Integrand size = 89, antiderivative size = 31 \[ \int \frac {48 (i \pi +\log (3))+48 (i \pi +\log (3)) \log (5 x)-15 x^2 (i \pi +\log (3))^2 \log ^2(5 x)}{512+320 x^2 (i \pi +\log (3)) \log (5 x)+50 x^4 (i \pi +\log (3))^2 \log ^2(5 x)} \, dx=\frac {3}{2 \left (5 x+\frac {16}{x (i \pi +\log (3)) \log (5 x)}\right )} \] Output:
3/(32/ln(5*x)/x/(ln(3)+I*Pi)+10*x)
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {48 (i \pi +\log (3))+48 (i \pi +\log (3)) \log (5 x)-15 x^2 (i \pi +\log (3))^2 \log ^2(5 x)}{512+320 x^2 (i \pi +\log (3)) \log (5 x)+50 x^4 (i \pi +\log (3))^2 \log ^2(5 x)} \, dx=\frac {3 x (\pi -i \log (3)) \log (5 x)}{2 \left (-16 i+5 x^2 (\pi -i \log (3)) \log (5 x)\right )} \] Input:
Integrate[(48*(I*Pi + Log[3]) + 48*(I*Pi + Log[3])*Log[5*x] - 15*x^2*(I*Pi + Log[3])^2*Log[5*x]^2)/(512 + 320*x^2*(I*Pi + Log[3])*Log[5*x] + 50*x^4* (I*Pi + Log[3])^2*Log[5*x]^2),x]
Output:
(3*x*(Pi - I*Log[3])*Log[5*x])/(2*(-16*I + 5*x^2*(Pi - I*Log[3])*Log[5*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.
| \(\displaystyle \int \frac {-15 x^2 (\log (3)+i \pi )^2 \log ^2(5 x)+48 (\log (3)+i \pi ) \log (5 x)+48 (\log (3)+i \pi )}{50 x^4 (\log (3)+i \pi )^2 \log ^2(5 x)+320 x^2 (\log (3)+i \pi ) \log (5 x)+512} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {15 x^2 (\log (3)+i \pi )^2 \log ^2(5 x)-48 (\log (3)+i \pi ) \log (5 x)-48 (\log (3)+i \pi )}{2 \left (16 i-5 \pi x^2 \left (1-\frac {i \log (3)}{\pi }\right ) \log (5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int -\frac {3 \left (-5 x^2 (i \pi +\log (3))^2 \log ^2(5 x)+16 (i \pi +\log (3)) \log (5 x)+16 (i \pi +\log (3))\right )}{\left (16 i-5 x^2 (\pi -i \log (3)) \log (5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{2} \int \frac {-5 x^2 (i \pi +\log (3))^2 \log ^2(5 x)+16 (i \pi +\log (3)) \log (5 x)+16 (i \pi +\log (3))}{\left (16 i-5 x^2 (\pi -i \log (3)) \log (5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {3}{2} \int \frac {(\pi -i \log (3)) \left (5 \pi x^2 \left (1-\frac {i \log (3)}{\pi }\right ) \log ^2(5 x)+16 i \log (5 x)+16 i\right )}{\left (16 i-5 x^2 (\pi -i \log (3)) \log (5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{2} (\pi -i \log (3)) \int \frac {5 x^2 (\pi -i \log (3)) \log ^2(5 x)+16 i \log (5 x)+16 i}{\left (16 i-5 x^2 (\pi -i \log (3)) \log (5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} (\pi -i \log (3)) \int \left (\frac {16 \left (5 x^2 (i \pi +\log (3))-32\right )}{5 x^2 (\pi -i \log (3)) \left (16 i-5 \pi x^2 \left (1-\frac {i \log (3)}{\pi }\right ) \log (5 x)\right )^2}+\frac {48 i}{5 x^2 (-\pi +i \log (3)) \left (16 i-5 \pi x^2 \left (1-\frac {i \log (3)}{\pi }\right ) \log (5 x)\right )}+\frac {1}{5 x^2 (\pi -i \log (3))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{2} (\pi -i \log (3)) \left (\frac {16 (\log (243)+5 i \pi ) \int \frac {1}{\left (16 i-5 \pi x^2 \left (1-\frac {i \log (3)}{\pi }\right ) \log (5 x)\right )^2}dx}{5 (\pi -i \log (3))}-\frac {512 \int \frac {1}{x^2 \left (16 i-5 \pi x^2 \left (1-\frac {i \log (3)}{\pi }\right ) \log (5 x)\right )^2}dx}{5 (\pi -i \log (3))}+\frac {48 \int \frac {1}{x^2 \left (16 i-5 \pi x^2 \left (1-\frac {i \log (3)}{\pi }\right ) \log (5 x)\right )}dx}{5 (\log (3)+i \pi )}-\frac {1}{5 x (\pi -i \log (3))}\right )\) |
Input:
Int[(48*(I*Pi + Log[3]) + 48*(I*Pi + Log[3])*Log[5*x] - 15*x^2*(I*Pi + Log [3])^2*Log[5*x]^2)/(512 + 320*x^2*(I*Pi + Log[3])*Log[5*x] + 50*x^4*(I*Pi + Log[3])^2*Log[5*x]^2),x]
Output:
$Aborted
Time = 0.97 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23
| method | result | size |
| risch | \(\frac {3}{10 x}-\frac {24}{5 x \left (5 i \pi \ln \left (5 x \right ) x^{2}+5 \ln \left (3\right ) \ln \left (5 x \right ) x^{2}+16\right )}\) | \(38\) |
| derivativedivides | \(\frac {15 \left (\ln \left (3\right )+i \pi \right ) \ln \left (5 x \right ) x}{2 \left (25 i \pi \ln \left (5 x \right ) x^{2}+25 \ln \left (3\right ) \ln \left (5 x \right ) x^{2}+80\right )}\) | \(41\) |
| default | \(\frac {15 \left (\ln \left (3\right )+i \pi \right ) \ln \left (5 x \right ) x}{2 \left (25 i \pi \ln \left (5 x \right ) x^{2}+25 \ln \left (3\right ) \ln \left (5 x \right ) x^{2}+80\right )}\) | \(41\) |
| parallelrisch | \(\frac {48 \ln \left (3\right ) \ln \left (5 x \right ) x +48 i \pi \ln \left (5 x \right ) x}{160 i \pi \ln \left (5 x \right ) x^{2}+160 \ln \left (3\right ) \ln \left (5 x \right ) x^{2}+512}\) | \(48\) |
Input:
int((-15*x^2*(ln(3)+I*Pi)^2*ln(5*x)^2+48*(ln(3)+I*Pi)*ln(5*x)+48*ln(3)+48* I*Pi)/(50*x^4*(ln(3)+I*Pi)^2*ln(5*x)^2+320*x^2*(ln(3)+I*Pi)*ln(5*x)+512),x ,method=_RETURNVERBOSE)
Output:
3/10/x-24/5/x/(5*I*Pi*ln(5*x)*x^2+5*ln(3)*ln(5*x)*x^2+16)
Time = 0.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {48 (i \pi +\log (3))+48 (i \pi +\log (3)) \log (5 x)-15 x^2 (i \pi +\log (3))^2 \log ^2(5 x)}{512+320 x^2 (i \pi +\log (3)) \log (5 x)+50 x^4 (i \pi +\log (3))^2 \log ^2(5 x)} \, dx=\frac {3 \, {\left (-i \, \pi x - x \log \left (3\right )\right )} \log \left (5 \, x\right )}{2 \, {\left (5 \, {\left (-i \, \pi x^{2} - x^{2} \log \left (3\right )\right )} \log \left (5 \, x\right ) - 16\right )}} \] Input:
integrate((-15*x^2*(log(3)+I*pi)^2*log(5*x)^2+48*(log(3)+I*pi)*log(5*x)+48 *log(3)+48*I*pi)/(50*x^4*(log(3)+I*pi)^2*log(5*x)^2+320*x^2*(log(3)+I*pi)* log(5*x)+512),x, algorithm="fricas")
Output:
3/2*(-I*pi*x - x*log(3))*log(5*x)/(5*(-I*pi*x^2 - x^2*log(3))*log(5*x) - 1 6)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (19) = 38\).
Time = 8.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {48 (i \pi +\log (3))+48 (i \pi +\log (3)) \log (5 x)-15 x^2 (i \pi +\log (3))^2 \log ^2(5 x)}{512+320 x^2 (i \pi +\log (3)) \log (5 x)+50 x^4 (i \pi +\log (3))^2 \log ^2(5 x)} \, dx=\frac {24}{- 25 x^{3} \log {\left (3 \right )} \log {\left (5 \right )} - 25 i \pi x^{3} \log {\left (5 \right )} - 80 x + \left (- 25 x^{3} \log {\left (3 \right )} - 25 i \pi x^{3}\right ) \log {\left (x \right )}} + \frac {3}{10 x} \] Input:
integrate((-15*x**2*(ln(3)+I*pi)**2*ln(5*x)**2+48*(ln(3)+I*pi)*ln(5*x)+48* ln(3)+48*I*pi)/(50*x**4*(ln(3)+I*pi)**2*ln(5*x)**2+320*x**2*(ln(3)+I*pi)*l n(5*x)+512),x)
Output:
24/(-25*x**3*log(3)*log(5) - 25*I*pi*x**3*log(5) - 80*x + (-25*x**3*log(3) - 25*I*pi*x**3)*log(x)) + 3/(10*x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90 \[ \int \frac {48 (i \pi +\log (3))+48 (i \pi +\log (3)) \log (5 x)-15 x^2 (i \pi +\log (3))^2 \log ^2(5 x)}{512+320 x^2 (i \pi +\log (3)) \log (5 x)+50 x^4 (i \pi +\log (3))^2 \log ^2(5 x)} \, dx=\frac {3 \, {\left ({\left (\pi - i \, \log \left (3\right )\right )} x \log \left (x\right ) + {\left (\pi \log \left (5\right ) - i \, \log \left (5\right ) \log \left (3\right )\right )} x\right )}}{2 \, {\left (5 \, {\left (\pi - i \, \log \left (3\right )\right )} x^{2} \log \left (x\right ) + 5 \, {\left (\pi \log \left (5\right ) - i \, \log \left (5\right ) \log \left (3\right )\right )} x^{2} - 16 i\right )}} \] Input:
integrate((-15*x^2*(log(3)+I*pi)^2*log(5*x)^2+48*(log(3)+I*pi)*log(5*x)+48 *log(3)+48*I*pi)/(50*x^4*(log(3)+I*pi)^2*log(5*x)^2+320*x^2*(log(3)+I*pi)* log(5*x)+512),x, algorithm="maxima")
Output:
3/2*((pi - I*log(3))*x*log(x) + (pi*log(5) - I*log(5)*log(3))*x)/(5*(pi - I*log(3))*x^2*log(x) + 5*(pi*log(5) - I*log(5)*log(3))*x^2 - 16*I)
Time = 0.13 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {48 (i \pi +\log (3))+48 (i \pi +\log (3)) \log (5 x)-15 x^2 (i \pi +\log (3))^2 \log ^2(5 x)}{512+320 x^2 (i \pi +\log (3)) \log (5 x)+50 x^4 (i \pi +\log (3))^2 \log ^2(5 x)} \, dx=-\frac {24}{25 i \, \pi x^{3} \log \left (5 \, x\right ) + 25 \, x^{3} \log \left (3\right ) \log \left (5 \, x\right ) + 80 \, x} + \frac {3}{10 \, x} \] Input:
integrate((-15*x^2*(log(3)+I*pi)^2*log(5*x)^2+48*(log(3)+I*pi)*log(5*x)+48 *log(3)+48*I*pi)/(50*x^4*(log(3)+I*pi)^2*log(5*x)^2+320*x^2*(log(3)+I*pi)* log(5*x)+512),x, algorithm="giac")
Output:
-24/(25*I*pi*x^3*log(5*x) + 25*x^3*log(3)*log(5*x) + 80*x) + 3/10/x
Timed out. \[ \int \frac {48 (i \pi +\log (3))+48 (i \pi +\log (3)) \log (5 x)-15 x^2 (i \pi +\log (3))^2 \log ^2(5 x)}{512+320 x^2 (i \pi +\log (3)) \log (5 x)+50 x^4 (i \pi +\log (3))^2 \log ^2(5 x)} \, dx=\int \frac {\Pi \,48{}\mathrm {i}+48\,\ln \left (3\right )+48\,\ln \left (5\,x\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )-15\,x^2\,{\ln \left (5\,x\right )}^2\,{\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )}^2}{50\,x^4\,{\ln \left (5\,x\right )}^2\,{\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )}^2+320\,x^2\,\ln \left (5\,x\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )+512} \,d x \] Input:
int((Pi*48i + 48*log(3) + 48*log(5*x)*(Pi*1i + log(3)) - 15*x^2*log(5*x)^2 *(Pi*1i + log(3))^2)/(50*x^4*log(5*x)^2*(Pi*1i + log(3))^2 + 320*x^2*log(5 *x)*(Pi*1i + log(3)) + 512),x)
Output:
int((Pi*48i + 48*log(3) + 48*log(5*x)*(Pi*1i + log(3)) - 15*x^2*log(5*x)^2 *(Pi*1i + log(3))^2)/(50*x^4*log(5*x)^2*(Pi*1i + log(3))^2 + 320*x^2*log(5 *x)*(Pi*1i + log(3)) + 512), x)
\[ \int \frac {48 (i \pi +\log (3))+48 (i \pi +\log (3)) \log (5 x)-15 x^2 (i \pi +\log (3))^2 \log ^2(5 x)}{512+320 x^2 (i \pi +\log (3)) \log (5 x)+50 x^4 (i \pi +\log (3))^2 \log ^2(5 x)} \, dx=\int \frac {-15 x^{2} \left (\mathrm {log}\left (3\right )+i \pi \right )^{2} \mathrm {log}\left (5 x \right )^{2}+48 \left (\mathrm {log}\left (3\right )+i \pi \right ) \mathrm {log}\left (5 x \right )+48 \,\mathrm {log}\left (3\right )+48 i \pi }{50 x^{4} \left (\mathrm {log}\left (3\right )+i \pi \right )^{2} \mathrm {log}\left (5 x \right )^{2}+320 x^{2} \left (\mathrm {log}\left (3\right )+i \pi \right ) \mathrm {log}\left (5 x \right )+512}d x \] Input:
int((-15*x^2*(log(3)+I*Pi)^2*log(5*x)^2+48*(log(3)+I*Pi)*log(5*x)+48*log(3 )+48*I*Pi)/(50*x^4*(log(3)+I*Pi)^2*log(5*x)^2+320*x^2*(log(3)+I*Pi)*log(5* x)+512),x)
Output:
int((-15*x^2*(log(3)+I*Pi)^2*log(5*x)^2+48*(log(3)+I*Pi)*log(5*x)+48*log(3 )+48*I*Pi)/(50*x^4*(log(3)+I*Pi)^2*log(5*x)^2+320*x^2*(log(3)+I*Pi)*log(5* x)+512),x)