Integrand size = 135, antiderivative size = 27 \[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=e-\frac {1-e^{\left (-x+(-2+x+\log (2))^2\right )^2}}{x}+x \] Output:
exp(1)+x-(1-exp(((ln(2)-2+x)^2-x)^2))/x
\[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=\int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx \] Input:
Integrate[(1 + x^2 + E^(16 - 40*x + 33*x^2 - 10*x^3 + x^4 + (-32 + 56*x - 28*x^2 + 4*x^3)*Log[2] + (24 - 26*x + 6*x^2)*Log[2]^2 + (-8 + 4*x)*Log[2]^ 3 + Log[2]^4)*(-1 - 40*x + 66*x^2 - 30*x^3 + 4*x^4 + (56*x - 56*x^2 + 12*x ^3)*Log[2] + (-26*x + 12*x^2)*Log[2]^2 + 4*x*Log[2]^3))/x^2,x]
Output:
Integrate[(1 + x^2 + E^(16 - 40*x + 33*x^2 - 10*x^3 + x^4 + (-32 + 56*x - 28*x^2 + 4*x^3)*Log[2] + (24 - 26*x + 6*x^2)*Log[2]^2 + (-8 + 4*x)*Log[2]^ 3 + Log[2]^4)*(-1 - 40*x + 66*x^2 - 30*x^3 + 4*x^4 + (56*x - 56*x^2 + 12*x ^3)*Log[2] + (-26*x + 12*x^2)*Log[2]^2 + 4*x*Log[2]^3))/x^2, x]
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.
| \(\displaystyle \int \frac {\left (4 x^4-30 x^3+66 x^2+\left (12 x^2-26 x\right ) \log ^2(2)+\left (12 x^3-56 x^2+56 x\right ) \log (2)-40 x+4 x \log ^3(2)-1\right ) \exp \left (x^4-10 x^3+33 x^2+\left (6 x^2-26 x+24\right ) \log ^2(2)+\left (4 x^3-28 x^2+56 x-32\right ) \log (2)-40 x+(4 x-8) \log ^3(2)+16+\log ^4(2)\right )+x^2+1}{x^2} \, dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (\frac {16^{x^3-7 x^2+14 x-8} \left (4 x^4-6 x^3 (5-\log (4))+2 x^2 \left (33+6 \log ^2(2)-28 \log (2)\right )-2 x (2-\log (2))^2 (5-\log (4))-1\right ) \exp \left (x^4-10 x^3+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20-2 \log ^3(2)+13 \log ^2(2)\right )+16+\log ^4(2)-8 \log ^3(2)+24 \log ^2(2)\right )}{x^2}+\frac {x^2+1}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {16^{x^3-7 x^2+14 x-8} \left (4 x^4-6 x^3 (5-\log (4))+2 x^2 \left (33+6 \log ^2(2)-28 \log (2)\right )-2 x (2-\log (2))^2 (5-\log (4))-1\right ) \exp \left (x^4-10 x^3+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20-2 \log ^3(2)+13 \log ^2(2)\right )+16+\log ^4(2)-8 \log ^3(2)+24 \log ^2(2)\right )}{x^2}+\frac {x^2+1}{x^2}\right )dx\) |
Input:
Int[(1 + x^2 + E^(16 - 40*x + 33*x^2 - 10*x^3 + x^4 + (-32 + 56*x - 28*x^2 + 4*x^3)*Log[2] + (24 - 26*x + 6*x^2)*Log[2]^2 + (-8 + 4*x)*Log[2]^3 + Lo g[2]^4)*(-1 - 40*x + 66*x^2 - 30*x^3 + 4*x^4 + (56*x - 56*x^2 + 12*x^3)*Lo g[2] + (-26*x + 12*x^2)*Log[2]^2 + 4*x*Log[2]^3))/x^2,x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(27)=54\).
