Integrand size = 93, antiderivative size = 28 \[ \int \frac {-126+e^x (-72-36 x)+\left (-63-36 e^x\right ) \log \left (-35 x^2-20 e^x x^2\right )+\left (-7-4 e^x\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )}{\left (7 x^2+4 e^x x^2\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )} \, dx=\frac {1}{x}+\frac {9}{x \log \left (5 \left (x^2-4 \left (2+e^x\right ) x^2\right )\right )} \] Output:
9/x/ln(5*x^2-20*(2+exp(x))*x^2)+1/x
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {-126+e^x (-72-36 x)+\left (-63-36 e^x\right ) \log \left (-35 x^2-20 e^x x^2\right )+\left (-7-4 e^x\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )}{\left (7 x^2+4 e^x x^2\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )} \, dx=\frac {1}{x}+\frac {9}{x \log \left (-5 \left (7+4 e^x\right ) x^2\right )} \] Input:
Integrate[(-126 + E^x*(-72 - 36*x) + (-63 - 36*E^x)*Log[-35*x^2 - 20*E^x*x ^2] + (-7 - 4*E^x)*Log[-35*x^2 - 20*E^x*x^2]^2)/((7*x^2 + 4*E^x*x^2)*Log[- 35*x^2 - 20*E^x*x^2]^2),x]
Output:
x^(-1) + 9/(x*Log[-5*(7 + 4*E^x)*x^2])
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.
\(\displaystyle \int \frac {\left (-4 e^x-7\right ) \log ^2\left (-20 e^x x^2-35 x^2\right )+\left (-36 e^x-63\right ) \log \left (-20 e^x x^2-35 x^2\right )+e^x (-36 x-72)-126}{\left (4 e^x x^2+7 x^2\right ) \log ^2\left (-20 e^x x^2-35 x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-4 e^x-7\right ) \log ^2\left (-20 e^x x^2-35 x^2\right )+\left (-36 e^x-63\right ) \log \left (-20 e^x x^2-35 x^2\right )+e^x (-36 x-72)-126}{\left (4 e^x x^2+7 x^2\right ) \log ^2\left (-5 \left (4 e^x+7\right ) x^2\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-\log ^2\left (-5 \left (4 e^x+7\right ) x^2\right )-9 \log \left (-5 \left (4 e^x+7\right ) x^2\right )-9 x-18}{x^2 \log ^2\left (-5 \left (4 e^x+7\right ) x^2\right )}+\frac {63}{\left (4 e^x+7\right ) x \log ^2\left (-5 \left (4 e^x+7\right ) x^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -18 \int \frac {1}{x^2 \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )}dx-9 \int \frac {1}{x \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )}dx+63 \int \frac {1}{\left (7+4 e^x\right ) x \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )}dx-9 \int \frac {1}{x^2 \log \left (-5 \left (7+4 e^x\right ) x^2\right )}dx+\frac {1}{x}\) |
Input:
Int[(-126 + E^x*(-72 - 36*x) + (-63 - 36*E^x)*Log[-35*x^2 - 20*E^x*x^2] + (-7 - 4*E^x)*Log[-35*x^2 - 20*E^x*x^2]^2)/((7*x^2 + 4*E^x*x^2)*Log[-35*x^2 - 20*E^x*x^2]^2),x]
Output:
$Aborted
Time = 0.35 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29
method | result | size |
parallelrisch | \(-\frac {-504-56 \ln \left (-5 x^{2} \left (4 \,{\mathrm e}^{x}+7\right )\right )}{56 x \ln \left (-5 x^{2} \left (4 \,{\mathrm e}^{x}+7\right )\right )}\) | \(36\) |
norman | \(\frac {9+\ln \left (-20 \,{\mathrm e}^{x} x^{2}-35 x^{2}\right )}{x \ln \left (-20 \,{\mathrm e}^{x} x^{2}-35 x^{2}\right )}\) | \(37\) |
