Integrand size = 187, antiderivative size = 27 \[ \int \frac {164-36 e^4+32 x+e^x \left (48+4 x-2 x^2+e^4 (-12+2 x)\right )+\left (16-4 e^4+4 x\right ) \log \left (4-e^4+x\right )}{-256+e^{2 x} \left (-4+e^4-x\right )-192 x-48 x^2-4 x^3+e^4 \left (64+32 x+4 x^2\right )+e^x \left (-64-32 x-4 x^2+e^4 (16+4 x)\right )+\left (-128+e^x \left (-16+4 e^4-4 x\right )-64 x-8 x^2+e^4 (32+8 x)\right ) \log \left (4-e^4+x\right )+\left (-16+4 e^4-4 x\right ) \log ^2\left (4-e^4+x\right )} \, dx=\frac {5-x}{4+\frac {e^x}{2}+x+\log \left (4-e^4+x\right )} \]
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {164-36 e^4+32 x+e^x \left (48+4 x-2 x^2+e^4 (-12+2 x)\right )+\left (16-4 e^4+4 x\right ) \log \left (4-e^4+x\right )}{-256+e^{2 x} \left (-4+e^4-x\right )-192 x-48 x^2-4 x^3+e^4 \left (64+32 x+4 x^2\right )+e^x \left (-64-32 x-4 x^2+e^4 (16+4 x)\right )+\left (-128+e^x \left (-16+4 e^4-4 x\right )-64 x-8 x^2+e^4 (32+8 x)\right ) \log \left (4-e^4+x\right )+\left (-16+4 e^4-4 x\right ) \log ^2\left (4-e^4+x\right )} \, dx=\frac {2 (5-x)}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \]
Integrate[(164 - 36*E^4 + 32*x + E^x*(48 + 4*x - 2*x^2 + E^4*(-12 + 2*x)) + (16 - 4*E^4 + 4*x)*Log[4 - E^4 + x])/(-256 + E^(2*x)*(-4 + E^4 - x) - 19 2*x - 48*x^2 - 4*x^3 + E^4*(64 + 32*x + 4*x^2) + E^x*(-64 - 32*x - 4*x^2 + E^4*(16 + 4*x)) + (-128 + E^x*(-16 + 4*E^4 - 4*x) - 64*x - 8*x^2 + E^4*(3 2 + 8*x))*Log[4 - E^4 + x] + (-16 + 4*E^4 - 4*x)*Log[4 - E^4 + 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.
\(\displaystyle \int \frac {e^x \left (-2 x^2+4 x+e^4 (2 x-12)+48\right )+32 x+\left (4 x-4 e^4+16\right ) \log \left (x-e^4+4\right )-36 e^4+164}{-4 x^3-48 x^2+e^4 \left (4 x^2+32 x+64\right )+e^x \left (-4 x^2-32 x+e^4 (4 x+16)-64\right )+\left (-8 x^2-64 x+e^x \left (-4 x+4 e^4-16\right )+e^4 (8 x+32)-128\right ) \log \left (x-e^4+4\right )-192 x+e^{2 x} \left (-x+e^4-4\right )+\left (-4 x+4 e^4-16\right ) \log ^2\left (x-e^4+4\right )-256} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 \left (-e^x \left (-x^2+2 x+24\right )-e^{x+4} (x-6)-16 x-\left (2 x-2 e^4+8\right ) \log \left (x-e^4+4\right )-82 \left (1-\frac {9 e^4}{41}\right )\right )}{\left (x-e^4+4\right ) \left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {-e^{x+4} (6-x)+16 x+e^x \left (-x^2+2 x+24\right )+2 \left (x-e^4+4\right ) \log \left (x-e^4+4\right )+2 \left (41-9 e^4\right )}{\left (x-e^4+4\right ) \left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {-e^{x+4} (6-x)+16 x+e^x \left (-x^2+2 x+24\right )+2 \left (x-e^4+4\right ) \log \left (x-e^4+4\right )+2 \left (41-9 e^4\right )}{\left (x-e^4+4\right ) \left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {2 (5-x) \left (-x^2-\log \left (x-e^4+4\right ) x-7 \left (1-\frac {e^4}{7}\right ) x-4 \left (1-\frac {e^4}{4}\right ) \log \left (x-e^4+4\right )-11 \left (1-\frac {3 e^4}{11}\right )\right )}{\left (x-e^4+4\right ) \left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}-\frac {x-6}{2 x+e^x+2 \log \left (x-e^4+4\right )+8}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (2 \int \frac {x^2}{\left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx-2 \left (7-e^4\right ) \left (9-e^4\right ) \int \frac {1}{\left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx+2 \left (4-e^4\right ) \left (9-e^4\right ) \int \frac {1}{\left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx+2 \left (11-3 e^4\right ) \int \frac {1}{\left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx-2 \left (4-e^4\right ) \left (7-e^4\right ) \left (9-e^4\right ) \int \frac {1}{\left (-x+e^4-4\right ) \left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx+2 \left (4-e^4\right )^2 \left (9-e^4\right ) \int \frac {1}{\left (-x+e^4-4\right ) \left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx+2 \left (11-3 e^4\right ) \left (9-e^4\right ) \int \frac {1}{\left (-x+e^4-4\right ) \left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx-2 \left (9-e^4\right ) \int \frac {x}{\left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx+2 \left (7-e^4\right ) \int \frac {x}{\left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx-2 \left (9-e^4\right ) \int \frac {\log \left (x-e^4+4\right )}{\left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx+2 \left (4-e^4\right ) \int \frac {\log \left (x-e^4+4\right )}{\left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx+2 \int \frac {x \log \left (x-e^4+4\right )}{\left (2 x+e^x+2 \log \left (x-e^4+4\right )+8\right )^2}dx+6 \int \frac {1}{2 x+e^x+2 \log \left (x-e^4+4\right )+8}dx-\int \frac {x}{2 x+e^x+2 \log \left (x-e^4+4\right )+8}dx\right )\) |
Int[(164 - 36*E^4 + 32*x + E^x*(48 + 4*x - 2*x^2 + E^4*(-12 + 2*x)) + (16 - 4*E^4 + 4*x)*Log[4 - E^4 + x])/(-256 + E^(2*x)*(-4 + E^4 - x) - 192*x - 48*x^2 - 4*x^3 + E^4*(64 + 32*x + 4*x^2) + E^x*(-64 - 32*x - 4*x^2 + E^4*( 16 + 4*x)) + (-128 + E^x*(-16 + 4*E^4 - 4*x) - 64*x - 8*x^2 + E^4*(32 + 8* x))*Log[4 - E^4 + x] + (-16 + 4*E^4 - 4*x)*Log[4 - E^4 + x]^2),x]
3.30.6.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
method | result | size |
risch | \(-\frac {2 \left (-5+x \right )}{2 \ln \left (4-{\mathrm e}^{4}+x \right )+{\mathrm e}^{x}+2 x +8}\) | \(25\) |
parallelrisch | \(-\frac {4 x -20}{2 \left (2 \ln \left (4-{\mathrm e}^{4}+x \right )+{\mathrm e}^{x}+2 x +8\right )}\) | \(27\) |
