Integrand size = 101, antiderivative size = 24 \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=\log \left (\frac {e^{2 x}}{(-4+x)^2}-4 \left (x-\frac {x}{\log (16)}\right )\right ) \] Output:
ln(exp(x)^2/(-4+x)^2-4*x+x/ln(2))
Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(24)=48\).
Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17 \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=-2 \log (4-x)+\log \left (-64 x+32 x^2-4 x^3-e^{2 x} \log (16)+64 x \log (16)-32 x^2 \log (16)+4 x^3 \log (16)\right ) \] Input:
Integrate[(-256 + 192*x - 48*x^2 + 4*x^3 + E^(2*x)*(-10 + 2*x)*Log[16] + ( 256 - 192*x + 48*x^2 - 4*x^3)*Log[16])/(-256*x + 192*x^2 - 48*x^3 + 4*x^4 + E^(2*x)*(-4 + x)*Log[16] + (256*x - 192*x^2 + 48*x^3 - 4*x^4)*Log[16]),x ]
Output:
-2*Log[4 - x] + Log[-64*x + 32*x^2 - 4*x^3 - E^(2*x)*Log[16] + 64*x*Log[16 ] - 32*x^2*Log[16] + 4*x^3*Log[16]]
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.
\(\displaystyle \int \frac {4 x^3-48 x^2+\left (-4 x^3+48 x^2-192 x+256\right ) \log (16)+192 x+e^{2 x} (2 x-10) \log (16)-256}{4 x^4-48 x^3+192 x^2+\left (-4 x^4+48 x^3-192 x^2+256 x\right ) \log (16)-256 x+e^{2 x} (x-4) \log (16)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-4 x^3+48 x^2-\left (-4 x^3+48 x^2-192 x+256\right ) \log (16)-192 x-e^{2 x} (2 x-10) \log (16)+256}{(4-x) \left (4 x^3 (1-\log (16))-32 x^2 (1-\log (16))+64 x (1-\log (16))+e^{2 x} \log (16)\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 \left (-2 x^3+19 x^2-48 x+16\right ) (1-\log (16))}{4 x^3 (1-\log (16))-32 x^2 (1-\log (16))+64 x (1-\log (16))+e^{2 x} \log (16)}+\frac {2 (x-5)}{x-4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 192 (1-\log (16)) \int \frac {x}{-4 (1-\log (16)) x^3+32 (1-\log (16)) x^2-64 (1-\log (16)) x-e^{2 x} \log (16)}dx+8 (1-\log (16)) \int \frac {x^3}{-4 (1-\log (16)) x^3+32 (1-\log (16)) x^2-64 (1-\log (16)) x-e^{2 x} \log (16)}dx+64 (1-\log (16)) \int \frac {1}{4 (1-\log (16)) x^3-32 (1-\log (16)) x^2+64 (1-\log (16)) x+e^{2 x} \log (16)}dx+76 (1-\log (16)) \int \frac {x^2}{4 (1-\log (16)) x^3-32 (1-\log (16)) x^2+64 (1-\log (16)) x+e^{2 x} \log (16)}dx+2 x-2 \log (4-x)\) |
Input:
Int[(-256 + 192*x - 48*x^2 + 4*x^3 + E^(2*x)*(-10 + 2*x)*Log[16] + (256 - 192*x + 48*x^2 - 4*x^3)*Log[16])/(-256*x + 192*x^2 - 48*x^3 + 4*x^4 + E^(2 *x)*(-4 + x)*Log[16] + (256*x - 192*x^2 + 48*x^3 - 4*x^4)*Log[16]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(21)=42\).
