Integrand size = 41, antiderivative size = 25 \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=-e^{4 \left (16+2 \left (\frac {16}{e^6}-x+\log (4+x)\right )\right )}+x \] Output:
x-exp(64+128/exp(3)^2-8*x+8*ln(4+x))
Time = 7.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=-e^{-8 x} (4+x) \left (-e^{8 x}+e^{64+\frac {128}{e^6}} (4+x)^7\right ) \] Input:
Integrate[(4 + x + E^((128 + E^6*(64 - 8*x) + 8*E^6*Log[4 + x])/E^6)*(24 + 8*x))/(4 + x),x]
Output:
-(((4 + x)*(-E^(8*x) + E^(64 + 128/E^6)*(4 + x)^7))/E^(8*x))
Time = 0.77 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {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.
| \(\displaystyle \int \frac {x+(8 x+24) e^{\frac {e^6 (64-8 x)+8 e^6 \log (x+4)+128}{e^6}}+4}{x+4} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (8 e^{-8 x+\frac {128}{e^6}+64} (x+3) (x+4)^7+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x-e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (x+4)^8\) |
Input:
Int[(4 + x + E^((128 + E^6*(64 - 8*x) + 8*E^6*Log[4 + x])/E^6)*(24 + 8*x)) /(4 + x),x]
Output:
x - E^(64*(1 + 2/E^6) - 8*x)*(4 + x)^8
Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32
| method | result | size |
| default | \(x -{\mathrm e}^{\left (8 \,{\mathrm e}^{6} \ln \left (4+x \right )+\left (-8 x +64\right ) {\mathrm e}^{6}+128\right ) {\mathrm e}^{-6}}\) | \(33\) |
| norman | \(x -{\mathrm e}^{\left (8 \,{\mathrm e}^{6} \ln \left (4+x \right )+\left (-8 x +64\right ) {\mathrm e}^{6}+128\right ) {\mathrm e}^{-6}}\) | \(33\) |
| parts | \(x -{\mathrm e}^{\left (8 \,{\mathrm e}^{6} \ln \left (4+x \right )+\left (-8 x +64\right ) {\mathrm e}^{6}+128\right ) {\mathrm e}^{-6}}\) | \(33\) |
| parallelrisch | \(x -{\mathrm e}^{\left (8 \,{\mathrm e}^{6} \ln \left (4+x \right )+\left (-8 x +64\right ) {\mathrm e}^{6}+128\right ) {\mathrm e}^{-6}}-8\) | \(34\) |
| risch | \(x +\left (-x^{8}-32 x^{7}-448 x^{6}-3584 x^{5}-17920 x^{4}-57344 x^{3}-114688 x^{2}-131072 x -65536\right ) {\mathrm e}^{-8 x +64+128 \,{\mathrm e}^{-6}}\) | \(54\) |
| orering | \(\frac {\left (x -\frac {249}{8}\right ) \left (\left (8 x +24\right ) {\mathrm e}^{\left (8 \,{\mathrm e}^{6} \ln \left (4+x \right )+\left (-8 x +64\right ) {\mathrm e}^{6}+128\right ) {\mathrm e}^{-6}}+4+x \right )}{4+x}+\frac {\left (8 x^{3}-192 x^{2}-1639 x -2972\right ) \left (\frac {8 \,{\mathrm e}^{\left (8 \,{\mathrm e}^{6} \ln \left (4+x \right )+\left (-8 x +64\right ) {\mathrm e}^{6}+128\right ) {\mathrm e}^{-6}}+\left (8 x +24\right ) \left (\frac {8 \,{\mathrm e}^{6}}{4+x}-8 \,{\mathrm e}^{6}\right ) {\mathrm e}^{-6} {\mathrm e}^{\left (8 \,{\mathrm e}^{6} \ln \left (4+x \right )+\left (-8 x +64\right ) {\mathrm e}^{6}+128\right ) {\mathrm e}^{-6}}+1}{4+x}-\frac {\left (8 x +24\right ) {\mathrm e}^{\left (8 \,{\mathrm e}^{6} \ln \left (4+x \right )+\left (-8 x +64\right ) {\mathrm e}^{6}+128\right ) {\mathrm e}^{-6}}+4+x}{\left (4+x \right )^{2}}\right )}{64 x^{2}+384 x +568}\) | \(216\) |
Input:
int(((8*x+24)*exp((8*exp(3)^2*ln(4+x)+(-8*x+64)*exp(3)^2+128)/exp(3)^2)+4+ x)/(4+x),x,method=_RETURNVERBOSE)
Output:
x-exp((8*exp(3)^2*ln(4+x)+(-8*x+64)*exp(3)^2+128)/exp(3)^2)
Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=x - e^{\left (-8 \, {\left ({\left (x - 8\right )} e^{6} - e^{6} \log \left (x + 4\right ) - 16\right )} e^{\left (-6\right )}\right )} \] Input:
integrate(((8*x+24)*exp((8*exp(3)^2*log(4+x)+(-8*x+64)*exp(3)^2+128)/exp(3 )^2)+4+x)/(4+x),x, algorithm="fricas")
Output:
x - e^(-8*((x - 8)*e^6 - e^6*log(x + 4) - 16)*e^(-6))
Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (17) = 34\).
