Integrand size = 117, antiderivative size = 23 \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=\left (e^x-x\right ) x \left (-x+\log (2)+\log \left (2+e^{7 x}\right )\right ) \] Output:
(exp(x)-x)*(ln(exp(7*x)+2)+ln(2)-x)*x
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=-\left (\left (e^x-x\right ) x \left (x-\log \left (4+2 e^{7 x}\right )\right )\right ) \] Input:
Integrate[(6*x^2 - 4*x*Log[2] + E^x*(-4*x - 2*x^2 + (2 + 2*x)*Log[2]) + E^ (7*x)*(-4*x^2 - 2*x*Log[2] + E^x*(5*x - x^2 + (1 + x)*Log[2])) + (-4*x + E ^x*(2 + 2*x) + E^(7*x)*(-2*x + E^x*(1 + x)))*Log[2 + E^(7*x)])/(2 + E^(7*x )),x]
Output:
-((E^x - x)*x*(x - Log[4 + 2*E^(7*x)]))
Leaf count is larger than twice the leaf count of optimal. \(987\) vs. \(2(23)=46\).
Time = 3.49 (sec) , antiderivative size = 987, normalized size of antiderivative = 42.91, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, 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.
| \(\displaystyle \int \frac {6 x^2+e^x \left (-2 x^2-4 x+(2 x+2) \log (2)\right )+e^{7 x} \left (-4 x^2+e^x \left (-x^2+5 x+(x+1) \log (2)\right )-2 x \log (2)\right )-4 x \log (2)+\left (-4 x+e^x (2 x+2)+e^{7 x} \left (e^x (x+1)-2 x\right )\right ) \log \left (e^{7 x}+2\right )}{e^{7 x}+2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (e^x \left (-x^2+x \log \left (e^{7 x}+2\right )+5 x \left (1+\frac {\log (2)}{5}\right )+\log \left (2 \left (e^{7 x}+2\right )\right )\right )-\frac {14 \left (e^x-x\right ) x}{e^{7 x}+2}-2 x \left (2 x+\log \left (e^{7 x}+2\right )+\log (2)\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x^3-e^x x^2-\log \left (2+e^{7 x}\right ) x^2-\log (2) x^2-\frac {(-1)^{5/7} x^2}{2^{6/7}}+\frac {(-1)^{4/7} x^2}{2^{6/7}}-\frac {(-1)^{3/7} x^2}{2^{6/7}}+\frac {(-1)^{2/7} x^2}{2^{6/7}}-\frac {\sqrt [7]{-1} x^2}{2^{6/7}}+\frac {x^2}{2^{6/7}}+\left (-\frac {1}{2}\right )^{6/7} x^2+2 e^x x-(-1)^{6/7} \sqrt [7]{2} \log \left (1-\sqrt [7]{-\frac {1}{2}} e^x\right ) x-\sqrt [7]{2} \log \left (1+\frac {e^x}{\sqrt [7]{2}}\right ) x+(-1)^{5/7} \sqrt [7]{2} \log \left (1+\frac {(-1)^{2/7} e^x}{\sqrt [7]{2}}\right ) x-(-1)^{4/7} \sqrt [7]{2} \log \left (1-\frac {(-1)^{3/7} e^x}{\sqrt [7]{2}}\right ) x+(-1)^{3/7} \sqrt [7]{2} \log \left (1+\frac {(-1)^{4/7} e^x}{\sqrt [7]{2}}\right ) x-(-1)^{2/7} \sqrt [7]{2} \log \left (1-\frac {(-1)^{5/7} e^x}{\sqrt [7]{2}}\right ) x+\sqrt [7]{-2} \log \left (1+\frac {(-1)^{6/7} e^x}{\sqrt [7]{2}}\right ) x+e^x \log \left (2+e^{7 x}\right ) x+e^x (5+\log (2)) x-2 e^x+\frac {(-1)^{5/7} (1-x)^2}{2^{6/7}}-\frac {(-1)^{4/7} (1-x)^2}{2^{6/7}}+\frac {(-1)^{3/7} (1-x)^2}{2^{6/7}}-\frac {(-1)^{2/7} (1-x)^2}{2^{6/7}}+\frac {\sqrt [7]{-1} (1-x)^2}{2^{6/7}}-\frac {(1-x)^2}{2^{6/7}}-\left (-\frac {1}{2}\right )^{6/7} (1-x)^2+7 e^x (1-x)+\sqrt [7]{2} \log \left (\sqrt [7]{2}+e^x\right )-(-1)^{6/7} \sqrt [7]{2} (1-x) \log \left (1-\sqrt [7]{-\frac {1}{2}} e^x\right )-\sqrt [7]{2} (1-x) \log \left (1+\frac {e^x}{\sqrt [7]{2}}\right )+(-1)^{5/7} \sqrt [7]{2} (1-x) \log \left (1+\frac {(-1)^{2/7} e^x}{\sqrt [7]{2}}\right )-(-1)^{4/7} \sqrt [7]{2} (1-x) \log \left (1-\frac {(-1)^{3/7} e^x}{\sqrt [7]{2}}\right )+(-1)^{3/7} \sqrt [7]{2} (1-x) \log \left (1+\frac {(-1)^{4/7} e^x}{\sqrt [7]{2}}\right )-(-1)^{2/7} \sqrt [7]{2} (1-x) \log \left (1-\frac {(-1)^{5/7} e^x}{\sqrt [7]{2}}\right )+\sqrt [7]{-2} (1-x) \log \left (1+\frac {(-1)^{6/7} e^x}{\sqrt [7]{2}}\right )-e^x \log \left (2+e^{7 x}\right )+e^x \log \left (2 \left (2+e^{7 x}\right )\right )-\sqrt [7]{2} \cos \left (\frac {\pi }{7}\right ) \log \left (2^{2/7}+e^{2 x}-2 \sqrt [7]{2} e^x \cos \left (\frac {\pi }{7}\right )\right )+\sqrt [7]{2} \log \left (2^{2/7}+e^{2 x}+2 \sqrt [7]{2} e^x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right )-2 \sqrt [7]{2} \arctan \left (\cot \left (\frac {\pi }{7}\right )-\frac {e^x \csc \left (\frac {\pi }{7}\right )}{\sqrt [7]{2}}\right ) \sin \left (\frac {\pi }{7}\right )-\sqrt [7]{2} \log \left (2^{2/7}+e^{2 x}-2 \sqrt [7]{2} e^x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )-e^x (5+\log (2))+2 \sqrt [7]{2} \arctan \left (\frac {e^x \sec \left (\frac {3 \pi }{14}\right )}{\sqrt [7]{2}}+\tan \left (\frac {3 \pi }{14}\right )\right ) \cos \left (\frac {3 \pi }{14}\right )+2 \sqrt [7]{2} \arctan \left (\frac {e^x \sec \left (\frac {\pi }{14}\right )}{\sqrt [7]{2}}-\tan \left (\frac {\pi }{14}\right )\right ) \cos \left (\frac {\pi }{14}\right )\) |
Input:
Int[(6*x^2 - 4*x*Log[2] + E^x*(-4*x - 2*x^2 + (2 + 2*x)*Log[2]) + E^(7*x)* (-4*x^2 - 2*x*Log[2] + E^x*(5*x - x^2 + (1 + x)*Log[2])) + (-4*x + E^x*(2 + 2*x) + E^(7*x)*(-2*x + E^x*(1 + x)))*Log[2 + E^(7*x)])/(2 + E^(7*x)),x]
Output:
-2*E^x + 7*E^x*(1 - x) - (-1/2)^(6/7)*(1 - x)^2 - (1 - x)^2/2^(6/7) + ((-1 )^(1/7)*(1 - x)^2)/2^(6/7) - ((-1)^(2/7)*(1 - x)^2)/2^(6/7) + ((-1)^(3/7)* (1 - x)^2)/2^(6/7) - ((-1)^(4/7)*(1 - x)^2)/2^(6/7) + ((-1)^(5/7)*(1 - x)^ 2)/2^(6/7) + 2*E^x*x + (-1/2)^(6/7)*x^2 + x^2/2^(6/7) - ((-1)^(1/7)*x^2)/2 ^(6/7) + ((-1)^(2/7)*x^2)/2^(6/7) - ((-1)^(3/7)*x^2)/2^(6/7) + ((-1)^(4/7) *x^2)/2^(6/7) - ((-1)^(5/7)*x^2)/2^(6/7) - E^x*x^2 + x^3 + 2*2^(1/7)*ArcTa