Time = 0.59 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96
method | result | size |
risch | \(-2 \ln \left (x -4\right )+\ln \left ({\mathrm e}^{2 x}-\frac {x \left (4 x^{2} \ln \left (2\right )-32 x \ln \left (2\right )-x^{2}+64 \ln \left (2\right )+8 x -16\right )}{\ln \left (2\right )}\right )\) | \(47\) |
norman | \(-2 \ln \left (x -4\right )+\ln \left (4 x^{3} \ln \left (2\right )-32 x^{2} \ln \left (2\right )-\ln \left (2\right ) {\mathrm e}^{2 x}-x^{3}+64 x \ln \left (2\right )+8 x^{2}-16 x \right )\) | \(50\) |
parallelrisch | \(\ln \left (\frac {4 x^{3} \ln \left (2\right )-32 x^{2} \ln \left (2\right )-\ln \left (2\right ) {\mathrm e}^{2 x}-x^{3}+64 x \ln \left (2\right )+8 x^{2}-16 x}{4 \ln \left (2\right )-1}\right )-2 \ln \left (x -4\right )\) | \(59\) |
Input:
int((4*(2*x-10)*ln(2)*exp(x)^2+4*(-4*x^3+48*x^2-192*x+256)*ln(2)+4*x^3-48* x^2+192*x-256)/(4*(x-4)*ln(2)*exp(x)^2+4*(-4*x^4+48*x^3-192*x^2+256*x)*ln( 2)+4*x^4-48*x^3+192*x^2-256*x),x,method=_RETURNVERBOSE)
Output:
-2*ln(x-4)+ln(exp(2*x)-x*(4*x^2*ln(2)-32*x*ln(2)-x^2+64*ln(2)+8*x-16)/ln(2 ))
Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=\log \left (x^{3} - 8 \, x^{2} - 4 \, {\left (x^{3} - 8 \, x^{2} + 16 \, x\right )} \log \left (2\right ) + e^{\left (2 \, x\right )} \log \left (2\right ) + 16 \, x\right ) - 2 \, \log \left (x - 4\right ) \] Input:
integrate((4*(2*x-10)*log(2)*exp(x)^2+4*(-4*x^3+48*x^2-192*x+256)*log(2)+4 *x^3-48*x^2+192*x-256)/(4*(-4+x)*log(2)*exp(x)^2+4*(-4*x^4+48*x^3-192*x^2+ 256*x)*log(2)+4*x^4-48*x^3+192*x^2-256*x),x, algorithm="fricas")
Output:
log(x^3 - 8*x^2 - 4*(x^3 - 8*x^2 + 16*x)*log(2) + e^(2*x)*log(2) + 16*x) - 2*log(x - 4)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
Time = 0.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=- 2 \log {\left (x - 4 \right )} + \log {\left (\frac {- 4 x^{3} \log {\left (2 \right )} + x^{3} - 8 x^{2} + 32 x^{2} \log {\left (2 \right )} - 64 x \log {\left (2 \right )} + 16 x}{\log {\left (2 \right )}} + e^{2 x} \right )} \] Input:
integrate((4*(2*x-10)*ln(2)*exp(x)**2+4*(-4*x**3+48*x**2-192*x+256)*ln(2)+ 4*x**3-48*x**2+192*x-256)/(4*(-4+x)*ln(2)*exp(x)**2+4*(-4*x**4+48*x**3-192 *x**2+256*x)*ln(2)+4*x**4-48*x**3+192*x**2-256*x),x)
Output:
-2*log(x - 4) + log((-4*x**3*log(2) + x**3 - 8*x**2 + 32*x**2*log(2) - 64* x*log(2) + 16*x)/log(2) + exp(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (21) = 42\).
Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.21 \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=-2 \, \log \left (x - 4\right ) + \log \left (-\frac {x^{3} {\left (4 \, \log \left (2\right ) - 1\right )} - 8 \, x^{2} {\left (4 \, \log \left (2\right ) - 1\right )} + 16 \, x {\left (4 \, \log \left (2\right ) - 1\right )} - e^{\left (2 \, x\right )} \log \left (2\right )}{\log \left (2\right )}\right ) \] Input:
integrate((4*(2*x-10)*log(2)*exp(x)^2+4*(-4*x^3+48*x^2-192*x+256)*log(2)+4 *x^3-48*x^2+192*x-256)/(4*(-4+x)*log(2)*exp(x)^2+4*(-4*x^4+48*x^3-192*x^2+ 256*x)*log(2)+4*x^4-48*x^3+192*x^2-256*x),x, algorithm="maxima")
Output:
-2*log(x - 4) + log(-(x^3*(4*log(2) - 1) - 8*x^2*(4*log(2) - 1) + 16*x*(4* log(2) - 1) - e^(2*x)*log(2))/log(2))
