Integrand size = 85, antiderivative size = 31 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=x+4 \left (x (2+x)+e^{e^4} \left (-2 x+e^5 \left (4+x^2\right )\right )\right )^2 \] Output:
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)))
Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(31)=62\).
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.00 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=x \left (1+16 x+16 e^{2 e^4} x+16 x^2+4 x^3-16 e^{e^4} x (2+x)-16 e^{5+2 e^4} \left (4+x^2\right )+4 e^{2 \left (5+e^4\right )} x \left (8+x^2\right )+8 e^{5+e^4} \left (8+4 x+2 x^2+x^3\right )\right ) \] Input:
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]
Output:
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))
Leaf count is larger than twice the leaf count of optimal. \(149\) vs. \(2(31)=62\).
Time = 0.37 (sec) , antiderivative size = 149, normalized size of antiderivative = 4.81, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (16 x^3+48 x^2+e^{2 e^4} \left (e^{10} \left (16 x^3+64 x\right )+e^5 \left (-48 x^2-64\right )+32 x\right )+e^{e^4} \left (-48 x^2+e^5 \left (32 x^3+48 x^2+64 x+64\right )-64 x\right )+32 x+1\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 e^{2 \left (5+e^4\right )} x^4+8 e^{5+e^4} x^4+4 x^4-16 e^{5+2 e^4} x^3+16 e^{5+e^4} x^3-16 e^{e^4} x^3+16 x^3+32 e^{2 \left (5+e^4\right )} x^2+32 e^{5+e^4} x^2+16 e^{2 e^4} x^2-32 e^{e^4} x^2+16 x^2-64 e^{5+2 e^4} x+64 e^{5+e^4} x+x\) |
Input:
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]
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(50)=100\).
Time = 0.32 (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 | \(x \left (4 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{3}+32 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x -16 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}+8 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x^{3}+16 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x^{2}-64 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}}+32 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x +16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x -16 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+4 x^{3}+64 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-32 x \,{\mathrm e}^{{\mathrm e}^{4}}+16 x^{2}+16 x +1\right )\) | \(122\) |
default | \(4 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{4}+32 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{3}+8 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x^{4}+16 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x^{3}-64 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x +32 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x^{2}+16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 x^{3} {\mathrm e}^{{\mathrm e}^{4}}+4 x^{4}+64 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x -32 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+16 x^{3}+16 x^{2}+x\) | \(132\) |
parallelrisch | \(4 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{4}+32 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{3}+8 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x^{4}+16 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x^{3}-64 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x +32 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x^{2}+16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 x^{3} {\mathrm e}^{{\mathrm e}^{4}}+4 x^{4}+64 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x -32 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+16 x^{3}+16 x^{2}+x\) | \(132\) |
parts | \(4 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{4}+32 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{3}+8 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x^{4}+16 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x^{3}-64 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x +32 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x^{2}+16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 x^{3} {\mathrm e}^{{\mathrm e}^{4}}+4 x^{4}+64 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}} x -32 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+16 x^{3}+16 x^{2}+x\) | \(132\) |
risch | \(4 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{4}+32 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}-16 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{3}+8 x^{4} {\mathrm e}^{5+{\mathrm e}^{4}}-16 x^{3} {\mathrm e}^{{\mathrm e}^{4}}+16 x^{3} {\mathrm e}^{5+{\mathrm e}^{4}}+4 x^{4}+64 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}}-64 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x -32 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}+32 x^{2} {\mathrm e}^{5+{\mathrm e}^{4}}+16 x^{3}+64 x \,{\mathrm e}^{5+{\mathrm e}^{4}}+16 x^{2}+x\) | \(143\) |
Input:
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)
Output:
(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(exp(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*exp (5)*exp(exp(4))^2+64*exp(5)*exp(exp(4))+1)*x
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (28) = 56\).
Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.90 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=4 \, x^{4} + 16 \, x^{3} + 16 \, x^{2} + 4 \, {\left (4 \, x^{2} + {\left (x^{4} + 8 \, x^{2}\right )} e^{10} - 4 \, {\left (x^{3} + 4 \, x\right )} e^{5}\right )} e^{\left (2 \, e^{4}\right )} - 8 \, {\left (2 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 2 \, x^{3} + 4 \, x^{2} + 8 \, x\right )} e^{5}\right )} e^{\left (e^{4}\right )} + x \] Input:
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")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (51) = 102\).
Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.13 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=x^{4} \cdot \left (4 + 8 e^{5} e^{e^{4}} + 4 e^{10} e^{2 e^{4}}\right ) + x^{3} \left (- 16 e^{5} e^{2 e^{4}} - 16 e^{e^{4}} + 16 + 16 e^{5} e^{e^{4}}\right ) + x^{2} \left (- 32 e^{e^{4}} + 16 + 32 e^{5} e^{e^{4}} + 16 e^{2 e^{4}} + 32 e^{10} e^{2 e^{4}}\right ) + x \left (- 64 e^{5} e^{2 e^{4}} + 1 + 64 e^{5} e^{e^{4}}\right ) \] Input:
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)
Output:
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*(-3 2*exp(exp(4)) + 16 + 32*exp(5)*exp(exp(4)) + 16*exp(2*exp(4)) + 32*exp(10) *exp(2*exp(4))) + x*(-64*exp(5)*exp(2*exp(4)) + 1 + 64*exp(5)*exp(exp(4)))
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (28) = 56\).
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.90 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=4 \, x^{4} + 16 \, x^{3} + 16 \, x^{2} + 4 \, {\left (4 \, x^{2} + {\left (x^{4} + 8 \, x^{2}\right )} e^{10} - 4 \, {\left (x^{3} + 4 \, x\right )} e^{5}\right )} e^{\left (2 \, e^{4}\right )} - 8 \, {\left (2 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 2 \, x^{3} + 4 \, x^{2} + 8 \, x\right )} e^{5}\right )} e^{\left (e^{4}\right )} + x \] Input:
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")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (28) = 56\).
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.90 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=4 \, x^{4} + 16 \, x^{3} + 16 \, x^{2} + 4 \, {\left (4 \, x^{2} + {\left (x^{4} + 8 \, x^{2}\right )} e^{10} - 4 \, {\left (x^{3} + 4 \, x\right )} e^{5}\right )} e^{\left (2 \, e^{4}\right )} - 8 \, {\left (2 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 2 \, x^{3} + 4 \, x^{2} + 8 \, x\right )} e^{5}\right )} e^{\left (e^{4}\right )} + x \] Input:
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")
Output:
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
Time = 2.48 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.19 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=\left (8\,{\mathrm {e}}^{{\mathrm {e}}^4+5}+4\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+10}+4\right )\,x^4+\left (\frac {{\mathrm {e}}^{{\mathrm {e}}^4}\,\left (48\,{\mathrm {e}}^5-48\right )}{3}-16\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+5}+16\right )\,x^3+\left (\frac {{\mathrm {e}}^{{\mathrm {e}}^4}\,\left (64\,{\mathrm {e}}^5-64\right )}{2}+\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^4}\,\left (64\,{\mathrm {e}}^{10}+32\right )}{2}+16\right )\,x^2+\left (64\,{\mathrm {e}}^{{\mathrm {e}}^4+5}-64\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+5}+1\right )\,x \] Input:
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)
Output:
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))*(64*exp(10) + 32))/2 + 16) + x*(64*exp(e xp(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)
Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 4.74 \[ \int \left (1+32 x+48 x^2+16 x^3+e^{2 e^4} \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right )+e^{e^4} \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right )\right ) \, dx=x \left (4 e^{2 e^{4}} e^{10} x^{3}+32 e^{2 e^{4}} e^{10} x -16 e^{2 e^{4}} e^{5} x^{2}-64 e^{2 e^{4}} e^{5}+16 e^{2 e^{4}} x +8 e^{e^{4}} e^{5} x^{3}+16 e^{e^{4}} e^{5} x^{2}+32 e^{e^{4}} e^{5} x +64 e^{e^{4}} e^{5}-16 e^{e^{4}} x^{2}-32 e^{e^{4}} x +4 x^{3}+16 x^{2}+16 x +1\right ) \] Input:
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)
Output:
x*(4*e**(2*e**4)*e**10*x**3 + 32*e**(2*e**4)*e**10*x - 16*e**(2*e**4)*e**5 *x**2 - 64*e**(2*e**4)*e**5 + 16*e**(2*e**4)*x + 8*e**(e**4)*e**5*x**3 + 1 6*e**(e**4)*e**5*x**2 + 32*e**(e**4)*e**5*x + 64*e**(e**4)*e**5 - 16*e**(e **4)*x**2 - 32*e**(e**4)*x + 4*x**3 + 16*x**2 + 16*x + 1)