n[(E^x*Sec[Pi/14])/2^(1/7) - Tan[Pi/14]]*Cos[Pi/14] + 2*2^(1/7)*ArcTan[(E^ x*Sec[(3*Pi)/14])/2^(1/7) + Tan[(3*Pi)/14]]*Cos[(3*Pi)/14] - x^2*Log[2] - E^x*(5 + Log[2]) + E^x*x*(5 + Log[2]) + 2^(1/7)*Log[2^(1/7) + E^x] - (-1)^ (6/7)*2^(1/7)*(1 - x)*Log[1 - (-1/2)^(1/7)*E^x] - (-1)^(6/7)*2^(1/7)*x*Log [1 - (-1/2)^(1/7)*E^x] - 2^(1/7)*(1 - x)*Log[1 + E^x/2^(1/7)] - 2^(1/7)*x* Log[1 + E^x/2^(1/7)] + (-1)^(5/7)*2^(1/7)*(1 - x)*Log[1 + ((-1)^(2/7)*E^x) /2^(1/7)] + (-1)^(5/7)*2^(1/7)*x*Log[1 + ((-1)^(2/7)*E^x)/2^(1/7)] - (-1)^ (4/7)*2^(1/7)*(1 - x)*Log[1 - ((-1)^(3/7)*E^x)/2^(1/7)] - (-1)^(4/7)*2^(1/ 7)*x*Log[1 - ((-1)^(3/7)*E^x)/2^(1/7)] + (-1)^(3/7)*2^(1/7)*(1 - x)*Log[1 + ((-1)^(4/7)*E^x)/2^(1/7)] + (-1)^(3/7)*2^(1/7)*x*Log[1 + ((-1)^(4/7)*E^x )/2^(1/7)] - (-1)^(2/7)*2^(1/7)*(1 - x)*Log[1 - ((-1)^(5/7)*E^x)/2^(1/7)] - (-1)^(2/7)*2^(1/7)*x*Log[1 - ((-1)^(5/7)*E^x)/2^(1/7)] + (-2)^(1/7)*(1 - x)*Log[1 + ((-1)^(6/7)*E^x)/2^(1/7)] + (-2)^(1/7)*x*Log[1 + ((-1)^(6/7)*E ^x)/2^(1/7)] - E^x*Log[2 + E^(7*x)] + E^x*x*Log[2 + E^(7*x)] - x^2*Log[...
Time = 0.71 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87
| method | result | size |
| risch | \(\left ({\mathrm e}^{x} x -x^{2}\right ) \ln \left ({\mathrm e}^{7 x}+2\right )-x^{2} \ln \left (2\right )+x \ln \left (2\right ) {\mathrm e}^{x}+x^{3}-{\mathrm e}^{x} x^{2}\) | \(43\) |
| parallelrisch | \({\mathrm e}^{x} \ln \left ({\mathrm e}^{7 x}+2\right ) x +x \ln \left (2\right ) {\mathrm e}^{x}-{\mathrm e}^{x} x^{2}-x^{2} \ln \left ({\mathrm e}^{7 x}+2\right )-x^{2} \ln \left (2\right )+x^{3}\) | \(48\) |
Input:
int(((((1+x)*exp(x)-2*x)*exp(7*x)+(2+2*x)*exp(x)-4*x)*ln(exp(7*x)+2)+((ln( 2)*(1+x)-x^2+5*x)*exp(x)-2*x*ln(2)-4*x^2)*exp(7*x)+((2+2*x)*ln(2)-2*x^2-4* x)*exp(x)-4*x*ln(2)+6*x^2)/(exp(7*x)+2),x,method=_RETURNVERBOSE)
Output:
(exp(x)*x-x^2)*ln(exp(7*x)+2)-x^2*ln(2)+x*ln(2)*exp(x)+x^3-exp(x)*x^2
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=x^{3} - x^{2} \log \left (2\right ) - {\left (x^{2} - x \log \left (2\right )\right )} e^{x} - {\left (x^{2} - x e^{x}\right )} \log \left (e^{\left (7 \, x\right )} + 2\right ) \] Input:
integrate(((((1+x)*exp(x)-2*x)*exp(7*x)+(2+2*x)*exp(x)-4*x)*log(exp(7*x)+2 )+((log(2)*(1+x)-x^2+5*x)*exp(x)-2*x*log(2)-4*x^2)*exp(7*x)+((2+2*x)*log(2 )-2*x^2-4*x)*exp(x)-4*x*log(2)+6*x^2)/(exp(7*x)+2),x, algorithm="fricas")
Output:
x^3 - x^2*log(2) - (x^2 - x*log(2))*e^x - (x^2 - x*e^x)*log(e^(7*x) + 2)
Timed out. \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=\text {Timed out} \] Input:
