Integrand size = 110, antiderivative size = 29 \[ \int \frac {33-48 x+18 x^2-2 x^3+e^5 \left (1-16 x+20 x^2-4 x^3\right )+e^{10} \left (2 x^2-2 x^3\right )+e^{x+x^2} \left (-5-7 x+2 x^2+e^5 \left (-1+x+2 x^2\right )\right )}{16-8 x+x^2+e^{10} x^2+e^5 \left (-8 x+2 x^2\right )} \, dx=2 x-x^2+\frac {-1+e^{x+x^2}}{-4+x+e^5 x} \]
Time = 5.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {33-48 x+18 x^2-2 x^3+e^5 \left (1-16 x+20 x^2-4 x^3\right )+e^{10} \left (2 x^2-2 x^3\right )+e^{x+x^2} \left (-5-7 x+2 x^2+e^5 \left (-1+x+2 x^2\right )\right )}{16-8 x+x^2+e^{10} x^2+e^5 \left (-8 x+2 x^2\right )} \, dx=2 x-x^2+\frac {-1+e^{x+x^2}}{-4+x+e^5 x} \]
Integrate[(33 - 48*x + 18*x^2 - 2*x^3 + E^5*(1 - 16*x + 20*x^2 - 4*x^3) + E^10*(2*x^2 - 2*x^3) + E^(x + x^2)*(-5 - 7*x + 2*x^2 + E^5*(-1 + x + 2*x^2 )))/(16 - 8*x + x^2 + E^10*x^2 + E^5*(-8*x + 2*x^2)),x]
Leaf count is larger than twice the leaf count of optimal. \(400\) vs. \(2(29)=58\).
Time = 1.47 (sec) , antiderivative size = 400, normalized size of antiderivative = 13.79, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {6, 7292, 7277, 27, 7293, 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.
\(\displaystyle \int \frac {-2 x^3+18 x^2+e^{x^2+x} \left (2 x^2+e^5 \left (2 x^2+x-1\right )-7 x-5\right )+e^5 \left (-4 x^3+20 x^2-16 x+1\right )+e^{10} \left (2 x^2-2 x^3\right )-48 x+33}{e^{10} x^2+x^2+e^5 \left (2 x^2-8 x\right )-8 x+16} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-2 x^3+18 x^2+e^{x^2+x} \left (2 x^2+e^5 \left (2 x^2+x-1\right )-7 x-5\right )+e^5 \left (-4 x^3+20 x^2-16 x+1\right )+e^{10} \left (2 x^2-2 x^3\right )-48 x+33}{\left (1+e^{10}\right ) x^2+e^5 \left (2 x^2-8 x\right )-8 x+16}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-2 x^3+18 x^2+e^{x^2+x} \left (2 x^2+e^5 \left (2 x^2+x-1\right )-7 x-5\right )+e^5 \left (-4 x^3+20 x^2-16 x+1\right )+e^{10} \left (2 x^2-2 x^3\right )-48 x+33}{\left (1+e^5\right )^2 x^2-8 \left (1+e^5\right ) x+16}dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 4 \left (1+e^5\right )^2 \int \frac {-2 x^3+18 x^2-48 x+e^5 \left (-4 x^3+20 x^2-16 x+1\right )+2 e^{10} \left (x^2-x^3\right )-e^{x^2+x} \left (-2 x^2+7 x+e^5 \left (-2 x^2-x+1\right )+5\right )+33}{4 \left (1+e^5\right )^2 \left (4-\left (1+e^5\right ) x\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {-2 x^3+18 x^2-e^{x^2+x} \left (-2 x^2+e^5 \left (-2 x^2-x+1\right )+7 x+5\right )+e^5 \left (-4 x^3+20 x^2-16 x+1\right )+2 e^{10} \left (x^2-x^3\right )-48 x+33}{\left (4-\left (1+e^5\right ) x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2 x^3}{\left (4-\left (1+e^5\right ) x\right )^2}+\frac {2 e^{10} (1-x) x^2}{\left (4-\left (1+e^5\right ) x\right )^2}+\frac {18 x^2}{\left (4-\left (1+e^5\right ) x\right )^2}+\frac {e^{x^2+x} \left (2 \left (1+e^5\right ) x^2-\left (7-e^5\right ) x-e^5-5\right )}{\left (4-\left (1+e^5\right ) x\right )^2}+\frac {e^5 \left (-4 x^3+20 x^2-16 x+1\right )}{\left (4-\left (1+e^5\right ) x\right )^2}-\frac {48 x}{\left (4-\left (1+e^5\right ) x\right )^2}+\frac {33}{\left (4-\left (1+e^5\right ) x\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e^{10} x^2}{\left (1+e^5\right )^2}-\frac {2 e^5 x^2}{\left (1+e^5\right )^2}-\frac {x^2}{\left (1+e^5\right )^2}-\frac {e^{x^2+x}}{4-\left (1+e^5\right ) x}+\frac {18 x}{\left (1+e^5\right )^2}-\frac {2 e^{10} \left (7-e^5\right ) x}{\left (1+e^5\right )^3}-\frac {4 e^5 \left (3-5 e^5\right ) x}{\left (1+e^5\right )^3}-\frac {16 x}{\left (1+e^5\right )^3}+\frac {e^5 \left (1+195 e^5-61 e^{10}+e^{15}\right )}{\left (1+e^5\right )^4 \left (4-\left (1+e^5\right ) x\right )}+\frac {33}{\left (1+e^5\right ) \left (4-\left (1+e^5\right ) x\right )}-\frac {192}{\left (1+e^5\right )^2 \left (4-\left (1+e^5\right ) x\right )}+\frac {288}{\left (1+e^5\right )^3 \left (4-\left (1+e^5\right ) x\right )}-\frac {32 e^{10} \left (3-e^5\right )}{\left (1+e^5\right )^4 \left (4-\left (1+e^5\right ) x\right )}-\frac {128}{\left (1+e^5\right )^4 \left (4-\left (1+e^5\right ) x\right )}-\frac {16 e^5 \left (3-8 e^5+e^{10}\right ) \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^4}-\frac {48 \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^2}+\frac {144 \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^3}-\frac {16 e^{10} \left (5-e^5\right ) \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^4}-\frac {96 \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^4}\) |
Int[(33 - 48*x + 18*x^2 - 2*x^3 + E^5*(1 - 16*x + 20*x^2 - 4*x^3) + E^10*( 2*x^2 - 2*x^3) + E^(x + x^2)*(-5 - 7*x + 2*x^2 + E^5*(-1 + x + 2*x^2)))/(1 6 - 8*x + x^2 + E^10*x^2 + E^5*(-8*x + 2*x^2)),x]
(-16*x)/(1 + E^5)^3 - (4*E^5*(3 - 5*E^5)*x)/(1 + E^5)^3 - (2*E^10*(7 - E^5 )*x)/(1 + E^5)^3 + (18*x)/(1 + E^5)^2 - x^2/(1 + E^5)^2 - (2*E^5*x^2)/(1 + E^5)^2 - (E^10*x^2)/(1 + E^5)^2 - E^(x + x^2)/(4 - (1 + E^5)*x) - 128/((1 + E^5)^4*(4 - (1 + E^5)*x)) - (32*E^10*(3 - E^5))/((1 + E^5)^4*(4 - (1 + E^5)*x)) + 288/((1 + E^5)^3*(4 - (1 + E^5)*x)) - 192/((1 + E^5)^2*(4 - (1 + E^5)*x)) + 33/((1 + E^5)*(4 - (1 + E^5)*x)) + (E^5*(1 + 195*E^5 - 61*E^1 0 + E^15))/((1 + E^5)^4*(4 - (1 + E^5)*x)) - (96*Log[4 - (1 + E^5)*x])/(1 + E^5)^4 - (16*E^10*(5 - E^5)*Log[4 - (1 + E^5)*x])/(1 + E^5)^4 + (144*Log [4 - (1 + E^5)*x])/(1 + E^5)^3 - (48*Log[4 - (1 + E^5)*x])/(1 + E^5)^2 - ( 16*E^5*(3 - 8*E^5 + E^10)*Log[4 - (1 + E^5)*x])/(1 + E^5)^4
