Integrand size = 75, antiderivative size = 32 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=5-e^{\left (5+(4+x)^2\right ) \left (\frac {4}{x}+\log (x)\right )}-\frac {1}{2 x}+2 x \] Output:
2*x-exp((ln(x)+4/x)*(5+(4+x)^2))+5-1/2/x
Time = 2.58 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=-\frac {1}{2 x}+2 x-e^{32+\frac {84}{x}+4 x} x^{21+x (8+x)} \] Input:
Integrate[(1 + 4*x^2 + E^((84 + 32*x + 4*x^2 + (21*x + 8*x^2 + x^3)*Log[x] )/x)*(168 - 42*x - 24*x^2 - 2*x^3 + (-16*x^2 - 4*x^3)*Log[x]))/(2*x^2),x]
Output:
-1/2*1/x + 2*x - E^(32 + 84/x + 4*x)*x^(21 + x*(8 + 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.
\(\displaystyle \int \frac {\left (-2 x^3-24 x^2+\left (-4 x^3-16 x^2\right ) \log (x)-42 x+168\right ) \exp \left (\frac {4 x^2+\left (x^3+8 x^2+21 x\right ) \log (x)+32 x+84}{x}\right )+4 x^2+1}{2 x^2} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {2 e^{\frac {4 \left (x^2+8 x+21\right )}{x}} \left (-x^3-12 x^2-21 x-2 \left (x^3+4 x^2\right ) \log (x)+84\right ) x^{\frac {x^3+8 x^2+21 x}{x}}+4 x^2+1}{x^2}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {1}{2} \int \left (\frac {4 x^2+1}{x^2}-2 e^{4 x+32+\frac {84}{x}} x^{x^2+8 x+19} \left (2 \log (x) x^3+x^3+8 \log (x) x^2+12 x^2+21 x-84\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (168 \int e^{4 x+32+\frac {84}{x}} x^{x^2+8 x+19}dx-42 \int e^{4 x+32+\frac {84}{x}} x^{x^2+8 x+20}dx-24 \int e^{4 x+32+\frac {84}{x}} x^{x^2+8 x+21}dx-2 \int e^{4 x+32+\frac {84}{x}} x^{x^2+8 x+22}dx+16 \int \frac {\int e^{4 \left (x+8+\frac {21}{x}\right )} x^{x^2+8 x+21}dx}{x}dx+4 \int \frac {\int e^{4 \left (x+8+\frac {21}{x}\right )} x^{x^2+8 x+22}dx}{x}dx-16 \log (x) \int e^{4 x+32+\frac {84}{x}} x^{x^2+8 x+21}dx-4 \log (x) \int e^{4 x+32+\frac {84}{x}} x^{x^2+8 x+22}dx+4 x-\frac {1}{x}\right )\) |
Input:
Int[(1 + 4*x^2 + E^((84 + 32*x + 4*x^2 + (21*x + 8*x^2 + x^3)*Log[x])/x)*( 168 - 42*x - 24*x^2 - 2*x^3 + (-16*x^2 - 4*x^3)*Log[x]))/(2*x^2),x]
Output:
$Aborted
Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97
method | result | size |
risch | \(-{\mathrm e}^{\frac {\left (x^{2}+8 x +21\right ) \left (x \ln \left (x \right )+4\right )}{x}}+2 x -\frac {1}{2 x}\) | \(31\) |
default | \(-{\mathrm e}^{\frac {\left (x^{3}+8 x^{2}+21 x \right ) \ln \left (x \right )+4 x^{2}+32 x +84}{x}}+2 x -\frac {1}{2 x}\) | \(42\) |
parts | \(-{\mathrm e}^{\frac {\left (x^{3}+8 x^{2}+21 x \right ) \ln \left (x \right )+4 x^{2}+32 x +84}{x}}+2 x -\frac {1}{2 x}\) | \(42\) |
norman | \(\frac {-\frac {1}{2}+2 x^{2}-x \,{\mathrm e}^{\frac {\left (x^{3}+8 x^{2}+21 x \right ) \ln \left (x \right )+4 x^{2}+32 x +84}{x}}}{x}\) | \(45\) |
parallelrisch | \(-\frac {-4 x^{2}+2 x \,{\mathrm e}^{\frac {\left (x^{3}+8 x^{2}+21 x \right ) \ln \left (x \right )+4 x^{2}+32 x +84}{x}}+1}{2 x}\) | \(46\) |