integrate(((((1+x)*exp(x)-2*x)*exp(7*x)+(2+2*x)*exp(x)-4*x)*ln(exp(7*x)+2) +((ln(2)*(1+x)-x**2+5*x)*exp(x)-2*x*ln(2)-4*x**2)*exp(7*x)+((2+2*x)*ln(2)- 2*x**2-4*x)*exp(x)-4*x*ln(2)+6*x**2)/(exp(7*x)+2),x)
Output:
Timed out
Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=x^{3} - x^{2} \log \left (2\right ) - {\left (x^{2} - x \log \left (2\right )\right )} e^{x} - {\left (x^{2} - x e^{x}\right )} \log \left (e^{\left (7 \, x\right )} + 2\right ) \] Input:
integrate(((((1+x)*exp(x)-2*x)*exp(7*x)+(2+2*x)*exp(x)-4*x)*log(exp(7*x)+2 )+((log(2)*(1+x)-x^2+5*x)*exp(x)-2*x*log(2)-4*x^2)*exp(7*x)+((2+2*x)*log(2 )-2*x^2-4*x)*exp(x)-4*x*log(2)+6*x^2)/(exp(7*x)+2),x, algorithm="maxima")
Output:
x^3 - x^2*log(2) - (x^2 - x*log(2))*e^x - (x^2 - x*e^x)*log(e^(7*x) + 2)
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (23) = 46\).
Time = 0.15 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=x^{3} - x^{2} e^{x} - x^{2} \log \left (2\right ) + x e^{x} \log \left (2\right ) - x^{2} \log \left (e^{\left (7 \, x\right )} + 2\right ) + x e^{x} \log \left (e^{\left (7 \, x\right )} + 2\right ) \] Input:
integrate(((((1+x)*exp(x)-2*x)*exp(7*x)+(2+2*x)*exp(x)-4*x)*log(exp(7*x)+2 )+((log(2)*(1+x)-x^2+5*x)*exp(x)-2*x*log(2)-4*x^2)*exp(7*x)+((2+2*x)*log(2 )-2*x^2-4*x)*exp(x)-4*x*log(2)+6*x^2)/(exp(7*x)+2),x, algorithm="giac")
Output:
x^3 - x^2*e^x - x^2*log(2) + x*e^x*log(2) - x^2*log(e^(7*x) + 2) + x*e^x*l og(e^(7*x) + 2)
Time = 4.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx={\mathrm {e}}^x\,\left (x\,\ln \left (2\right )-x^2\right )-\frac {x^2\,\ln \left (4\right )}{2}+\ln \left ({\mathrm {e}}^{7\,x}+2\right )\,\left (x\,{\mathrm {e}}^x-x^2\right )+x^3 \] Input:
int(-(4*x*log(2) + exp(x)*(4*x - log(2)*(2*x + 2) + 2*x^2) + exp(7*x)*(2*x *log(2) - exp(x)*(5*x + log(2)*(x + 1) - x^2) + 4*x^2) - 6*x^2 + log(exp(7 *x) + 2)*(4*x - exp(x)*(2*x + 2) + exp(7*x)*(2*x - exp(x)*(x + 1))))/(exp( 7*x) + 2),x)
Output:
exp(x)*(x*log(2) - x^2) - (x^2*log(4))/2 + log(exp(7*x) + 2)*(x*exp(x) - x ^2) + x^3
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=x \left (e^{x} \mathrm {log}\left (e^{7 x}+2\right )+e^{x} \mathrm {log}\left (2\right )-e^{x} x -\mathrm {log}\left (e^{7 x}+2\right ) x -\mathrm {log}\left (2\right ) x +x^{2}\right ) \] Input:
int(((((1+x)*exp(x)-2*x)*exp(7*x)+(2+2*x)*exp(x)-4*x)*log(exp(7*x)+2)+((lo g(2)*(1+x)-x^2+5*x)*exp(x)-2*x*log(2)-4*x^2)*exp(7*x)+((2+2*x)*log(2)-2*x^ 2-4*x)*exp(x)-4*x*log(2)+6*x^2)/(exp(7*x)+2),x)
Output:
x*(e**x*log(e**(7*x) + 2) + e**x*log(2) - e**x*x - log(e**(7*x) + 2)*x - l og(2)*x + x**2)