Time = 6.70 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.81
| method | result | size |
| norman | \(\frac {-1+x^{2}+{\mathrm e}^{\ln \left (2\right )^{4}+\left (4 x -8\right ) \ln \left (2\right )^{3}+\left (6 x^{2}-26 x +24\right ) \ln \left (2\right )^{2}+\left (4 x^{3}-28 x^{2}+56 x -32\right ) \ln \left (2\right )+x^{4}-10 x^{3}+33 x^{2}-40 x +16}}{x}\) | \(76\) |
| parallelrisch | \(\frac {-1+x^{2}+{\mathrm e}^{\ln \left (2\right )^{4}+\left (4 x -8\right ) \ln \left (2\right )^{3}+\left (6 x^{2}-26 x +24\right ) \ln \left (2\right )^{2}+\left (4 x^{3}-28 x^{2}+56 x -32\right ) \ln \left (2\right )+x^{4}-10 x^{3}+33 x^{2}-40 x +16}}{x}\) | \(76\) |
| parts | \(x -\frac {1}{x}+\frac {{\mathrm e}^{\ln \left (2\right )^{4}+\left (4 x -8\right ) \ln \left (2\right )^{3}+\left (6 x^{2}-26 x +24\right ) \ln \left (2\right )^{2}+\left (4 x^{3}-28 x^{2}+56 x -32\right ) \ln \left (2\right )+x^{4}-10 x^{3}+33 x^{2}-40 x +16}}{x}\) | \(78\) |
| risch | \(x -\frac {1}{x}+\frac {16^{\left (-1+x \right ) \left (-2+x \right ) \left (x -4\right )} {\mathrm e}^{\ln \left (2\right )^{4}+4 x \ln \left (2\right )^{3}+6 x^{2} \ln \left (2\right )^{2}+x^{4}-8 \ln \left (2\right )^{3}-26 x \ln \left (2\right )^{2}-10 x^{3}+24 \ln \left (2\right )^{2}+33 x^{2}-40 x +16}}{x}\) | \(82\) |
Input:
int(((4*x*ln(2)^3+(12*x^2-26*x)*ln(2)^2+(12*x^3-56*x^2+56*x)*ln(2)+4*x^4-3 0*x^3+66*x^2-40*x-1)*exp(ln(2)^4+(4*x-8)*ln(2)^3+(6*x^2-26*x+24)*ln(2)^2+( 4*x^3-28*x^2+56*x-32)*ln(2)+x^4-10*x^3+33*x^2-40*x+16)+x^2+1)/x^2,x,method =_RETURNVERBOSE)
Output:
(-1+x^2+exp(ln(2)^4+(4*x-8)*ln(2)^3+(6*x^2-26*x+24)*ln(2)^2+(4*x^3-28*x^2+ 56*x-32)*ln(2)+x^4-10*x^3+33*x^2-40*x+16))/x
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (24) = 48\).
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.74 \[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=\frac {x^{2} + e^{\left (x^{4} + 4 \, {\left (x - 2\right )} \log \left (2\right )^{3} + \log \left (2\right )^{4} - 10 \, x^{3} + 2 \, {\left (3 \, x^{2} - 13 \, x + 12\right )} \log \left (2\right )^{2} + 33 \, x^{2} + 4 \, {\left (x^{3} - 7 \, x^{2} + 14 \, x - 8\right )} \log \left (2\right ) - 40 \, x + 16\right )} - 1}{x} \] Input:
integrate(((4*x*log(2)^3+(12*x^2-26*x)*log(2)^2+(12*x^3-56*x^2+56*x)*log(2 )+4*x^4-30*x^3+66*x^2-40*x-1)*exp(log(2)^4+(4*x-8)*log(2)^3+(6*x^2-26*x+24 )*log(2)^2+(4*x^3-28*x^2+56*x-32)*log(2)+x^4-10*x^3+33*x^2-40*x+16)+x^2+1) /x^2,x, algorithm="fricas")
Output:
(x^2 + e^(x^4 + 4*(x - 2)*log(2)^3 + log(2)^4 - 10*x^3 + 2*(3*x^2 - 13*x + 12)*log(2)^2 + 33*x^2 + 4*(x^3 - 7*x^2 + 14*x - 8)*log(2) - 40*x + 16) - 1)/x