Input:
int(1/2*(((-4*x^3-16*x^2)*ln(x)-2*x^3-24*x^2-42*x+168)*exp(((x^3+8*x^2+21* x)*ln(x)+4*x^2+32*x+84)/x)+4*x^2+1)/x^2,x,method=_RETURNVERBOSE)
Output:
-exp((x^2+8*x+21)*(x*ln(x)+4)/x)+2*x-1/2/x
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=\frac {4 \, x^{2} - 2 \, x e^{\left (\frac {4 \, x^{2} + {\left (x^{3} + 8 \, x^{2} + 21 \, x\right )} \log \left (x\right ) + 32 \, x + 84}{x}\right )} - 1}{2 \, x} \] Input:
integrate(1/2*(((-4*x^3-16*x^2)*log(x)-2*x^3-24*x^2-42*x+168)*exp(((x^3+8* x^2+21*x)*log(x)+4*x^2+32*x+84)/x)+4*x^2+1)/x^2,x, algorithm="fricas")
Output:
1/2*(4*x^2 - 2*x*e^((4*x^2 + (x^3 + 8*x^2 + 21*x)*log(x) + 32*x + 84)/x) - 1)/x
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=2 x - e^{\frac {4 x^{2} + 32 x + \left (x^{3} + 8 x^{2} + 21 x\right ) \log {\left (x \right )} + 84}{x}} - \frac {1}{2 x} \] Input:
integrate(1/2*(((-4*x**3-16*x**2)*ln(x)-2*x**3-24*x**2-42*x+168)*exp(((x** 3+8*x**2+21*x)*ln(x)+4*x**2+32*x+84)/x)+4*x**2+1)/x**2,x)
Output:
2*x - exp((4*x**2 + 32*x + (x**3 + 8*x**2 + 21*x)*log(x) + 84)/x) - 1/(2*x )
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=-x^{21} e^{\left (x^{2} \log \left (x\right ) + 8 \, x \log \left (x\right ) + 4 \, x + \frac {84}{x} + 32\right )} + 2 \, x - \frac {1}{2 \, x} \] Input:
integrate(1/2*(((-4*x^3-16*x^2)*log(x)-2*x^3-24*x^2-42*x+168)*exp(((x^3+8* x^2+21*x)*log(x)+4*x^2+32*x+84)/x)+4*x^2+1)/x^2,x, algorithm="maxima")
Output:
-x^21*e^(x^2*log(x) + 8*x*log(x) + 4*x + 84/x + 32) + 2*x - 1/2/x
Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=2 \, x - \frac {1}{2 \, x} - e^{\left (x^{2} \log \left (x\right ) + 8 \, x \log \left (x\right ) + 4 \, x + \frac {84}{x} + 21 \, \log \left (x\right ) + 32\right )} \] Input:
integrate(1/2*(((-4*x^3-16*x^2)*log(x)-2*x^3-24*x^2-42*x+168)*exp(((x^3+8* x^2+21*x)*log(x)+4*x^2+32*x+84)/x)+4*x^2+1)/x^2,x, algorithm="giac")
Output:
2*x - 1/2/x - e^(x^2*log(x) + 8*x*log(x) + 4*x + 84/x + 21*log(x) + 32)
Time = 3.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=2\,x-\frac {1}{2\,x}-x^{8\,x}\,x^{x^2}\,x^{21}\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{32}\,{\mathrm {e}}^{84/x} \] Input:
int((2*x^2 - (exp((32*x + log(x)*(21*x + 8*x^2 + x^3) + 4*x^2 + 84)/x)*(42 *x + log(x)*(16*x^2 + 4*x^3) + 24*x^2 + 2*x^3 - 168))/2 + 1/2)/x^2,x)
Output:
2*x - 1/(2*x) - x^(8*x)*x^(x^2)*x^21*exp(4*x)*exp(32)*exp(84/x)
Time = 4.62 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=\frac {-2 x^{x^{2}+8 x} e^{\frac {4 x^{2}+84}{x}} e^{32} x^{22}+4 x^{2}-1}{2 x} \] Input:
int(1/2*(((-4*x^3-16*x^2)*log(x)-2*x^3-24*x^2-42*x+168)*exp(((x^3+8*x^2+21 *x)*log(x)+4*x^2+32*x+84)/x)+4*x^2+1)/x^2,x)
Output:
( - 2*x**(x**2 + 8*x)*e**((4*x**2 + 84)/x)*e**32*x**22 + 4*x**2 - 1)/(2*x)