Integrand size = 17, antiderivative size = 65 \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\cos \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right )}{b d n \pi }+\frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n} \] Output:
cos(1/2*d^2*Pi*(a+b*ln(c*x^n))^2)/b/d/n/Pi+FresnelS(d*(a+b*ln(c*x^n)))*(a+ b*ln(c*x^n))/b/n
Leaf count is larger than twice the leaf count of optimal. \(164\) vs. \(2(65)=130\).
Time = 0.07 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.52 \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\cos \left (\frac {1}{2} a^2 d^2 \pi \right ) \cos \left (a b d^2 \pi \log \left (c x^n\right )+\frac {1}{2} b^2 d^2 \pi \log ^2\left (c x^n\right )\right )}{b d n \pi }+\frac {a \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}+\frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \log \left (c x^n\right )}{n}-\frac {\sin \left (\frac {1}{2} a^2 d^2 \pi \right ) \sin \left (a b d^2 \pi \log \left (c x^n\right )+\frac {1}{2} b^2 d^2 \pi \log ^2\left (c x^n\right )\right )}{b d n \pi } \] Input:
Integrate[FresnelS[d*(a + b*Log[c*x^n])]/x,x]
Output:
(Cos[(a^2*d^2*Pi)/2]*Cos[a*b*d^2*Pi*Log[c*x^n] + (b^2*d^2*Pi*Log[c*x^n]^2) /2])/(b*d*n*Pi) + (a*FresnelS[d*(a + b*Log[c*x^n])])/(b*n) + (FresnelS[d*( a + b*Log[c*x^n])]*Log[c*x^n])/n - (Sin[(a^2*d^2*Pi)/2]*Sin[a*b*d^2*Pi*Log [c*x^n] + (b^2*d^2*Pi*Log[c*x^n]^2)/2])/(b*d*n*Pi)
Time = 0.29 (sec) , antiderivative size = 64, 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, 6972}
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.
\(\displaystyle \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle \frac {\int \operatorname {FresnelS}\left (a d+b \log \left (c x^n\right ) d\right )d\left (a d+b \log \left (c x^n\right ) d\right )}{b d n}\) |
\(\Big \downarrow \) 6972 |
\(\displaystyle \frac {\left (a d+b d \log \left (c x^n\right )\right ) \operatorname {FresnelS}\left (a d+b \log \left (c x^n\right ) d\right )+\frac {\cos \left (\frac {1}{2} \pi \left (a d+b d \log \left (c x^n\right )\right )^2\right )}{\pi }}{b d n}\) |
Input:
Int[FresnelS[d*(a + b*Log[c*x^n])]/x,x]
Output:
(Cos[(Pi*(a*d + b*d*Log[c*x^n])^2)/2]/Pi + FresnelS[a*d + b*d*Log[c*x^n]]* (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[FresnelS[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(FresnelS[a + b*x]/b), x] + Simp[Cos[(Pi/2)*(a + b*x)^2]/(b*Pi), 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 = 1.84 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {\operatorname {FresnelS}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )+\frac {\cos \left (\frac {\pi {\left (a d +b d \ln \left (c \,x^{n}\right )\right )}^{2}}{2}\right )}{\pi }}{n b d}\) | \(63\) |
default | \(\frac {\operatorname {FresnelS}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )+\frac {\cos \left (\frac {\pi {\left (a d +b d \ln \left (c \,x^{n}\right )\right )}^{2}}{2}\right )}{\pi }}{n b d}\) | \(63\) |
Input:
int(FresnelS(d*(a+b*ln(c*x^n)))/x,x,method=_RETURNVERBOSE)
Output:
1/n/b/d*(FresnelS(a*d+b*d*ln(c*x^n))*(a*d+b*d*ln(c*x^n))+1/Pi*cos(1/2*Pi*( a*d+b*d*ln(c*x^n))^2))
Time = 0.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.83 \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (\pi b d n \log \left (x\right ) + \pi b d \log \left (c\right ) + \pi a d\right )} \operatorname {S}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \cos \left (\frac {1}{2} \, \pi b^{2} d^{2} n^{2} \log \left (x\right )^{2} + \pi b^{2} d^{2} n \log \left (c\right ) \log \left (x\right ) + \frac {1}{2} \, \pi b^{2} d^{2} \log \left (c\right )^{2} + \pi a b d^{2} n \log \left (x\right ) + \pi a b d^{2} \log \left (c\right ) + \frac {1}{2} \, \pi a^{2} d^{2}\right )}{\pi b d n} \] Input:
integrate(fresnel_sin(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_sin(b*d*log(c*x^n) + a *d) + cos(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)
\[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {S\left (a d + b d \log {\left (c x^{n} \right )}\right )}{x}\, dx \] Input:
integrate(fresnels(d*(a+b*ln(c*x**n)))/x,x)
Output:
Integral(fresnels(a*d + b*d*log(c*x**n))/x, x)
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.25 \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} d \operatorname {S}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) + \frac {\cos \left (\frac {1}{2} \, \pi b^{2} d^{2} \log \left (c x^{n}\right )^{2} + \pi a b d^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, \pi a^{2} d^{2}\right )}{\pi }}{b d n} \] Input:
integrate(fresnel_sin(d*(a+b*log(c*x^n)))/x,x, algorithm="maxima")
Output:
((b*log(c*x^n) + a)*d*fresnel_sin((b*log(c*x^n) + a)*d) + cos(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)
\[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int { \frac {\operatorname {S}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x} \,d x } \] Input:
integrate(fresnel_sin(d*(a+b*log(c*x^n)))/x,x, algorithm="giac")
Output:
integrate(fresnel_sin((b*log(c*x^n) + a)*d)/x, x)
Timed out. \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {\mathrm {FresnelS}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x} \,d x \] Input:
int(FresnelS(d*(a + b*log(c*x^n)))/x,x)
Output:
int(FresnelS(d*(a + b*log(c*x^n)))/x, x)
\[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {\mathrm {FresnelS}\left (d \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )\right )}{x}d x \] Input:
int(FresnelS(d*(a+b*log(c*x^n)))/x,x)
Output:
int(FresnelS(d*(a+b*log(c*x^n)))/x,x)