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.78 \[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=x + \frac {e^{x^{4} - 10 x^{3} + 33 x^{2} - 40 x + \left (4 x - 8\right ) \log {\left (2 \right )}^{3} + \left (6 x^{2} - 26 x + 24\right ) \log {\left (2 \right )}^{2} + \left (4 x^{3} - 28 x^{2} + 56 x - 32\right ) \log {\left (2 \right )} + \log {\left (2 \right )}^{4} + 16}}{x} - \frac {1}{x} \] Input:
integrate(((4*x*ln(2)**3+(12*x**2-26*x)*ln(2)**2+(12*x**3-56*x**2+56*x)*ln (2)+4*x**4-30*x**3+66*x**2-40*x-1)*exp(ln(2)**4+(4*x-8)*ln(2)**3+(6*x**2-2 6*x+24)*ln(2)**2+(4*x**3-28*x**2+56*x-32)*ln(2)+x**4-10*x**3+33*x**2-40*x+ 16)+x**2+1)/x**2,x)
Output:
x + exp(x**4 - 10*x**3 + 33*x**2 - 40*x + (4*x - 8)*log(2)**3 + (6*x**2 - 26*x + 24)*log(2)**2 + (4*x**3 - 28*x**2 + 56*x - 32)*log(2) + log(2)**4 + 16)/x - 1/x
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (24) = 48\).
Time = 0.32 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.30 \[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=x + \frac {e^{\left (x^{4} + 4 \, x^{3} \log \left (2\right ) + 6 \, x^{2} \log \left (2\right )^{2} + 4 \, x \log \left (2\right )^{3} + \log \left (2\right )^{4} - 10 \, x^{3} - 28 \, x^{2} \log \left (2\right ) - 26 \, x \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3} + 33 \, x^{2} + 56 \, x \log \left (2\right ) + 24 \, \log \left (2\right )^{2} - 40 \, x + 16\right )}}{4294967296 \, x} - \frac {1}{x} \] Input:
integrate(((4*x*log(2)^3+(12*x^2-26*x)*log(2)^2+(12*x^3-56*x^2+56*x)*log(2 )+4*x^4-30*x^3+66*x^2-40*x-1)*exp(log(2)^4+(4*x-8)*log(2)^3+(6*x^2-26*x+24 )*log(2)^2+(4*x^3-28*x^2+56*x-32)*log(2)+x^4-10*x^3+33*x^2-40*x+16)+x^2+1) /x^2,x, algorithm="maxima")
Output:
x + 1/4294967296*e^(x^4 + 4*x^3*log(2) + 6*x^2*log(2)^2 + 4*x*log(2)^3 + l og(2)^4 - 10*x^3 - 28*x^2*log(2) - 26*x*log(2)^2 - 8*log(2)^3 + 33*x^2 + 5 6*x*log(2) + 24*log(2)^2 - 40*x + 16)/x - 1/x
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (24) = 48\).
Time = 0.38 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.30 \[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=\frac {4294967296 \, x^{2} + e^{\left (x^{4} + 4 \, x^{3} \log \left (2\right ) + 6 \, x^{2} \log \left (2\right )^{2} + 4 \, x \log \left (2\right )^{3} + \log \left (2\right )^{4} - 10 \, x^{3} - 28 \, x^{2} \log \left (2\right ) - 26 \, x \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3} + 33 \, x^{2} + 56 \, x \log \left (2\right ) + 24 \, \log \left (2\right )^{2} - 40 \, x + 16\right )} - 4294967296}{4294967296 \, x} \] Input:
integrate(((4*x*log(2)^3+(12*x^2-26*x)*log(2)^2+(12*x^3-56*x^2+56*x)*log(2 )+4*x^4-30*x^3+66*x^2-40*x-1)*exp(log(2)^4+(4*x-8)*log(2)^3+(6*x^2-26*x+24 )*log(2)^2+(4*x^3-28*x^2+56*x-32)*log(2)+x^4-10*x^3+33*x^2-40*x+16)+x^2+1) /x^2,x, algorithm="giac")
