Integrand size = 75, antiderivative size = 24 \[ \int \frac {-98+68 x-2 x^2-62 x^3+16 x^4-2 x^6}{\left (49 x-14 x^2+x^3-14 x^4+2 x^5+x^7\right ) \log ^2\left (e^{-\frac {15-5 x}{-7+x+x^3}} x\right )} \, dx=\frac {2}{\log \left (e^{-\frac {5 (3-x)}{-7+x+x^3}} x\right )} \] Output:
2/ln(x/exp(5*(3-x)/(x^3+x-7)))
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-98+68 x-2 x^2-62 x^3+16 x^4-2 x^6}{\left (49 x-14 x^2+x^3-14 x^4+2 x^5+x^7\right ) \log ^2\left (e^{-\frac {15-5 x}{-7+x+x^3}} x\right )} \, dx=\frac {2}{\log \left (e^{\frac {5 (-3+x)}{-7+x+x^3}} x\right )} \] Input:
Integrate[(-98 + 68*x - 2*x^2 - 62*x^3 + 16*x^4 - 2*x^6)/((49*x - 14*x^2 + x^3 - 14*x^4 + 2*x^5 + x^7)*Log[x/E^((15 - 5*x)/(-7 + x + x^3))]^2),x]
Output:
2/Log[E^((5*(-3 + x))/(-7 + x + x^3))*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.
| \(\displaystyle \int \frac {-2 x^6+16 x^4-62 x^3-2 x^2+68 x-98}{\left (x^7+2 x^5-14 x^4+x^3-14 x^2+49 x\right ) \log ^2\left (e^{-\frac {15-5 x}{x^3+x-7}} x\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-2 x^6+16 x^4-62 x^3-2 x^2+68 x-98}{x \left (x^6+2 x^4-14 x^3+x^2-14 x+49\right ) \log ^2\left (e^{-\frac {15-5 x}{x^3+x-7}} x\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \frac {-2 x^6+16 x^4-62 x^3-2 x^2+68 x-98}{x \left (x^3+x-7\right )^2 \log ^2\left (e^{-\frac {15-5 x}{x^3+x-7}} x\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 \left (-x^6+8 x^4-31 x^3-x^2+34 x-49\right )}{x \left (-x^3-x+7\right )^2 \log ^2\left (e^{-\frac {15-5 x}{x^3+x-7}} x\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {x^6-8 x^4+31 x^3+x^2-34 x+49}{x \left (-x^3-x+7\right )^2 \log ^2\left (e^{\frac {5 (3-x)}{-x^3-x+7}} x\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {x^6-8 x^4+31 x^3+x^2-34 x+49}{x \left (-x^3-x+7\right )^2 \log ^2\left (e^{\frac {5 (3-x)}{-x^3-x+7}} x\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {5 \left (9 x^2+2 x-18\right )}{\left (x^3+x-7\right )^2 \log ^2\left (e^{\frac {5 (x-3)}{x^3+x-7}} x\right )}+\frac {1}{x \log ^2\left (e^{\frac {5 (x-3)}{x^3+x-7}} x\right )}-\frac {10}{\left (x^3+x-7\right ) \log ^2\left (e^{\frac {5 (x-3)}{x^3+x-7}} x\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\int \frac {1}{x \log ^2\left (e^{\frac {5 (x-3)}{x^3+x-7}} x\right )}dx-90 \int \frac {1}{\left (x^3+x-7\right )^2 \log ^2\left (e^{\frac {5 (x-3)}{x^3+x-7}} x\right )}dx+10 \int \frac {x}{\left (x^3+x-7\right )^2 \log ^2\left (e^{\frac {5 (x-3)}{x^3+x-7}} x\right )}dx-10 \int \frac {1}{\left (x^3+x-7\right ) \log ^2\left (e^{\frac {5 (x-3)}{x^3+x-7}} x\right )}dx+45 \int \frac {x^2}{\left (x^3+x-7\right )^2 \log ^2\left (e^{\frac {5 (x-3)}{x^3+x-7}} x\right )}dx\right )\) |
Input:
Int[(-98 + 68*x - 2*x^2 - 62*x^3 + 16*x^4 - 2*x^6)/((49*x - 14*x^2 + x^3 - 14*x^4 + 2*x^5 + x^7)*Log[x/E^((15 - 5*x)/(-7 + x + x^3))]^2),x]
Output:
$Aborted
Time = 8.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75
| method | result | size |