Time = 4.54 (sec) , antiderivative size = 590, normalized size of antiderivative = 23.60 \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=x + 819200 \left (- \frac {x e^{- 8 x}}{8} - \frac {e^{- 8 x}}{64}\right ) e^{64} e^{\frac {128}{e^{6}}} + 745472 \left (- \frac {x^{2} e^{- 8 x}}{8} - \frac {x e^{- 8 x}}{32} - \frac {e^{- 8 x}}{256}\right ) e^{64} e^{\frac {128}{e^{6}}} + 387072 \left (- \frac {x^{3} e^{- 8 x}}{8} - \frac {3 x^{2} e^{- 8 x}}{64} - \frac {3 x e^{- 8 x}}{256} - \frac {3 e^{- 8 x}}{2048}\right ) e^{64} e^{\frac {128}{e^{6}}} + 125440 \left (- \frac {x^{4} e^{- 8 x}}{8} - \frac {x^{3} e^{- 8 x}}{16} - \frac {3 x^{2} e^{- 8 x}}{128} - \frac {3 x e^{- 8 x}}{512} - \frac {3 e^{- 8 x}}{4096}\right ) e^{64} e^{\frac {128}{e^{6}}} + 25984 \left (- \frac {x^{5} e^{- 8 x}}{8} - \frac {5 x^{4} e^{- 8 x}}{64} - \frac {5 x^{3} e^{- 8 x}}{128} - \frac {15 x^{2} e^{- 8 x}}{1024} - \frac {15 x e^{- 8 x}}{4096} - \frac {15 e^{- 8 x}}{32768}\right ) e^{64} e^{\frac {128}{e^{6}}} + 3360 \left (- \frac {x^{6} e^{- 8 x}}{8} - \frac {3 x^{5} e^{- 8 x}}{32} - \frac {15 x^{4} e^{- 8 x}}{256} - \frac {15 x^{3} e^{- 8 x}}{512} - \frac {45 x^{2} e^{- 8 x}}{4096} - \frac {45 x e^{- 8 x}}{16384} - \frac {45 e^{- 8 x}}{131072}\right ) e^{64} e^{\frac {128}{e^{6}}} + 248 \left (- \frac {x^{7} e^{- 8 x}}{8} - \frac {7 x^{6} e^{- 8 x}}{64} - \frac {21 x^{5} e^{- 8 x}}{256} - \frac {105 x^{4} e^{- 8 x}}{2048} - \frac {105 x^{3} e^{- 8 x}}{4096} - \frac {315 x^{2} e^{- 8 x}}{32768} - \frac {315 x e^{- 8 x}}{131072} - \frac {315 e^{- 8 x}}{1048576}\right ) e^{64} e^{\frac {128}{e^{6}}} + 8 \left (- \frac {x^{8} e^{- 8 x}}{8} - \frac {x^{7} e^{- 8 x}}{8} - \frac {7 x^{6} e^{- 8 x}}{64} - \frac {21 x^{5} e^{- 8 x}}{256} - \frac {105 x^{4} e^{- 8 x}}{2048} - \frac {105 x^{3} e^{- 8 x}}{4096} - \frac {315 x^{2} e^{- 8 x}}{32768} - \frac {315 x e^{- 8 x}}{131072} - \frac {315 e^{- 8 x}}{1048576}\right ) e^{64} e^{\frac {128}{e^{6}}} - 49152 e^{64} e^{- 8 x} e^{\frac {128}{e^{6}}} \] Input:
integrate(((8*x+24)*exp((8*exp(3)**2*ln(4+x)+(-8*x+64)*exp(3)**2+128)/exp( 3)**2)+4+x)/(4+x),x)
Output:
x + 819200*(-x*exp(-8*x)/8 - exp(-8*x)/64)*exp(64)*exp(128*exp(-6)) + 7454 72*(-x**2*exp(-8*x)/8 - x*exp(-8*x)/32 - exp(-8*x)/256)*exp(64)*exp(128*ex p(-6)) + 387072*(-x**3*exp(-8*x)/8 - 3*x**2*exp(-8*x)/64 - 3*x*exp(-8*x)/2 56 - 3*exp(-8*x)/2048)*exp(64)*exp(128*exp(-6)) + 125440*(-x**4*exp(-8*x)/ 8 - x**3*exp(-8*x)/16 - 3*x**2*exp(-8*x)/128 - 3*x*exp(-8*x)/512 - 3*exp(- 8*x)/4096)*exp(64)*exp(128*exp(-6)) + 25984*(-x**5*exp(-8*x)/8 - 5*x**4*ex p(-8*x)/64 - 5*x**3*exp(-8*x)/128 - 15*x**2*exp(-8*x)/1024 - 15*x*exp(-8*x )/4096 - 15*exp(-8*x)/32768)*exp(64)*exp(128*exp(-6)) + 3360*(-x**6*exp(-8 *x)/8 - 3*x**5*exp(-8*x)/32 - 15*x**4*exp(-8*x)/256 - 15*x**3*exp(-8*x)/51 2 - 45*x**2*exp(-8*x)/4096 - 45*x*exp(-8*x)/16384 - 45*exp(-8*x)/131072)*e xp(64)*exp(128*exp(-6)) + 248*(-x**7*exp(-8*x)/8 - 7*x**6*exp(-8*x)/64 - 2 1*x**5*exp(-8*x)/256 - 105*x**4*exp(-8*x)/2048 - 105*x**3*exp(-8*x)/4096 - 315*x**2*exp(-8*x)/32768 - 315*x*exp(-8*x)/131072 - 315*exp(-8*x)/1048576 )*exp(64)*exp(128*exp(-6)) + 8*(-x**8*exp(-8*x)/8 - x**7*exp(-8*x)/8 - 7*x **6*exp(-8*x)/64 - 21*x**5*exp(-8*x)/256 - 105*x**4*exp(-8*x)/2048 - 105*x **3*exp(-8*x)/4096 - 315*x**2*exp(-8*x)/32768 - 315*x*exp(-8*x)/131072 - 3 15*exp(-8*x)/1048576)*exp(64)*exp(128*exp(-6)) - 49152*exp(64)*exp(-8*x)*e xp(128*exp(-6))
\[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=\int { \frac {8 \, {\left (x + 3\right )} e^{\left (-8 \, {\left ({\left (x - 8\right )} e^{6} - e^{6} \log \left (x + 4\right ) - 16\right )} e^{\left (-6\right )}\right )} + x + 4}{x + 4} \,d x } \] Input:
integrate(((8*x+24)*exp((8*exp(3)^2*log(4+x)+(-8*x+64)*exp(3)^2+128)/exp(3 )^2)+4+x)/(4+x),x, algorithm="maxima")
Output:
-1572864*e^(64*(e^6 + 2)*e^(-6) + 32)*exp_integral_e(1, 8*x + 32) + x - (x ^9*e^(128*e^(-6) + 64) + 36*x^8*e^(128*e^(-6) + 64) + 576*x^7*e^(128*e^(-6 ) + 64) + 5376*x^6*e^(128*e^(-6) + 64) + 32256*x^5*e^(128*e^(-6) + 64) + 1 29024*x^4*e^(128*e^(-6) + 64) + 344064*x^3*e^(128*e^(-6) + 64) + 589824*x^ 2*e^(128*e^(-6) + 64) + 589824*x*e^(128*e^(-6) + 64))*e^(-8*x)/(x + 4) + i ntegrate(262144*(2*x*e^(128*e^(-6) + 64) + 9*e^(128*e^(-6) + 64))*e^(-8*x) /(x^2 + 8*x + 16), x)
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (20) = 40\).
