Integrand size = 17, antiderivative size = 73 \[ \int \frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \log (x)}{8}-\frac {3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{8 b n}-\frac {\cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n} \]
3/8*ln(x)-3/8*cos(a+b*ln(c*x^n))*sin(a+b*ln(c*x^n))/b/n-1/4*cos(a+b*ln(c*x ^n))*sin(a+b*ln(c*x^n))^3/b/n
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {12 \left (a+b \log \left (c x^n\right )\right )-8 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n} \]
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3039, 3042, 3115, 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.
\(\displaystyle \int \frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \sin ^4\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sin \left (a+b \log \left (c x^n\right )\right )^4d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\frac {3}{4} \int \sin ^2\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b}}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {3}{4} \int \sin \left (a+b \log \left (c x^n\right )\right )^2d\log \left (c x^n\right )-\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b}}{n}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \int 1d\log \left (c x^n\right )-\frac {\sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 b}\right )-\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b}}{n}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \log \left (c x^n\right )-\frac {\sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 b}\right )-\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b}}{n}\) |
(-1/4*(Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/b + (3*(Log[c*x^n]/2 - (Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(2*b)))/4)/n
3.1.22.3.1 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[((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 = 6.52 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.63
method | result | size |
parallelrisch | \(\frac {12 \ln \left (x \right ) b n +\sin \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )-8 \sin \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )}{32 b n}\) | \(46\) |
derivativedivides | \(\frac {-\frac {\left ({\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}+\frac {3 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}\right ) \cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{4}+\frac {3 b \ln \left (c \,x^{n}\right )}{8}+\frac {3 a}{8}}{n b}\) | \(61\) |
default | \(\frac {-\frac {\left ({\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}+\frac {3 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}\right ) \cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{4}+\frac {3 b \ln \left (c \,x^{n}\right )}{8}+\frac {3 a}{8}}{n b}\) | \(61\) |
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \, b n \log \left (x\right ) + {\left (2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 5 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{8 \, b n} \]
1/8*(3*b*n*log(x) + (2*cos(b*n*log(x) + b*log(c) + a)^3 - 5*cos(b*n*log(x) + b*log(c) + a))*sin(b*n*log(x) + b*log(c) + a))/(b*n)
Time = 11.45 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.37 \[ \int \frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=- \frac {\begin {cases} \log {\left (x \right )} \cos {\left (2 a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cos {\left (2 a + 2 b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\sin {\left (2 a + 2 b \log {\left (c x^{n} \right )} \right )}}{2 b n} & \text {otherwise} \end {cases}}{2} + \frac {\begin {cases} \log {\left (x \right )} \cos {\left (4 a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cos {\left (4 a + 4 b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\sin {\left (4 a + 4 b \log {\left (c x^{n} \right )} \right )}}{4 b n} & \text {otherwise} \end {cases}}{8} + \frac {3 \log {\left (x \right )}}{8} \]
-Piecewise((log(x)*cos(2*a), Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log(x)*co s(2*a + 2*b*log(c)), Eq(n, 0)), (sin(2*a + 2*b*log(c*x**n))/(2*b*n), True) )/2 + Piecewise((log(x)*cos(4*a), Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log( x)*cos(4*a + 4*b*log(c)), Eq(n, 0)), (sin(4*a + 4*b*log(c*x**n))/(4*b*n), True))/8 + 3*log(x)/8
Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.27 \[ \int \frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {12 \, b n \log \left (x\right ) + \cos \left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right ) \sin \left (4 \, b \log \left (c\right )\right ) - 8 \, \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) \sin \left (2 \, b \log \left (c\right )\right ) + \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right ) - 8 \, \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}{32 \, b n} \]
1/32*(12*b*n*log(x) + cos(4*b*log(x^n) + 4*a)*sin(4*b*log(c)) - 8*cos(2*b* log(x^n) + 2*a)*sin(2*b*log(c)) + cos(4*b*log(c))*sin(4*b*log(x^n) + 4*a) - 8*cos(2*b*log(c))*sin(2*b*log(x^n) + 2*a))/(b*n)
\[ \int \frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x} \,d x } \]
Time = 29.63 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3\,\ln \left (x^n\right )}{8\,n}-\frac {\frac {\sin \left (2\,a+2\,b\,\ln \left (c\,x^n\right )\right )}{4}-\frac {\sin \left (4\,a+4\,b\,\ln \left (c\,x^n\right )\right )}{32}}{b\,n} \]