| default | \(-\frac {-2940 x^{3}-2940 x +20580}{1470 \ln \left (x \,{\mathrm e}^{\frac {5 x -15}{x^{3}+x -7}}\right ) \left (x^{3}+x -7\right )}\) | \(42\) |
| parallelrisch | \(-\frac {-5880 x^{3}-5880 x +41160}{2940 \ln \left (x \,{\mathrm e}^{\frac {5 x -15}{x^{3}+x -7}}\right ) \left (x^{3}+x -7\right )}\) | \(42\) |
| risch | \(-\frac {4 i}{\pi \,\operatorname {csgn}\left (i {\mathrm e}^{\frac {5 x -15}{x^{3}+x -7}}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{\frac {5 x -15}{x^{3}+x -7}}\right )^{2}-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{\frac {5 x -15}{x^{3}+x -7}}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{\frac {5 x -15}{x^{3}+x -7}}\right ) \operatorname {csgn}\left (i x \right )-\pi \operatorname {csgn}\left (i x \,{\mathrm e}^{\frac {5 x -15}{x^{3}+x -7}}\right )^{3}+\pi \operatorname {csgn}\left (i x \,{\mathrm e}^{\frac {5 x -15}{x^{3}+x -7}}\right )^{2} \operatorname {csgn}\left (i x \right )-2 i \ln \left (x \right )+2 i \ln \left ({\mathrm e}^{-\frac {5 \left (-3+x \right )}{x^{3}+x -7}}\right )}\) | \(168\) |
Input:
int((-2*x^6+16*x^4-62*x^3-2*x^2+68*x-98)/(x^7+2*x^5-14*x^4+x^3-14*x^2+49*x )/ln(x/exp((15-5*x)/(x^3+x-7)))^2,x,method=_RETURNVERBOSE)
Output:
-1/1470*(-2940*x^3-2940*x+20580)/ln(x/exp(-5*(-3+x)/(x^3+x-7)))/(x^3+x-7)
Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {-98+68 x-2 x^2-62 x^3+16 x^4-2 x^6}{\left (49 x-14 x^2+x^3-14 x^4+2 x^5+x^7\right ) \log ^2\left (e^{-\frac {15-5 x}{-7+x+x^3}} x\right )} \, dx=\frac {2}{\log \left (x e^{\left (\frac {5 \, {\left (x - 3\right )}}{x^{3} + x - 7}\right )}\right )} \] Input:
integrate((-2*x^6+16*x^4-62*x^3-2*x^2+68*x-98)/(x^7+2*x^5-14*x^4+x^3-14*x^ 2+49*x)/log(x/exp((15-5*x)/(x^3+x-7)))^2,x, algorithm="fricas")
Output:
2/log(x*e^(5*(x - 3)/(x^3 + x - 7)))
Time = 0.14 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {-98+68 x-2 x^2-62 x^3+16 x^4-2 x^6}{\left (49 x-14 x^2+x^3-14 x^4+2 x^5+x^7\right ) \log ^2\left (e^{-\frac {15-5 x}{-7+x+x^3}} x\right )} \, dx=\frac {2}{\log {\left (x e^{- \frac {15 - 5 x}{x^{3} + x - 7}} \right )}} \] Input:
integrate((-2*x**6+16*x**4-62*x**3-2*x**2+68*x-98)/(x**7+2*x**5-14*x**4+x* *3-14*x**2+49*x)/ln(x/exp((15-5*x)/(x**3+x-7)))**2,x)
Output:
2/log(x*exp(-(15 - 5*x)/(x**3 + x - 7)))
Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-98+68 x-2 x^2-62 x^3+16 x^4-2 x^6}{\left (49 x-14 x^2+x^3-14 x^4+2 x^5+x^7\right ) \log ^2\left (e^{-\frac {15-5 x}{-7+x+x^3}} x\right )} \, dx=\frac {2 \, {\left (x^{3} + x - 7\right )}}{{\left (x^{3} + x - 7\right )} \log \left (x\right ) + 5 \, x - 15} \] Input:
integrate((-2*x^6+16*x^4-62*x^3-2*x^2+68*x-98)/(x^7+2*x^5-14*x^4+x^3-14*x^ 2+49*x)/log(x/exp((15-5*x)/(x^3+x-7)))^2,x, algorithm="maxima")
Output:
2*(x^3 + x - 7)/((x^3 + x - 7)*log(x) + 5*x - 15)
Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {-98+68 x-2 x^2-62 x^3+16 x^4-2 x^6}{\left (49 x-14 x^2+x^3-14 x^4+2 x^5+x^7\right ) \log ^2\left (e^{-\frac {15-5 x}{-7+x+x^3}} x\right )} \, dx=\frac {2 \, {\left (x^{3} + x - 7\right )}}{x^{3} \log \left (x\right ) + x \log \left (x\right ) + 5 \, x - 7 \, \log \left (x\right ) - 15} \] Input:
integrate((-2*x^6+16*x^4-62*x^3-2*x^2+68*x-98)/(x^7+2*x^5-14*x^4+x^3-14*x^ 2+49*x)/log(x/exp((15-5*x)/(x^3+x-7)))^2,x, algorithm="giac")
Output:
2*(x^3 + x - 7)/(x^3*log(x) + x*log(x) + 5*x - 7*log(x) - 15)
Time = 3.71 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {-98+68 x-2 x^2-62 x^3+16 x^4-2 x^6}{\left (49 x-14 x^2+x^3-14 x^4+2 x^5+x^7\right ) \log ^2\left (e^{-\frac {15-5 x}{-7+x+x^3}} x\right )} \, dx=\frac {2\,x^3+2\,x-14}{5\,x-7\,\ln \left (x\right )+x^3\,\ln \left (x\right )+x\,\ln \left (x\right )-15} \] Input:
int(-(2*x^2 - 68*x + 62*x^3 - 16*x^4 + 2*x^6 + 98)/(log(x*exp((5*x - 15)/( x + x^3 - 7)))^2*(49*x - 14*x^2 + x^3 - 14*x^4 + 2*x^5 + x^7)),x)
Output:
(2*x + 2*x^3 - 14)/(5*x - 7*log(x) + x^3*log(x) + x*log(x) - 15)
\[ \int \frac {-98+68 x-2 x^2-62 x^3+16 x^4-2 x^6}{\left (49 x-14 x^2+x^3-14 x^4+2 x^5+x^7\right ) \log ^2\left (e^{-\frac {15-5 x}{-7+x+x^3}} x\right )} \, dx =\text {Too large to display} \] Input:
int((-2*x^6+16*x^4-62*x^3-2*x^2+68*x-98)/(x^7+2*x^5-14*x^4+x^3-14*x^2+49*x )/log(x/exp((15-5*x)/(x^3+x-7)))^2,x)
Output:
2*( - int(x**5/(log((e**((5*x)/(x**3 + x - 7))*x)/e**(15/(x**3 + x - 7)))* *2*x**6 + 2*log((e**((5*x)/(x**3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2*x **4 - 14*log((e**((5*x)/(x**3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2*x**3 + log((e**((5*x)/(x**3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2*x**2 - 14* log((e**((5*x)/(x**3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2*x + 49*log((e **((5*x)/(x**3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2),x) + 8*int(x**3/(l og((e**((5*x)/(x**3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2*x**6 + 2*log(( e**((5*x)/(x**3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2*x**4 - 14*log((e** ((5*x)/(x**3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2*x**3 + log((e**((5*x) /(x**3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2*x**2 - 14*log((e**((5*x)/(x **3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2*x + 49*log((e**((5*x)/(x**3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2),x) - 31*int(x**2/(log((e**((5*x)/(x **3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2*x**6 + 2*log((e**((5*x)/(x**3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2*x**4 - 14*log((e**((5*x)/(x**3 + x - 7))*x)/e**(15/(x**3 + x - 7)))**2*x**3 + log((e**((5*x)/(x**3 + x - 7)) *x)/e**(15/(x**3 + x - 7)))**2*x**2 - 14*log((e**((5*x)/(x**3 + x - 7))*x) /e**(15/(x**3 + x - 7)))**2*x + 49*log((e**((5*x)/(x**3 + x - 7))*x)/e**(1 5/(x**3 + x - 7)))**2),x) - int(x/(log((e**((5*x)/(x**3 + x - 7))*x)/e**(1 5/(x**3 + x - 7)))**2*x**6 + 2*log((e**((5*x)/(x**3 + x - 7))*x)/e**(15/(x **3 + x - 7)))**2*x**4 - 14*log((e**((5*x)/(x**3 + x - 7))*x)/e**(15/(x...