Time = 0.11 (sec) , antiderivative size = 159, normalized size of antiderivative = 6.36 \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=-x^{8} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 32 \, x^{7} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 448 \, x^{6} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 3584 \, x^{5} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 17920 \, x^{4} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 57344 \, x^{3} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 114688 \, x^{2} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 131072 \, x e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} + x - 65536 \, e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} \] Input:
integrate(((8*x+24)*exp((8*exp(3)^2*log(4+x)+(-8*x+64)*exp(3)^2+128)/exp(3 )^2)+4+x)/(4+x),x, algorithm="giac")
Output:
-x^8*e^(64*(e^6 + 2)*e^(-6) - 8*x) - 32*x^7*e^(64*(e^6 + 2)*e^(-6) - 8*x) - 448*x^6*e^(64*(e^6 + 2)*e^(-6) - 8*x) - 3584*x^5*e^(64*(e^6 + 2)*e^(-6) - 8*x) - 17920*x^4*e^(64*(e^6 + 2)*e^(-6) - 8*x) - 57344*x^3*e^(64*(e^6 + 2)*e^(-6) - 8*x) - 114688*x^2*e^(64*(e^6 + 2)*e^(-6) - 8*x) - 131072*x*e^( 64*(e^6 + 2)*e^(-6) - 8*x) + x - 65536*e^(64*(e^6 + 2)*e^(-6) - 8*x)
Time = 4.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 5.28 \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=x-65536\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-131072\,x\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-114688\,x^2\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-57344\,x^3\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-17920\,x^4\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-3584\,x^5\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-448\,x^6\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-32\,x^7\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-x^8\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64} \] Input:
int((x + exp(exp(-6)*(8*log(x + 4)*exp(6) - exp(6)*(8*x - 64) + 128))*(8*x + 24) + 4)/(x + 4),x)
Output:
x - 65536*exp(128*exp(-6) - 8*x + 64) - 131072*x*exp(128*exp(-6) - 8*x + 6 4) - 114688*x^2*exp(128*exp(-6) - 8*x + 64) - 57344*x^3*exp(128*exp(-6) - 8*x + 64) - 17920*x^4*exp(128*exp(-6) - 8*x + 64) - 3584*x^5*exp(128*exp(- 6) - 8*x + 64) - 448*x^6*exp(128*exp(-6) - 8*x + 64) - 32*x^7*exp(128*exp( -6) - 8*x + 64) - x^8*exp(128*exp(-6) - 8*x + 64)
Time = 0.19 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.84 \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=\frac {-e^{\frac {128}{e^{6}}} e^{64} x^{8}-32 e^{\frac {128}{e^{6}}} e^{64} x^{7}-448 e^{\frac {128}{e^{6}}} e^{64} x^{6}-3584 e^{\frac {128}{e^{6}}} e^{64} x^{5}-17920 e^{\frac {128}{e^{6}}} e^{64} x^{4}-57344 e^{\frac {128}{e^{6}}} e^{64} x^{3}-114688 e^{\frac {128}{e^{6}}} e^{64} x^{2}-131072 e^{\frac {128}{e^{6}}} e^{64} x -65536 e^{\frac {128}{e^{6}}} e^{64}+e^{8 x} x}{e^{8 x}} \] Input:
int(((8*x+24)*exp((8*exp(3)^2*log(4+x)+(-8*x+64)*exp(3)^2+128)/exp(3)^2)+4 +x)/(4+x),x)
Output:
( - e**(128/e**6)*e**64*x**8 - 32*e**(128/e**6)*e**64*x**7 - 448*e**(128/e **6)*e**64*x**6 - 3584*e**(128/e**6)*e**64*x**5 - 17920*e**(128/e**6)*e**6 4*x**4 - 57344*e**(128/e**6)*e**64*x**3 - 114688*e**(128/e**6)*e**64*x**2 - 131072*e**(128/e**6)*e**64*x - 65536*e**(128/e**6)*e**64 + e**(8*x)*x)/e **(8*x)