Integrand size = 57, antiderivative size = 18 \[ \int \frac {6+4^{4 x+x^2} (-48-24 x) \log (4)}{484+4^{1+8 x+2 x^2}+4^{4 x+x^2} (88-4 x)-44 x+x^2} \, dx=\frac {3}{11+4^{x (4+x)}-\frac {x}{2}} \] Output:
3/(11+exp(2*x*(4+x)*ln(2))-1/2*x)
Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {6+4^{4 x+x^2} (-48-24 x) \log (4)}{484+4^{1+8 x+2 x^2}+4^{4 x+x^2} (88-4 x)-44 x+x^2} \, dx=\frac {6}{22+2^{1+8 x+2 x^2}-x} \] Input:
Integrate[(6 + 4^(4*x + x^2)*(-48 - 24*x)*Log[4])/(484 + 4^(1 + 8*x + 2*x^ 2) + 4^(4*x + x^2)*(88 - 4*x) - 44*x + x^2),x]
Output:
6/(22 + 2^(1 + 8*x + 2*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.
\(\displaystyle \int \frac {4^{x^2+4 x} (-24 x-48) \log (4)+6}{x^2+4^{2 x^2+8 x+1}+4^{x^2+4 x} (88-4 x)-44 x+484} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {4^{x^2+4 x} (-24 x-48) \log (4)+6}{\left (2^{2 x^2+8 x+1}-x+22\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {12 (x+2) \log (4)}{2^{2 x^2+8 x+1}-x+22}-\frac {6 \left (2 x^2 \log (4)-40 x \log (4)-1-88 \log (4)\right )}{\left (2^{2 x^2+8 x+1}-x+22\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 6 (1+88 \log (4)) \int \frac {1}{\left (-x+2^{2 x^2+8 x+1}+22\right )^2}dx-24 \log (4) \int \frac {1}{-x+2^{2 x^2+8 x+1}+22}dx+240 \log (4) \int \frac {x}{\left (-x+2^{2 x^2+8 x+1}+22\right )^2}dx-12 \log (4) \int \frac {x}{-x+2^{2 x^2+8 x+1}+22}dx-12 \log (4) \int \frac {x^2}{\left (-x+2^{2 x^2+8 x+1}+22\right )^2}dx\) |
Input:
Int[(6 + 4^(4*x + x^2)*(-48 - 24*x)*Log[4])/(484 + 4^(1 + 8*x + 2*x^2) + 4 ^(4*x + x^2)*(88 - 4*x) - 44*x + x^2),x]
Output:
$Aborted
Time = 0.32 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94
method | result | size |
risch | \(-\frac {6}{x -2 \,4^{\left (4+x \right ) x}-22}\) | \(17\) |
parallelrisch | \(-\frac {6}{x -2 \,{\mathrm e}^{2 x \left (4+x \right ) \ln \left (2\right )}-22}\) | \(19\) |
norman | \(-\frac {6}{x -2 \,{\mathrm e}^{2 \left (x^{2}+4 x \right ) \ln \left (2\right )}-22}\) | \(22\) |
Input:
int((2*(-24*x-48)*ln(2)*exp(2*(x^2+4*x)*ln(2))+6)/(4*exp(2*(x^2+4*x)*ln(2) )^2+(-4*x+88)*exp(2*(x^2+4*x)*ln(2))+x^2-44*x+484),x,method=_RETURNVERBOSE )
Output:
-6/(x-2*4^((4+x)*x)-22)
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {6+4^{4 x+x^2} (-48-24 x) \log (4)}{484+4^{1+8 x+2 x^2}+4^{4 x+x^2} (88-4 x)-44 x+x^2} \, dx=\frac {6}{2 \cdot 2^{2 \, x^{2} + 8 \, x} - x + 22} \] Input:
integrate((2*(-24*x-48)*log(2)*exp(2*(x^2+4*x)*log(2))+6)/(4*exp(2*(x^2+4* x)*log(2))^2+(-4*x+88)*exp(2*(x^2+4*x)*log(2))+x^2-44*x+484),x, algorithm= "fricas")
Output:
6/(2*2^(2*x^2 + 8*x) - x + 22)
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {6+4^{4 x+x^2} (-48-24 x) \log (4)}{484+4^{1+8 x+2 x^2}+4^{4 x+x^2} (88-4 x)-44 x+x^2} \, dx=\frac {3}{- \frac {x}{2} + e^{\left (2 x^{2} + 8 x\right ) \log {\left (2 \right )}} + 11} \] Input:
integrate((2*(-24*x-48)*ln(2)*exp(2*(x**2+4*x)*ln(2))+6)/(4*exp(2*(x**2+4* x)*ln(2))**2+(-4*x+88)*exp(2*(x**2+4*x)*ln(2))+x**2-44*x+484),x)
Output:
3/(-x/2 + exp((2*x**2 + 8*x)*log(2)) + 11)
Time = 0.14 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {6+4^{4 x+x^2} (-48-24 x) \log (4)}{484+4^{1+8 x+2 x^2}+4^{4 x+x^2} (88-4 x)-44 x+x^2} \, dx=-\frac {6}{x - 2 \, e^{\left (2 \, x^{2} \log \left (2\right ) + 8 \, x \log \left (2\right )\right )} - 22} \] Input:
integrate((2*(-24*x-48)*log(2)*exp(2*(x^2+4*x)*log(2))+6)/(4*exp(2*(x^2+4* x)*log(2))^2+(-4*x+88)*exp(2*(x^2+4*x)*log(2))+x^2-44*x+484),x, algorithm= "maxima")
Output:
-6/(x - 2*e^(2*x^2*log(2) + 8*x*log(2)) - 22)
\[ \int \frac {6+4^{4 x+x^2} (-48-24 x) \log (4)}{484+4^{1+8 x+2 x^2}+4^{4 x+x^2} (88-4 x)-44 x+x^2} \, dx=\int { \frac {6 \, {\left (8 \cdot 2^{2 \, x^{2} + 8 \, x} {\left (x + 2\right )} \log \left (2\right ) - 1\right )}}{4 \cdot 2^{2 \, x^{2} + 8 \, x} {\left (x - 22\right )} - x^{2} - 4 \cdot 2^{4 \, x^{2} + 16 \, x} + 44 \, x - 484} \,d x } \] Input:
integrate((2*(-24*x-48)*log(2)*exp(2*(x^2+4*x)*log(2))+6)/(4*exp(2*(x^2+4* x)*log(2))^2+(-4*x+88)*exp(2*(x^2+4*x)*log(2))+x^2-44*x+484),x, algorithm= "giac")
Output:
integrate(6*(8*2^(2*x^2 + 8*x)*(x + 2)*log(2) - 1)/(4*2^(2*x^2 + 8*x)*(x - 22) - x^2 - 4*2^(4*x^2 + 16*x) + 44*x - 484), x)
Timed out. \[ \int \frac {6+4^{4 x+x^2} (-48-24 x) \log (4)}{484+4^{1+8 x+2 x^2}+4^{4 x+x^2} (88-4 x)-44 x+x^2} \, dx=\int -\frac {2\,{\mathrm {e}}^{2\,\ln \left (2\right )\,\left (x^2+4\,x\right )}\,\ln \left (2\right )\,\left (24\,x+48\right )-6}{4\,{\mathrm {e}}^{4\,\ln \left (2\right )\,\left (x^2+4\,x\right )}-44\,x-{\mathrm {e}}^{2\,\ln \left (2\right )\,\left (x^2+4\,x\right )}\,\left (4\,x-88\right )+x^2+484} \,d x \] Input:
int(-(2*exp(2*log(2)*(4*x + x^2))*log(2)*(24*x + 48) - 6)/(4*exp(4*log(2)* (4*x + x^2)) - 44*x - exp(2*log(2)*(4*x + x^2))*(4*x - 88) + x^2 + 484),x)
Output:
int(-(2*exp(2*log(2)*(4*x + x^2))*log(2)*(24*x + 48) - 6)/(4*exp(4*log(2)* (4*x + x^2)) - 44*x - exp(2*log(2)*(4*x + x^2))*(4*x - 88) + x^2 + 484), x )
\[ \int \frac {6+4^{4 x+x^2} (-48-24 x) \log (4)}{484+4^{1+8 x+2 x^2}+4^{4 x+x^2} (88-4 x)-44 x+x^2} \, dx=\int \frac {2 \left (-24 x -48\right ) \mathrm {log}\left (2\right ) {\mathrm e}^{2 \left (x^{2}+4 x \right ) \mathrm {log}\left (2\right )}+6}{4 \left ({\mathrm e}^{2 \left (x^{2}+4 x \right ) \mathrm {log}\left (2\right )}\right )^{2}+\left (-4 x +88\right ) {\mathrm e}^{2 \left (x^{2}+4 x \right ) \mathrm {log}\left (2\right )}+x^{2}-44 x +484}d x \] Input:
int((2*(-24*x-48)*log(2)*exp(2*(x^2+4*x)*log(2))+6)/(4*exp(2*(x^2+4*x)*log (2))^2+(-4*x+88)*exp(2*(x^2+4*x)*log(2))+x^2-44*x+484),x)
Output:
int((2*(-24*x-48)*log(2)*exp(2*(x^2+4*x)*log(2))+6)/(4*exp(2*(x^2+4*x)*log (2))^2+(-4*x+88)*exp(2*(x^2+4*x)*log(2))+x^2-44*x+484),x)