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=\log \left (-4 \, x^{3} \log \left (2\right ) + x^{3} + 32 \, x^{2} \log \left (2\right ) - 8 \, x^{2} - 64 \, x \log \left (2\right ) + e^{\left (2 \, x\right )} \log \left (2\right ) + 16 \, x\right ) - 2 \, \log \left (x - 4\right ) \] Input:
integrate((4*(2*x-10)*log(2)*exp(x)^2+4*(-4*x^3+48*x^2-192*x+256)*log(2)+4 *x^3-48*x^2+192*x-256)/(4*(-4+x)*log(2)*exp(x)^2+4*(-4*x^4+48*x^3-192*x^2+ 256*x)*log(2)+4*x^4-48*x^3+192*x^2-256*x),x, algorithm="giac")
Output:
log(-4*x^3*log(2) + x^3 + 32*x^2*log(2) - 8*x^2 - 64*x*log(2) + e^(2*x)*lo g(2) + 16*x) - 2*log(x - 4)
Timed out. \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=\int \frac {192\,x-4\,\ln \left (2\right )\,\left (4\,x^3-48\,x^2+192\,x-256\right )-48\,x^2+4\,x^3+4\,{\mathrm {e}}^{2\,x}\,\ln \left (2\right )\,\left (2\,x-10\right )-256}{4\,\ln \left (2\right )\,\left (-4\,x^4+48\,x^3-192\,x^2+256\,x\right )-256\,x+192\,x^2-48\,x^3+4\,x^4+4\,{\mathrm {e}}^{2\,x}\,\ln \left (2\right )\,\left (x-4\right )} \,d x \] Input:
int((192*x - 4*log(2)*(192*x - 48*x^2 + 4*x^3 - 256) - 48*x^2 + 4*x^3 + 4* exp(2*x)*log(2)*(2*x - 10) - 256)/(4*log(2)*(256*x - 192*x^2 + 48*x^3 - 4* x^4) - 256*x + 192*x^2 - 48*x^3 + 4*x^4 + 4*exp(2*x)*log(2)*(x - 4)),x)
Output:
int((192*x - 4*log(2)*(192*x - 48*x^2 + 4*x^3 - 256) - 48*x^2 + 4*x^3 + 4* exp(2*x)*log(2)*(2*x - 10) - 256)/(4*log(2)*(256*x - 192*x^2 + 48*x^3 - 4* x^4) - 256*x + 192*x^2 - 48*x^3 + 4*x^4 + 4*exp(2*x)*log(2)*(x - 4)), x)
\[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx =\text {Too large to display} \] Input:
int((4*(2*x-10)*log(2)*exp(x)^2+4*(-4*x^3+48*x^2-192*x+256)*log(2)+4*x^3-4 8*x^2+192*x-256)/(4*(-4+x)*log(2)*exp(x)^2+4*(-4*x^4+48*x^3-192*x^2+256*x) *log(2)+4*x^4-48*x^3+192*x^2-256*x),x)
Output:
- 10*int(e**(2*x)/(e**(2*x)*log(2)*x - 4*e**(2*x)*log(2) - 4*log(2)*x**4 + 48*log(2)*x**3 - 192*log(2)*x**2 + 256*log(2)*x + x**4 - 12*x**3 + 48*x* *2 - 64*x),x)*log(2) - 4*int(x**3/(e**(2*x)*log(2)*x - 4*e**(2*x)*log(2) - 4*log(2)*x**4 + 48*log(2)*x**3 - 192*log(2)*x**2 + 256*log(2)*x + x**4 - 12*x**3 + 48*x**2 - 64*x),x)*log(2) + int(x**3/(e**(2*x)*log(2)*x - 4*e**( 2*x)*log(2) - 4*log(2)*x**4 + 48*log(2)*x**3 - 192*log(2)*x**2 + 256*log(2 )*x + x**4 - 12*x**3 + 48*x**2 - 64*x),x) + 48*int(x**2/(e**(2*x)*log(2)*x - 4*e**(2*x)*log(2) - 4*log(2)*x**4 + 48*log(2)*x**3 - 192*log(2)*x**2 + 256*log(2)*x + x**4 - 12*x**3 + 48*x**2 - 64*x),x)*log(2) - 12*int(x**2/(e **(2*x)*log(2)*x - 4*e**(2*x)*log(2) - 4*log(2)*x**4 + 48*log(2)*x**3 - 19 2*log(2)*x**2 + 256*log(2)*x + x**4 - 12*x**3 + 48*x**2 - 64*x),x) + 2*int ((e**(2*x)*x)/(e**(2*x)*log(2)*x - 4*e**(2*x)*log(2) - 4*log(2)*x**4 + 48* log(2)*x**3 - 192*log(2)*x**2 + 256*log(2)*x + x**4 - 12*x**3 + 48*x**2 - 64*x),x)*log(2) - 192*int(x/(e**(2*x)*log(2)*x - 4*e**(2*x)*log(2) - 4*log (2)*x**4 + 48*log(2)*x**3 - 192*log(2)*x**2 + 256*log(2)*x + x**4 - 12*x** 3 + 48*x**2 - 64*x),x)*log(2) + 48*int(x/(e**(2*x)*log(2)*x - 4*e**(2*x)*l og(2) - 4*log(2)*x**4 + 48*log(2)*x**3 - 192*log(2)*x**2 + 256*log(2)*x + x**4 - 12*x**3 + 48*x**2 - 64*x),x) + 256*int(1/(e**(2*x)*log(2)*x - 4*e** (2*x)*log(2) - 4*log(2)*x**4 + 48*log(2)*x**3 - 192*log(2)*x**2 + 256*log( 2)*x + x**4 - 12*x**3 + 48*x**2 - 64*x),x)*log(2) - 64*int(1/(e**(2*x)*...