Integrand size = 14, antiderivative size = 29 \[ \int \frac {\cos ^2(\log (x)) \sin ^2(\log (x))}{x} \, dx=\frac {\log (x)}{8}+\frac {1}{8} \cos (\log (x)) \sin (\log (x))-\frac {1}{4} \cos ^3(\log (x)) \sin (\log (x)) \]
Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.55 \[ \int \frac {\cos ^2(\log (x)) \sin ^2(\log (x))}{x} \, dx=\frac {\log (x)}{8}-\frac {1}{32} \sin (4 \log (x)) \]
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3039, 3042, 3048, 3042, 3115, 24}
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.
\(\displaystyle \int \frac {\sin ^2(\log (x)) \cos ^2(\log (x))}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \int \sin ^2(\log (x)) \cos ^2(\log (x))d\log (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (\log (x))^2 \cos (\log (x))^2d\log (x)\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {1}{4} \int \cos ^2(\log (x))d\log (x)-\frac {1}{4} \sin (\log (x)) \cos ^3(\log (x))\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \int \sin \left (\log (x)+\frac {\pi }{2}\right )^2d\log (x)-\frac {1}{4} \sin (\log (x)) \cos ^3(\log (x))\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{4} \left (\frac {\int 1d\log (x)}{2}+\frac {1}{2} \sin (\log (x)) \cos (\log (x))\right )-\frac {1}{4} \sin (\log (x)) \cos ^3(\log (x))\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{4} \left (\frac {\log (x)}{2}+\frac {1}{2} \sin (\log (x)) \cos (\log (x))\right )-\frac {1}{4} \sin (\log (x)) \cos ^3(\log (x))\) |
3.9.54.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Time = 0.30 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.45
method | result | size |
parallelrisch | \(\ln \left (x^{\frac {1}{8}}\right )-\frac {\sin \left (4 \ln \left (x \right )\right )}{32}\) | \(13\) |
derivativedivides | \(\frac {\ln \left (x \right )}{8}+\frac {\cos \left (\ln \left (x \right )\right ) \sin \left (\ln \left (x \right )\right )}{8}-\frac {\cos \left (\ln \left (x \right )\right )^{3} \sin \left (\ln \left (x \right )\right )}{4}\) | \(24\) |
default | \(\frac {\ln \left (x \right )}{8}+\frac {\cos \left (\ln \left (x \right )\right ) \sin \left (\ln \left (x \right )\right )}{8}-\frac {\cos \left (\ln \left (x \right )\right )^{3} \sin \left (\ln \left (x \right )\right )}{4}\) | \(24\) |
risch | \(\frac {\ln \left (x \right )}{8}+\frac {i x^{4 i}}{64}-\frac {i x^{-4 i}}{64}\) | \(24\) |
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^2(\log (x)) \sin ^2(\log (x))}{x} \, dx=-\frac {1}{8} \, {\left (2 \, \cos \left (\log \left (x\right )\right )^{3} - \cos \left (\log \left (x\right )\right )\right )} \sin \left (\log \left (x\right )\right ) + \frac {1}{8} \, \log \left (x\right ) \]
Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (29) = 58\).
Time = 8.13 (sec) , antiderivative size = 476, normalized size of antiderivative = 16.41 \[ \int \frac {\cos ^2(\log (x)) \sin ^2(\log (x))}{x} \, dx=\frac {\log {\left (x \right )} \tan ^{8}{\left (\frac {\log {\left (x \right )}}{2} \right )}}{8 \tan ^{8}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{6}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 48 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 8} + \frac {4 \log {\left (x \right )} \tan ^{6}{\left (\frac {\log {\left (x \right )}}{2} \right )}}{8 \tan ^{8}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{6}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 48 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 8} + \frac {6 \log {\left (x \right )} \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )}}{8 \tan ^{8}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{6}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 48 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 8} + \frac {4 \log {\left (x \right )} \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )}}{8 \tan ^{8}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{6}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 48 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 8} + \frac {\log {\left (x \right )}}{8 \tan ^{8}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{6}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 48 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 8} + \frac {2 \tan ^{7}{\left (\frac {\log {\left (x \right )}}{2} \right )}}{8 \tan ^{8}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{6}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 48 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 8} - \frac {14 \tan ^{5}{\left (\frac {\log {\left (x \right )}}{2} \right )}}{8 \tan ^{8}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{6}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 48 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 8} + \frac {14 \tan ^{3}{\left (\frac {\log {\left (x \right )}}{2} \right )}}{8 \tan ^{8}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{6}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 48 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 8} - \frac {2 \tan {\left (\frac {\log {\left (x \right )}}{2} \right )}}{8 \tan ^{8}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{6}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 48 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 32 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 8} \]
log(x)*tan(log(x)/2)**8/(8*tan(log(x)/2)**8 + 32*tan(log(x)/2)**6 + 48*tan (log(x)/2)**4 + 32*tan(log(x)/2)**2 + 8) + 4*log(x)*tan(log(x)/2)**6/(8*ta n(log(x)/2)**8 + 32*tan(log(x)/2)**6 + 48*tan(log(x)/2)**4 + 32*tan(log(x) /2)**2 + 8) + 6*log(x)*tan(log(x)/2)**4/(8*tan(log(x)/2)**8 + 32*tan(log(x )/2)**6 + 48*tan(log(x)/2)**4 + 32*tan(log(x)/2)**2 + 8) + 4*log(x)*tan(lo g(x)/2)**2/(8*tan(log(x)/2)**8 + 32*tan(log(x)/2)**6 + 48*tan(log(x)/2)**4 + 32*tan(log(x)/2)**2 + 8) + log(x)/(8*tan(log(x)/2)**8 + 32*tan(log(x)/2 )**6 + 48*tan(log(x)/2)**4 + 32*tan(log(x)/2)**2 + 8) + 2*tan(log(x)/2)**7 /(8*tan(log(x)/2)**8 + 32*tan(log(x)/2)**6 + 48*tan(log(x)/2)**4 + 32*tan( log(x)/2)**2 + 8) - 14*tan(log(x)/2)**5/(8*tan(log(x)/2)**8 + 32*tan(log(x )/2)**6 + 48*tan(log(x)/2)**4 + 32*tan(log(x)/2)**2 + 8) + 14*tan(log(x)/2 )**3/(8*tan(log(x)/2)**8 + 32*tan(log(x)/2)**6 + 48*tan(log(x)/2)**4 + 32* tan(log(x)/2)**2 + 8) - 2*tan(log(x)/2)/(8*tan(log(x)/2)**8 + 32*tan(log(x )/2)**6 + 48*tan(log(x)/2)**4 + 32*tan(log(x)/2)**2 + 8)
Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.41 \[ \int \frac {\cos ^2(\log (x)) \sin ^2(\log (x))}{x} \, dx=\frac {1}{8} \, \log \left (x\right ) - \frac {1}{32} \, \sin \left (4 \, \log \left (x\right )\right ) \]
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.41 \[ \int \frac {\cos ^2(\log (x)) \sin ^2(\log (x))}{x} \, dx=\frac {1}{8} \, \log \left (x\right ) - \frac {1}{32} \, \sin \left (4 \, \log \left (x\right )\right ) \]
Time = 26.87 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.41 \[ \int \frac {\cos ^2(\log (x)) \sin ^2(\log (x))}{x} \, dx=\frac {\ln \left (x\right )}{8}-\frac {\sin \left (4\,\ln \left (x\right )\right )}{32} \]