Output:
1/4294967296*(4294967296*x^2 + e^(x^4 + 4*x^3*log(2) + 6*x^2*log(2)^2 + 4* x*log(2)^3 + log(2)^4 - 10*x^3 - 28*x^2*log(2) - 26*x*log(2)^2 - 8*log(2)^ 3 + 33*x^2 + 56*x*log(2) + 24*log(2)^2 - 40*x + 16) - 4294967296)/x
Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.70 \[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=x-\frac {1}{x}+\frac {2^{56\,x}\,2^{4\,x^3}\,{\mathrm {e}}^{4\,x\,{\ln \left (2\right )}^3}\,{\mathrm {e}}^{-26\,x\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{{\ln \left (2\right )}^4}\,{\mathrm {e}}^{-40\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{6\,x^2\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{-8\,{\ln \left (2\right )}^3}\,{\mathrm {e}}^{24\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{-10\,x^3}\,{\mathrm {e}}^{33\,x^2}}{4294967296\,2^{28\,x^2}\,x} \] Input:
int((x^2 - exp(log(2)^3*(4*x - 8) - 40*x + log(2)*(56*x - 28*x^2 + 4*x^3 - 32) + log(2)^2*(6*x^2 - 26*x + 24) + log(2)^4 + 33*x^2 - 10*x^3 + x^4 + 1 6)*(40*x - log(2)*(56*x - 56*x^2 + 12*x^3) + log(2)^2*(26*x - 12*x^2) - 4* x*log(2)^3 - 66*x^2 + 30*x^3 - 4*x^4 + 1) + 1)/x^2,x)
Output:
x - 1/x + (2^(56*x)*2^(4*x^3)*exp(4*x*log(2)^3)*exp(-26*x*log(2)^2)*exp(lo g(2)^4)*exp(-40*x)*exp(x^4)*exp(16)*exp(6*x^2*log(2)^2)*exp(-8*log(2)^3)*e xp(24*log(2)^2)*exp(-10*x^3)*exp(33*x^2))/(4294967296*2^(28*x^2)*x)
Time = 0.58 (sec) , antiderivative size = 163, normalized size of antiderivative = 6.04 \[ \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx=\frac {e^{\mathrm {log}\left (2\right )^{4}+4 \mathrm {log}\left (2\right )^{3} x +6 \mathrm {log}\left (2\right )^{2} x^{2}+24 \mathrm {log}\left (2\right )^{2}+x^{4}+33 x^{2}} 2^{56 x} 2^{4 x^{3}} e^{16}+4294967296 e^{8 \mathrm {log}\left (2\right )^{3}+26 \mathrm {log}\left (2\right )^{2} x +10 x^{3}+40 x} 2^{28 x^{2}} x^{2}-4294967296 e^{8 \mathrm {log}\left (2\right )^{3}+26 \mathrm {log}\left (2\right )^{2} x +10 x^{3}+40 x} 2^{28 x^{2}}}{4294967296 e^{8 \mathrm {log}\left (2\right )^{3}+26 \mathrm {log}\left (2\right )^{2} x +10 x^{3}+40 x} 2^{28 x^{2}} x} \] Input:
int(((4*x*log(2)^3+(12*x^2-26*x)*log(2)^2+(12*x^3-56*x^2+56*x)*log(2)+4*x^ 4-30*x^3+66*x^2-40*x-1)*exp(log(2)^4+(4*x-8)*log(2)^3+(6*x^2-26*x+24)*log( 2)^2+(4*x^3-28*x^2+56*x-32)*log(2)+x^4-10*x^3+33*x^2-40*x+16)+x^2+1)/x^2,x )
Output:
(e**(log(2)**4 + 4*log(2)**3*x + 6*log(2)**2*x**2 + 24*log(2)**2 + x**4 + 33*x**2)*2**(56*x)*2**(4*x**3)*e**16 + 4294967296*e**(8*log(2)**3 + 26*log (2)**2*x + 10*x**3 + 40*x)*2**(28*x**2)*x**2 - 4294967296*e**(8*log(2)**3 + 26*log(2)**2*x + 10*x**3 + 40*x)*2**(28*x**2))/(4294967296*e**(8*log(2)* *3 + 26*log(2)**2*x + 10*x**3 + 40*x)*2**(28*x**2)*x)