int(((-4*exp(4)+4*x+16)*ln(4-exp(4)+x)+((2*x-12)*exp(4)-2*x^2+4*x+48)*exp( x)-36*exp(4)+32*x+164)/((4*exp(4)-16-4*x)*ln(4-exp(4)+x)^2+((4*exp(4)-16-4 *x)*exp(x)+(8*x+32)*exp(4)-8*x^2-64*x-128)*ln(4-exp(4)+x)+(exp(4)-x-4)*exp (x)^2+((4*x+16)*exp(4)-4*x^2-32*x-64)*exp(x)+(4*x^2+32*x+64)*exp(4)-4*x^3- 48*x^2-192*x-256),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {164-36 e^4+32 x+e^x \left (48+4 x-2 x^2+e^4 (-12+2 x)\right )+\left (16-4 e^4+4 x\right ) \log \left (4-e^4+x\right )}{-256+e^{2 x} \left (-4+e^4-x\right )-192 x-48 x^2-4 x^3+e^4 \left (64+32 x+4 x^2\right )+e^x \left (-64-32 x-4 x^2+e^4 (16+4 x)\right )+\left (-128+e^x \left (-16+4 e^4-4 x\right )-64 x-8 x^2+e^4 (32+8 x)\right ) \log \left (4-e^4+x\right )+\left (-16+4 e^4-4 x\right ) \log ^2\left (4-e^4+x\right )} \, dx=-\frac {2 \, {\left (x - 5\right )}}{2 \, x + e^{x} + 2 \, \log \left (x - e^{4} + 4\right ) + 8} \]
integrate(((-4*exp(4)+4*x+16)*log(4-exp(4)+x)+((2*x-12)*exp(4)-2*x^2+4*x+4 8)*exp(x)-36*exp(4)+32*x+164)/((4*exp(4)-16-4*x)*log(4-exp(4)+x)^2+((4*exp (4)-16-4*x)*exp(x)+(8*x+32)*exp(4)-8*x^2-64*x-128)*log(4-exp(4)+x)+(exp(4) -x-4)*exp(x)^2+((4*x+16)*exp(4)-4*x^2-32*x-64)*exp(x)+(4*x^2+32*x+64)*exp( 4)-4*x^3-48*x^2-192*x-256),x, algorithm=\
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {164-36 e^4+32 x+e^x \left (48+4 x-2 x^2+e^4 (-12+2 x)\right )+\left (16-4 e^4+4 x\right ) \log \left (4-e^4+x\right )}{-256+e^{2 x} \left (-4+e^4-x\right )-192 x-48 x^2-4 x^3+e^4 \left (64+32 x+4 x^2\right )+e^x \left (-64-32 x-4 x^2+e^4 (16+4 x)\right )+\left (-128+e^x \left (-16+4 e^4-4 x\right )-64 x-8 x^2+e^4 (32+8 x)\right ) \log \left (4-e^4+x\right )+\left (-16+4 e^4-4 x\right ) \log ^2\left (4-e^4+x\right )} \, dx=\frac {10 - 2 x}{2 x + e^{x} + 2 \log {\left (x - e^{4} + 4 \right )} + 8} \]
integrate(((-4*exp(4)+4*x+16)*ln(4-exp(4)+x)+((2*x-12)*exp(4)-2*x**2+4*x+4 8)*exp(x)-36*exp(4)+32*x+164)/((4*exp(4)-16-4*x)*ln(4-exp(4)+x)**2+((4*exp (4)-16-4*x)*exp(x)+(8*x+32)*exp(4)-8*x**2-64*x-128)*ln(4-exp(4)+x)+(exp(4) -x-4)*exp(x)**2+((4*x+16)*exp(4)-4*x**2-32*x-64)*exp(x)+(4*x**2+32*x+64)*e xp(4)-4*x**3-48*x**2-192*x-256),x)
Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {164-36 e^4+32 x+e^x \left (48+4 x-2 x^2+e^4 (-12+2 x)\right )+\left (16-4 e^4+4 x\right ) \log \left (4-e^4+x\right )}{-256+e^{2 x} \left (-4+e^4-x\right )-192 x-48 x^2-4 x^3+e^4 \left (64+32 x+4 x^2\right )+e^x \left (-64-32 x-4 x^2+e^4 (16+4 x)\right )+\left (-128+e^x \left (-16+4 e^4-4 x\right )-64 x-8 x^2+e^4 (32+8 x)\right ) \log \left (4-e^4+x\right )+\left (-16+4 e^4-4 x\right ) \log ^2\left (4-e^4+x\right )} \, dx=-\frac {2 \, {\left (x - 5\right )}}{2 \, x + e^{x} + 2 \, \log \left (x - e^{4} + 4\right ) + 8} \]
integrate(((-4*exp(4)+4*x+16)*log(4-exp(4)+x)+((2*x-12)*exp(4)-2*x^2+4*x+4 8)*exp(x)-36*exp(4)+32*x+164)/((4*exp(4)-16-4*x)*log(4-exp(4)+x)^2+((4*exp (4)-16-4*x)*exp(x)+(8*x+32)*exp(4)-8*x^2-64*x-128)*log(4-exp(4)+x)+(exp(4) -x-4)*exp(x)^2+((4*x+16)*exp(4)-4*x^2-32*x-64)*exp(x)+(4*x^2+32*x+64)*exp( 4)-4*x^3-48*x^2-192*x-256),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 8653 vs. \(2 (24) = 48\).
