Integrand size = 17, antiderivative size = 54 \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n}+\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \] Output:
cos(d*(a+b*ln(c*x^n)))/b/d/n+(a+b*ln(c*x^n))*Si(d*(a+b*ln(c*x^n)))/b/n
Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.76 \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\cos (a d) \cos \left (b d \log \left (c x^n\right )\right )}{b d n}-\frac {\sin (a d) \sin \left (b d \log \left (c x^n\right )\right )}{b d n}+\frac {\log \left (c x^n\right ) \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{n}+\frac {a \text {Si}\left (a d+b d \log \left (c x^n\right )\right )}{b n} \] Input:
Integrate[SinIntegral[d*(a + b*Log[c*x^n])]/x,x]
Output:
(Cos[a*d]*Cos[b*d*Log[c*x^n]])/(b*d*n) - (Sin[a*d]*Sin[b*d*Log[c*x^n]])/(b *d*n) + (Log[c*x^n]*SinIntegral[d*(a + b*Log[c*x^n])])/n + (a*SinIntegral[ a*d + b*d*Log[c*x^n]])/(b*n)
Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3039, 7281, 7053}
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.
\(\displaystyle \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle \frac {\int \text {Si}\left (a d+b \log \left (c x^n\right ) d\right )d\left (a d+b \log \left (c x^n\right ) d\right )}{b d n}\) |
\(\Big \downarrow \) 7053 |
\(\displaystyle \frac {\left (a d+b d \log \left (c x^n\right )\right ) \text {Si}\left (a d+b \log \left (c x^n\right ) d\right )+\cos \left (a d+b d \log \left (c x^n\right )\right )}{b d n}\) |
Input:
Int[SinIntegral[d*(a + b*Log[c*x^n])]/x,x]
Output:
(Cos[a*d + b*d*Log[c*x^n]] + (a*d + b*d*Log[c*x^n])*SinIntegral[a*d + b*d* Log[c*x^n]])/(b*d*n)
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[SinIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(SinIntegr al[a + b*x]/b), x] + Simp[Cos[a + b*x]/b, x] /; FreeQ[{a, b}, x]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
Time = 0.72 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\operatorname {Si}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )+\cos \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n b d}\) | \(54\) |
default | \(\frac {\operatorname {Si}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )+\cos \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n b d}\) | \(54\) |
Input:
int(Si(d*(a+b*ln(c*x^n)))/x,x,method=_RETURNVERBOSE)
Output:
1/n/b/d*(Si(a*d+b*d*ln(c*x^n))*(a*d+b*d*ln(c*x^n))+cos(a*d+b*d*ln(c*x^n)))
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06 \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \operatorname {Si}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \cos \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}{b d n} \] Input:
integrate(sin_integral(d*(a+b*log(c*x^n)))/x,x, algorithm="fricas")
Output:
((b*d*n*log(x) + b*d*log(c) + a*d)*sin_integral(b*d*log(c*x^n) + a*d) + co s(b*d*n*log(x) + b*d*log(c) + a*d))/(b*d*n)
\[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {\operatorname {Si}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \] Input:
integrate(Si(d*(a+b*ln(c*x**n)))/x,x)
Output:
Integral(Si(a*d + b*d*log(c*x**n))/x, x)
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} d \operatorname {Si}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) + \cos \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{b d n} \] Input:
integrate(sin_integral(d*(a+b*log(c*x^n)))/x,x, algorithm="maxima")
Output:
((b*log(c*x^n) + a)*d*sin_integral((b*log(c*x^n) + a)*d) + cos((b*log(c*x^ n) + a)*d))/(b*d*n)
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.09 \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \operatorname {Si}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) + \cos \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}{b d n} \] Input:
integrate(sin_integral(d*(a+b*log(c*x^n)))/x,x, algorithm="giac")
Output:
((b*d*n*log(x) + b*d*log(c) + a*d)*sin_integral(b*d*n*log(x) + b*d*log(c) + a*d) + cos(b*d*n*log(x) + b*d*log(c) + a*d))/(b*d*n)
Timed out. \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\mathrm {sinint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,\ln \left (c\,x^n\right )}{n}+\frac {a\,\mathrm {sinint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,n}+\frac {\cos \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,d\,n} \] Input:
int(sinint(d*(a + b*log(c*x^n)))/x,x)
Output:
(sinint(d*(a + b*log(c*x^n)))*log(c*x^n))/n + (a*sinint(d*(a + b*log(c*x^n ))))/(b*n) + cos(d*(a + b*log(c*x^n)))/(b*d*n)
Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.20 \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\cos \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )+\mathrm {log}\left (x^{n} c \right ) \mathit {si} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right ) b d +\mathit {si} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right ) a d}{b d n} \] Input:
int(Si(d*(a+b*log(c*x^n)))/x,x)
Output:
(cos(log(x**n*c)*b*d + a*d) + log(x**n*c)*si(log(x**n*c)*b*d + a*d)*b*d + si(log(x**n*c)*b*d + a*d)*a*d)/(b*d*n)