16)