3.26.76.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Time = 0.86 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28
method | result | size |
risch | \(-x^{2}+2 x -\frac {1}{x +x \,{\mathrm e}^{5}-4}+\frac {{\mathrm e}^{\left (1+x \right ) x}}{x +x \,{\mathrm e}^{5}-4}\) | \(37\) |
norman | \(\frac {\left (-{\mathrm e}^{5}-1\right ) x^{3}+\left (2 \,{\mathrm e}^{5}+6\right ) x^{2}+\left (-\frac {33}{4}-\frac {{\mathrm e}^{5}}{4}\right ) x +{\mathrm e}^{x^{2}+x}}{x +x \,{\mathrm e}^{5}-4}\) | \(46\) |
parallelrisch | \(-\frac {4 x^{3} {\mathrm e}^{5}-8 x^{2} {\mathrm e}^{5}+4 x^{3}+x \,{\mathrm e}^{5}-24 x^{2}+33 x -4 \,{\mathrm e}^{x^{2}+x}}{4 \left (x +x \,{\mathrm e}^{5}-4\right )}\) | \(52\) |
parts | \(\frac {{\mathrm e}^{x^{2}+x}}{x +x \,{\mathrm e}^{5}-4}+\frac {\left (-{\mathrm e}^{5}-1\right ) x^{3}+\left (2 \,{\mathrm e}^{5}+6\right ) x^{2}+\left (-\frac {33}{4}-\frac {{\mathrm e}^{5}}{4}\right ) x}{x +x \,{\mathrm e}^{5}-4}\) | \(57\) |
int((((2*x^2+x-1)*exp(5)+2*x^2-7*x-5)*exp(x^2+x)+(-2*x^3+2*x^2)*exp(5)^2+( -4*x^3+20*x^2-16*x+1)*exp(5)-2*x^3+18*x^2-48*x+33)/(x^2*exp(5)^2+(2*x^2-8* x)*exp(5)+x^2-8*x+16),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {33-48 x+18 x^2-2 x^3+e^5 \left (1-16 x+20 x^2-4 x^3\right )+e^{10} \left (2 x^2-2 x^3\right )+e^{x+x^2} \left (-5-7 x+2 x^2+e^5 \left (-1+x+2 x^2\right )\right )}{16-8 x+x^2+e^{10} x^2+e^5 \left (-8 x+2 x^2\right )} \, dx=-\frac {x^{3} - 6 \, x^{2} + {\left (x^{3} - 2 \, x^{2}\right )} e^{5} + 8 \, x - e^{\left (x^{2} + x\right )} + 1}{x e^{5} + x - 4} \]
integrate((((2*x^2+x-1)*exp(5)+2*x^2-7*x-5)*exp(x^2+x)+(-2*x^3+2*x^2)*exp( 5)^2+(-4*x^3+20*x^2-16*x+1)*exp(5)-2*x^3+18*x^2-48*x+33)/(x^2*exp(5)^2+(2* x^2-8*x)*exp(5)+x^2-8*x+16),x, algorithm=\
Time = 0.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {33-48 x+18 x^2-2 x^3+e^5 \left (1-16 x+20 x^2-4 x^3\right )+e^{10} \left (2 x^2-2 x^3\right )+e^{x+x^2} \left (-5-7 x+2 x^2+e^5 \left (-1+x+2 x^2\right )\right )}{16-8 x+x^2+e^{10} x^2+e^5 \left (-8 x+2 x^2\right )} \, dx=- x^{2} + 2 x + \frac {e^{x^{2} + x}}{x + x e^{5} - 4} - \frac {1}{x \left (1 + e^{5}\right ) - 4} \]
integrate((((2*x**2+x-1)*exp(5)+2*x**2-7*x-5)*exp(x**2+x)+(-2*x**3+2*x**2) *exp(5)**2+(-4*x**3+20*x**2-16*x+1)*exp(5)-2*x**3+18*x**2-48*x+33)/(x**2*e xp(5)**2+(2*x**2-8*x)*exp(5)+x**2-8*x+16),x)
Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 703, normalized size of antiderivative = 24.24 \[ \int \frac {33-48 x+18 x^2-2 x^3+e^5 \left (1-16 x+20 x^2-4 x^3\right )+e^{10} \left (2 x^2-2 x^3\right )+e^{x+x^2} \left (-5-7 x+2 x^2+e^5 \left (-1+x+2 x^2\right )\right )}{16-8 x+x^2+e^{10} x^2+e^5 \left (-8 x+2 x^2\right )} \, dx =\text {Too large to display} \]