risch | \(\frac {1}{x}+\frac {18 i}{x \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+\pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right ) \operatorname {csgn}\left (i x^{2} \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right )-\pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{2} \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right )^{2}+2 \pi \operatorname {csgn}\left (i x^{2} \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right )^{2}-\pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right ) \operatorname {csgn}\left (i x^{2} \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right )^{2}-\pi \operatorname {csgn}\left (i x^{2} \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right )^{3}-2 \pi +2 i \ln \left (5\right )+4 i \ln \left (x \right )+2 i \ln \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right )}\) | \(186\) |
Input:
int(((-4*exp(x)-7)*ln(-20*exp(x)*x^2-35*x^2)^2+(-36*exp(x)-63)*ln(-20*exp( x)*x^2-35*x^2)+(-36*x-72)*exp(x)-126)/(4*exp(x)*x^2+7*x^2)/ln(-20*exp(x)*x ^2-35*x^2)^2,x,method=_RETURNVERBOSE)
Output:
-1/56*(-504-56*ln(-5*x^2*(4*exp(x)+7)))/x/ln(-5*x^2*(4*exp(x)+7))
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {-126+e^x (-72-36 x)+\left (-63-36 e^x\right ) \log \left (-35 x^2-20 e^x x^2\right )+\left (-7-4 e^x\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )}{\left (7 x^2+4 e^x x^2\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )} \, dx=\frac {\log \left (-20 \, x^{2} e^{x} - 35 \, x^{2}\right ) + 9}{x \log \left (-20 \, x^{2} e^{x} - 35 \, x^{2}\right )} \] Input:
integrate(((-4*exp(x)-7)*log(-20*exp(x)*x^2-35*x^2)^2+(-36*exp(x)-63)*log( -20*exp(x)*x^2-35*x^2)+(-36*x-72)*exp(x)-126)/(4*exp(x)*x^2+7*x^2)/log(-20 *exp(x)*x^2-35*x^2)^2,x, algorithm="fricas")
Output:
(log(-20*x^2*e^x - 35*x^2) + 9)/(x*log(-20*x^2*e^x - 35*x^2))
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {-126+e^x (-72-36 x)+\left (-63-36 e^x\right ) \log \left (-35 x^2-20 e^x x^2\right )+\left (-7-4 e^x\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )}{\left (7 x^2+4 e^x x^2\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )} \, dx=\frac {1}{x} + \frac {9}{x \log {\left (- 20 x^{2} e^{x} - 35 x^{2} \right )}} \] Input:
integrate(((-4*exp(x)-7)*ln(-20*exp(x)*x**2-35*x**2)**2+(-36*exp(x)-63)*ln (-20*exp(x)*x**2-35*x**2)+(-36*x-72)*exp(x)-126)/(4*exp(x)*x**2+7*x**2)/ln (-20*exp(x)*x**2-35*x**2)**2,x)
Output:
1/x + 9/(x*log(-20*x**2*exp(x) - 35*x**2))
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {-126+e^x (-72-36 x)+\left (-63-36 e^x\right ) \log \left (-35 x^2-20 e^x x^2\right )+\left (-7-4 e^x\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )}{\left (7 x^2+4 e^x x^2\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )} \, dx=\frac {i \, \pi + \log \left (5\right ) + 2 \, \log \left (x\right ) + \log \left (4 \, e^{x} + 7\right ) + 9}{{\left (i \, \pi + \log \left (5\right )\right )} x + 2 \, x \log \left (x\right ) + x \log \left (4 \, e^{x} + 7\right )} \] Input:
integrate(((-4*exp(x)-7)*log(-20*exp(x)*x^2-35*x^2)^2+(-36*exp(x)-63)*log( -20*exp(x)*x^2-35*x^2)+(-36*x-72)*exp(x)-126)/(4*exp(x)*x^2+7*x^2)/log(-20 *exp(x)*x^2-35*x^2)^2,x, algorithm="maxima")
Output:
(I*pi + log(5) + 2*log(x) + log(4*e^x + 7) + 9)/((I*pi + log(5))*x + 2*x*l og(x) + x*log(4*e^x + 7))
Time = 0.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {-126+e^x (-72-36 x)+\left (-63-36 e^x\right ) \log \left (-35 x^2-20 e^x x^2\right )+\left (-7-4 e^x\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )}{\left (7 x^2+4 e^x x^2\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )} \, dx=\frac {\log \left (-20 \, x^{2} e^{x} - 35 \, x^{2}\right ) + 9}{x \log \left (-20 \, x^{2} e^{x} - 35 \, x^{2}\right )} \] Input:
integrate(((-4*exp(x)-7)*log(-20*exp(x)*x^2-35*x^2)^2+(-36*exp(x)-63)*log( -20*exp(x)*x^2-35*x^2)+(-36*x-72)*exp(x)-126)/(4*exp(x)*x^2+7*x^2)/log(-20 *exp(x)*x^2-35*x^2)^2,x, algorithm="giac")
Output:
(log(-20*x^2*e^x - 35*x^2) + 9)/(x*log(-20*x^2*e^x - 35*x^2))
Timed out. \[ \int \frac {-126+e^x (-72-36 x)+\left (-63-36 e^x\right ) \log \left (-35 x^2-20 e^x x^2\right )+\left (-7-4 e^x\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )}{\left (7 x^2+4 e^x x^2\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )} \, dx=\int -\frac {\left (4\,{\mathrm {e}}^x+7\right )\,{\ln \left (-20\,x^2\,{\mathrm {e}}^x-35\,x^2\right )}^2+\left (36\,{\mathrm {e}}^x+63\right )\,\ln \left (-20\,x^2\,{\mathrm {e}}^x-35\,x^2\right )+{\mathrm {e}}^x\,\left (36\,x+72\right )+126}{{\ln \left (-20\,x^2\,{\mathrm {e}}^x-35\,x^2\right )}^2\,\left (4\,x^2\,{\mathrm {e}}^x+7\,x^2\right )} \,d x \] Input:
int(-(log(- 20*x^2*exp(x) - 35*x^2)*(36*exp(x) + 63) + log(- 20*x^2*exp(x) - 35*x^2)^2*(4*exp(x) + 7) + exp(x)*(36*x + 72) + 126)/(log(- 20*x^2*exp( x) - 35*x^2)^2*(4*x^2*exp(x) + 7*x^2)),x)
Output:
int(-(log(- 20*x^2*exp(x) - 35*x^2)*(36*exp(x) + 63) + log(- 20*x^2*exp(x) - 35*x^2)^2*(4*exp(x) + 7) + exp(x)*(36*x + 72) + 126)/(log(- 20*x^2*exp( x) - 35*x^2)^2*(4*x^2*exp(x) + 7*x^2)), x)
Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {-126+e^x (-72-36 x)+\left (-63-36 e^x\right ) \log \left (-35 x^2-20 e^x x^2\right )+\left (-7-4 e^x\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )}{\left (7 x^2+4 e^x x^2\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )} \, dx=\frac {\mathrm {log}\left (-20 e^{x} x^{2}-35 x^{2}\right )+9}{\mathrm {log}\left (-20 e^{x} x^{2}-35 x^{2}\right ) x} \] Input:
int(((-4*exp(x)-7)*log(-20*exp(x)*x^2-35*x^2)^2+(-36*exp(x)-63)*log(-20*ex p(x)*x^2-35*x^2)+(-36*x-72)*exp(x)-126)/(4*exp(x)*x^2+7*x^2)/log(-20*exp(x )*x^2-35*x^2)^2,x)
Output:
(log( - 20*e**x*x**2 - 35*x**2) + 9)/(log( - 20*e**x*x**2 - 35*x**2)*x)