Time = 0.46 (sec) , antiderivative size = 8653, normalized size of antiderivative = 320.48 \[ \int \frac {164-36 e^4+32 x+e^x \left (48+4 x-2 x^2+e^4 (-12+2 x)\right )+\left (16-4 e^4+4 x\right ) \log \left (4-e^4+x\right )}{-256+e^{2 x} \left (-4+e^4-x\right )-192 x-48 x^2-4 x^3+e^4 \left (64+32 x+4 x^2\right )+e^x \left (-64-32 x-4 x^2+e^4 (16+4 x)\right )+\left (-128+e^x \left (-16+4 e^4-4 x\right )-64 x-8 x^2+e^4 (32+8 x)\right ) \log \left (4-e^4+x\right )+\left (-16+4 e^4-4 x\right ) \log ^2\left (4-e^4+x\right )} \, dx=\text {Too large to display} \]
integrate(((-4*exp(4)+4*x+16)*log(4-exp(4)+x)+((2*x-12)*exp(4)-2*x^2+4*x+4 8)*exp(x)-36*exp(4)+32*x+164)/((4*exp(4)-16-4*x)*log(4-exp(4)+x)^2+((4*exp (4)-16-4*x)*exp(x)+(8*x+32)*exp(4)-8*x^2-64*x-128)*log(4-exp(4)+x)+(exp(4) -x-4)*exp(x)^2+((4*x+16)*exp(4)-4*x^2-32*x-64)*exp(x)+(4*x^2+32*x+64)*exp( 4)-4*x^3-48*x^2-192*x-256),x, algorithm=\
-2*(8*(x - e^4 + 4)^8*e^4 + 2*(x - e^4 + 4)^8*e^(2*x + 4) + 8*(x - e^4 + 4 )^8*e^(x + 4) + 24*(x - e^4 + 4)^7*e^4*log(x - e^4 + 4) + 6*(x - e^4 + 4)^ 7*e^(2*x + 4)*log(x - e^4 + 4) + 24*(x - e^4 + 4)^7*e^(x + 4)*log(x - e^4 + 4) + 24*(x - e^4 + 4)^6*e^4*log(x - e^4 + 4)^2 + 6*(x - e^4 + 4)^6*e^(2* x + 4)*log(x - e^4 + 4)^2 + 24*(x - e^4 + 4)^6*e^(x + 4)*log(x - e^4 + 4)^ 2 + 8*(x - e^4 + 4)^5*e^4*log(x - e^4 + 4)^3 + 2*(x - e^4 + 4)^5*e^(2*x + 4)*log(x - e^4 + 4)^3 + 8*(x - e^4 + 4)^5*e^(x + 4)*log(x - e^4 + 4)^3 + 3 2*(x - e^4 + 4)^7*e^8 - 72*(x - e^4 + 4)^7*e^4 + (x - e^4 + 4)^7*e^(3*x + 4) + 8*(x - e^4 + 4)^7*e^(2*x + 8) - 18*(x - e^4 + 4)^7*e^(2*x + 4) + 32*( x - e^4 + 4)^7*e^(x + 8) - 76*(x - e^4 + 4)^7*e^(x + 4) + 72*(x - e^4 + 4) ^6*e^8*log(x - e^4 + 4) - 200*(x - e^4 + 4)^6*e^4*log(x - e^4 + 4) + 2*(x - e^4 + 4)^6*e^(3*x + 4)*log(x - e^4 + 4) + 18*(x - e^4 + 4)^6*e^(2*x + 8) *log(x - e^4 + 4) - 54*(x - e^4 + 4)^6*e^(2*x + 4)*log(x - e^4 + 4) + 72*( x - e^4 + 4)^6*e^(x + 8)*log(x - e^4 + 4) - 216*(x - e^4 + 4)^6*e^(x + 4)* log(x - e^4 + 4) + 48*(x - e^4 + 4)^5*e^8*log(x - e^4 + 4)^2 - 184*(x - e^ 4 + 4)^5*e^4*log(x - e^4 + 4)^2 + (x - e^4 + 4)^5*e^(3*x + 4)*log(x - e^4 + 4)^2 + 12*(x - e^4 + 4)^5*e^(2*x + 8)*log(x - e^4 + 4)^2 - 54*(x - e^4 + 4)^5*e^(2*x + 4)*log(x - e^4 + 4)^2 + 48*(x - e^4 + 4)^5*e^(x + 8)*log(x - e^4 + 4)^2 - 204*(x - e^4 + 4)^5*e^(x + 4)*log(x - e^4 + 4)^2 + 8*(x - e ^4 + 4)^4*e^8*log(x - e^4 + 4)^3 - 56*(x - e^4 + 4)^4*e^4*log(x - e^4 +...
