Integrand size = 17, antiderivative size = 55 \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n} \] Output:
Ci(d*(a+b*ln(c*x^n)))*(a+b*ln(c*x^n))/b/n-sin(d*(a+b*ln(c*x^n)))/b/d/n
Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.75 \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {a \operatorname {CosIntegral}\left (a d+b d \log \left (c x^n\right )\right )}{b n}+\frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \log \left (c x^n\right )}{n}-\frac {\cos \left (b d \log \left (c x^n\right )\right ) \sin (a d)}{b d n}-\frac {\cos (a d) \sin \left (b d \log \left (c x^n\right )\right )}{b d n} \] Input:
Integrate[CosIntegral[d*(a + b*Log[c*x^n])]/x,x]
Output:
(a*CosIntegral[a*d + b*d*Log[c*x^n]])/(b*n) + (CosIntegral[d*(a + b*Log[c* x^n])]*Log[c*x^n])/n - (Cos[b*d*Log[c*x^n]]*Sin[a*d])/(b*d*n) - (Cos[a*d]* Sin[b*d*Log[c*x^n]])/(b*d*n)
Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3039, 7281, 7054}
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.
\(\displaystyle \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle \frac {\int \operatorname {CosIntegral}\left (a d+b \log \left (c x^n\right ) d\right )d\left (a d+b \log \left (c x^n\right ) d\right )}{b d n}\) |
\(\Big \downarrow \) 7054 |
\(\displaystyle \frac {\left (a d+b d \log \left (c x^n\right )\right ) \operatorname {CosIntegral}\left (a d+b \log \left (c x^n\right ) d\right )-\sin \left (a d+b d \log \left (c x^n\right )\right )}{b d n}\) |
Input:
Int[CosIntegral[d*(a + b*Log[c*x^n])]/x,x]
Output:
(CosIntegral[a*d + b*d*Log[c*x^n]]*(a*d + b*d*Log[c*x^n]) - Sin[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[CosIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(CosIntegr al[a + b*x]/b), x] - Simp[Sin[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 = 0.76 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {\operatorname {Ci}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )-\sin \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n b d}\) | \(56\) |
default | \(\frac {\operatorname {Ci}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )-\sin \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n b d}\) | \(56\) |
Input:
int(Ci(d*(a+b*ln(c*x^n)))/x,x,method=_RETURNVERBOSE)
Output:
1/n/b/d*(Ci(a*d+b*d*ln(c*x^n))*(a*d+b*d*ln(c*x^n))-sin(a*d+b*d*ln(c*x^n)))
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (55) = 110\).
Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.20 \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (\pi b d n \log \left (x\right ) + \pi b d \log \left (c\right ) + \pi a d\right )} \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \sin \left (\frac {1}{2} \, \pi b^{2} d^{2} n^{2} \log \left (x\right )^{2} + \pi b^{2} d^{2} n \log \left (c\right ) \log \left (x\right ) + \frac {1}{2} \, \pi b^{2} d^{2} \log \left (c\right )^{2} + \pi a b d^{2} n \log \left (x\right ) + \pi a b d^{2} \log \left (c\right ) + \frac {1}{2} \, \pi a^{2} d^{2}\right )}{\pi b d n} \] Input:
integrate(fresnel_cos(d*(a+b*log(c*x^n)))/x,x, algorithm="fricas")
Output:
((pi*b*d*n*log(x) + pi*b*d*log(c) + pi*a*d)*fresnel_cos(b*d*log(c*x^n) + a *d) - sin(1/2*pi*b^2*d^2*n^2*log(x)^2 + pi*b^2*d^2*n*log(c)*log(x) + 1/2*p i*b^2*d^2*log(c)^2 + pi*a*b*d^2*n*log(x) + pi*a*b*d^2*log(c) + 1/2*pi*a^2* d^2))/(pi*b*d*n)
\[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {\operatorname {Ci}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \] Input:
integrate(Ci(d*(a+b*ln(c*x**n)))/x,x)
Output:
Integral(Ci(a*d + b*d*log(c*x**n))/x, x)
Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.49 \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} d \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) - \frac {\sin \left (\frac {1}{2} \, \pi b^{2} d^{2} \log \left (c x^{n}\right )^{2} + \pi a b d^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, \pi a^{2} d^{2}\right )}{\pi }}{b d n} \] Input:
integrate(fresnel_cos(d*(a+b*log(c*x^n)))/x,x, algorithm="maxima")
Output:
((b*log(c*x^n) + a)*d*fresnel_cos((b*log(c*x^n) + a)*d) - sin(1/2*pi*b^2*d ^2*log(c*x^n)^2 + pi*a*b*d^2*log(c*x^n) + 1/2*pi*a^2*d^2)/pi)/(b*d*n)
\[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int { \frac {\operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x} \,d x } \] Input:
integrate(fresnel_cos(d*(a+b*log(c*x^n)))/x,x, algorithm="giac")
Output:
integrate(fresnel_cos((b*log(c*x^n) + a)*d)/x, x)
Timed out. \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\ln \left (c\,x^n\right )\,\mathrm {cosint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{n}+\frac {a\,\mathrm {cosint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,n}-\frac {\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,d\,n} \] Input:
int(cosint(d*(a + b*log(c*x^n)))/x,x)
Output:
(log(c*x^n)*cosint(d*(a + b*log(c*x^n))))/n + (a*cosint(d*(a + b*log(c*x^n ))))/(b*n) - sin(d*(a + b*log(c*x^n)))/(b*d*n)
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.22 \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\mathit {ci} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right ) \mathrm {log}\left (x^{n} c \right ) b d +\mathit {ci} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right ) a d -\sin \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}{b d n} \] Input:
int(Ci(d*(a+b*log(c*x^n)))/x,x)
Output:
(ci(log(x**n*c)*b*d + a*d)*log(x**n*c)*b*d + ci(log(x**n*c)*b*d + a*d)*a*d - sin(log(x**n*c)*b*d + a*d))/(b*d*n)