integrate((((2*x^2+x-1)*exp(5)+2*x^2-7*x-5)*exp(x^2+x)+(-2*x^3+2*x^2)*exp( 5)^2+(-4*x^3+20*x^2-16*x+1)*exp(5)-2*x^3+18*x^2-48*x+33)/(x^2*exp(5)^2+(2* x^2-8*x)*exp(5)+x^2-8*x+16),x, algorithm=\
-((x^2*(e^5 + 1) + 16*x)/(e^15 + 3*e^10 + 3*e^5 + 1) + 96*log(x*(e^5 + 1) - 4)/(e^20 + 4*e^15 + 6*e^10 + 4*e^5 + 1) - 128/(x*(e^25 + 5*e^20 + 10*e^1 5 + 10*e^10 + 5*e^5 + 1) - 4*e^20 - 16*e^15 - 24*e^10 - 16*e^5 - 4))*e^10 + 2*(x/(e^10 + 2*e^5 + 1) + 8*log(x*(e^5 + 1) - 4)/(e^15 + 3*e^10 + 3*e^5 + 1) - 16/(x*(e^20 + 4*e^15 + 6*e^10 + 4*e^5 + 1) - 4*e^15 - 12*e^10 - 12* e^5 - 4))*e^10 - 2*((x^2*(e^5 + 1) + 16*x)/(e^15 + 3*e^10 + 3*e^5 + 1) + 9 6*log(x*(e^5 + 1) - 4)/(e^20 + 4*e^15 + 6*e^10 + 4*e^5 + 1) - 128/(x*(e^25 + 5*e^20 + 10*e^15 + 10*e^10 + 5*e^5 + 1) - 4*e^20 - 16*e^15 - 24*e^10 - 16*e^5 - 4))*e^5 + 20*(x/(e^10 + 2*e^5 + 1) + 8*log(x*(e^5 + 1) - 4)/(e^15 + 3*e^10 + 3*e^5 + 1) - 16/(x*(e^20 + 4*e^15 + 6*e^10 + 4*e^5 + 1) - 4*e^ 15 - 12*e^10 - 12*e^5 - 4))*e^5 - 16*(log(x*(e^5 + 1) - 4)/(e^10 + 2*e^5 + 1) - 4/(x*(e^15 + 3*e^10 + 3*e^5 + 1) - 4*e^10 - 8*e^5 - 4))*e^5 - (x^2*( e^5 + 1) + 16*x)/(e^15 + 3*e^10 + 3*e^5 + 1) + 18*x/(e^10 + 2*e^5 + 1) - e ^5/(x*(e^10 + 2*e^5 + 1) - 4*e^5 - 4) + e^(x^2 + x)/(x*(e^5 + 1) - 4) - 96 *log(x*(e^5 + 1) - 4)/(e^20 + 4*e^15 + 6*e^10 + 4*e^5 + 1) + 144*log(x*(e^ 5 + 1) - 4)/(e^15 + 3*e^10 + 3*e^5 + 1) - 48*log(x*(e^5 + 1) - 4)/(e^10 + 2*e^5 + 1) + 128/(x*(e^25 + 5*e^20 + 10*e^15 + 10*e^10 + 5*e^5 + 1) - 4*e^ 20 - 16*e^15 - 24*e^10 - 16*e^5 - 4) - 288/(x*(e^20 + 4*e^15 + 6*e^10 + 4* e^5 + 1) - 4*e^15 - 12*e^10 - 12*e^5 - 4) + 192/(x*(e^15 + 3*e^10 + 3*e^5 + 1) - 4*e^10 - 8*e^5 - 4) - 33/(x*(e^10 + 2*e^5 + 1) - 4*e^5 - 4)
Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {33-48 x+18 x^2-2 x^3+e^5 \left (1-16 x+20 x^2-4 x^3\right )+e^{10} \left (2 x^2-2 x^3\right )+e^{x+x^2} \left (-5-7 x+2 x^2+e^5 \left (-1+x+2 x^2\right )\right )}{16-8 x+x^2+e^{10} x^2+e^5 \left (-8 x+2 x^2\right )} \, dx=-\frac {x^{3} e^{5} + x^{3} - 2 \, x^{2} e^{5} - 6 \, x^{2} + 8 \, x - e^{\left (x^{2} + x\right )} + 1}{x e^{5} + x - 4} \]
integrate((((2*x^2+x-1)*exp(5)+2*x^2-7*x-5)*exp(x^2+x)+(-2*x^3+2*x^2)*exp( 5)^2+(-4*x^3+20*x^2-16*x+1)*exp(5)-2*x^3+18*x^2-48*x+33)/(x^2*exp(5)^2+(2* x^2-8*x)*exp(5)+x^2-8*x+16),x, algorithm=\
Time = 11.79 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {33-48 x+18 x^2-2 x^3+e^5 \left (1-16 x+20 x^2-4 x^3\right )+e^{10} \left (2 x^2-2 x^3\right )+e^{x+x^2} \left (-5-7 x+2 x^2+e^5 \left (-1+x+2 x^2\right )\right )}{16-8 x+x^2+e^{10} x^2+e^5 \left (-8 x+2 x^2\right )} \, dx=2\,x+\frac {{\mathrm {e}}^{x^2+x}-1}{x\,\left ({\mathrm {e}}^5+1\right )-4}-x^2 \]