Timed out. \[ \int \frac {164-36 e^4+32 x+e^x \left (48+4 x-2 x^2+e^4 (-12+2 x)\right )+\left (16-4 e^4+4 x\right ) \log \left (4-e^4+x\right )}{-256+e^{2 x} \left (-4+e^4-x\right )-192 x-48 x^2-4 x^3+e^4 \left (64+32 x+4 x^2\right )+e^x \left (-64-32 x-4 x^2+e^4 (16+4 x)\right )+\left (-128+e^x \left (-16+4 e^4-4 x\right )-64 x-8 x^2+e^4 (32+8 x)\right ) \log \left (4-e^4+x\right )+\left (-16+4 e^4-4 x\right ) \log ^2\left (4-e^4+x\right )} \, dx=\int -\frac {32\,x-36\,{\mathrm {e}}^4+\ln \left (x-{\mathrm {e}}^4+4\right )\,\left (4\,x-4\,{\mathrm {e}}^4+16\right )+{\mathrm {e}}^x\,\left (4\,x-2\,x^2+{\mathrm {e}}^4\,\left (2\,x-12\right )+48\right )+164}{192\,x+\ln \left (x-{\mathrm {e}}^4+4\right )\,\left (64\,x+{\mathrm {e}}^x\,\left (4\,x-4\,{\mathrm {e}}^4+16\right )+8\,x^2-{\mathrm {e}}^4\,\left (8\,x+32\right )+128\right )-{\mathrm {e}}^4\,\left (4\,x^2+32\,x+64\right )+{\mathrm {e}}^{2\,x}\,\left (x-{\mathrm {e}}^4+4\right )+{\mathrm {e}}^x\,\left (32\,x+4\,x^2-{\mathrm {e}}^4\,\left (4\,x+16\right )+64\right )+{\ln \left (x-{\mathrm {e}}^4+4\right )}^2\,\left (4\,x-4\,{\mathrm {e}}^4+16\right )+48\,x^2+4\,x^3+256} \,d x \]
int(-(32*x - 36*exp(4) + log(x - exp(4) + 4)*(4*x - 4*exp(4) + 16) + exp(x )*(4*x - 2*x^2 + exp(4)*(2*x - 12) + 48) + 164)/(192*x + log(x - exp(4) + 4)*(64*x + exp(x)*(4*x - 4*exp(4) + 16) + 8*x^2 - exp(4)*(8*x + 32) + 128) - exp(4)*(32*x + 4*x^2 + 64) + exp(2*x)*(x - exp(4) + 4) + exp(x)*(32*x + 4*x^2 - exp(4)*(4*x + 16) + 64) + log(x - exp(4) + 4)^2*(4*x - 4*exp(4) + 16) + 48*x^2 + 4*x^3 + 256),x)
int(-(32*x - 36*exp(4) + log(x - exp(4) + 4)*(4*x - 4*exp(4) + 16) + exp(x )*(4*x - 2*x^2 + exp(4)*(2*x - 12) + 48) + 164)/(192*x + log(x - exp(4) + 4)*(64*x + exp(x)*(4*x - 4*exp(4) + 16) + 8*x^2 - exp(4)*(8*x + 32) + 128) - exp(4)*(32*x + 4*x^2 + 64) + exp(2*x)*(x - exp(4) + 4) + exp(x)*(32*x + 4*x^2 - exp(4)*(4*x + 16) + 64) + log(x - exp(4) + 4)^2*(4*x - 4*exp(4) + 16) + 48*x^2 + 4*x^3 + 256), x)