\(\int \operatorname {FresnelS}(d (a+b \log (c x^n))) \, dx\) [56]

Optimal result

Integrand size = 13, antiderivative size = 214 \[ \int \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\left (\frac {1}{4}-\frac {i}{4}\right ) e^{-\frac {2 a b n-\frac {i}{d^2 \pi }}{2 b^2 n^2}} x \left (c x^n\right )^{-1/n} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {1}{n}+i a b d^2 \pi +i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )+\left (\frac {1}{4}-\frac {i}{4}\right ) e^{-\frac {2 a b n+\frac {i}{d^2 \pi }}{2 b^2 n^2}} x \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {1}{n}-i a b d^2 \pi -i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )+x \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \] Output:

(1/4-1/4*I)*x*erf((1/2+1/2*I)*(1/n+I*a*b*d^2*Pi+I*b^2*d^2*Pi*ln(c*x^n))/b/ 
d/Pi^(1/2))/exp(1/2*(2*a*b*n-I/d^2/Pi)/b^2/n^2)/((c*x^n)^(1/n))+(1/4-1/4*I 
)*x*erfi((1/2+1/2*I)*(1/n-I*a*b*d^2*Pi-I*b^2*d^2*Pi*ln(c*x^n))/b/d/Pi^(1/2 
))/exp(1/2*(2*a*b*n+I/d^2/Pi)/b^2/n^2)/((c*x^n)^(1/n))+x*FresnelS(d*(a+b*l 
n(c*x^n)))
 

Mathematica [A] (verified)

Time = 4.26 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.48 \[ \int \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {\sqrt [4]{-1} e^{\frac {1}{2} \left (-\frac {2 a}{b n}-\frac {i}{b^2 d^2 n^2 \pi }-i a^2 d^2 \pi +2 i a b d^2 \pi \left (n \log (x)-\log \left (c x^n\right )\right )-i b^2 d^2 \pi \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right )} x \left (c x^n\right )^{-1/n} \left (e^{\frac {i}{b^2 d^2 n^2 \pi }} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )+i \text {erfi}\left (\frac {(-1)^{3/4} \left (i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {2 \pi }}\right )\right ) \left (\cos \left (\frac {1}{2} d^2 \pi \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2\right )+i \sin \left (\frac {1}{2} d^2 \pi \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2\right )\right )}{2 \sqrt {2}} \] Input:

Integrate[FresnelS[d*(a + b*Log[c*x^n])],x]
 

Output:

x*FresnelS[d*(a + b*Log[c*x^n])] + ((-1)^(1/4)*E^(((-2*a)/(b*n) - I/(b^2*d 
^2*n^2*Pi) - I*a^2*d^2*Pi + (2*I)*a*b*d^2*Pi*(n*Log[x] - Log[c*x^n]) - I*b 
^2*d^2*Pi*(-(n*Log[x]) + Log[c*x^n])^2)/2)*x*(E^(I/(b^2*d^2*n^2*Pi))*Erfi[ 
((1/2 + I/2)*(-I + a*b*d^2*n*Pi + b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[Pi 
])] + I*Erfi[((-1)^(3/4)*(I + a*b*d^2*n*Pi + b^2*d^2*n*Pi*Log[c*x^n]))/(b* 
d*n*Sqrt[2*Pi])])*(Cos[(d^2*Pi*(a - b*n*Log[x] + b*Log[c*x^n])^2)/2] + I*S 
in[(d^2*Pi*(a - b*n*Log[x] + b*Log[c*x^n])^2)/2]))/(2*Sqrt[2]*(c*x^n)^n^(- 
1))
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {7022, 5126, 2710, 2706, 2664, 2633, 2634}

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.

\(\displaystyle \int \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\)

\(\Big \downarrow \) 7022

\(\displaystyle x \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \int \sin \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right )dx\)

\(\Big \downarrow \) 5126

\(\displaystyle x \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \left (\frac {1}{2} i \int e^{-\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2}dx-\frac {1}{2} i \int e^{\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2}dx\right )\)

\(\Big \downarrow \) 2710

\(\displaystyle x \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \left (\frac {1}{2} i x^{i \pi a b d^2 n} \left (c x^n\right )^{-i \pi a b d^2} \int \exp \left (-\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2-\frac {1}{2} i a^2 \pi d^2\right ) x^{-i a b d^2 n \pi }dx-\frac {1}{2} i x^{-i \pi a b d^2 n} \left (c x^n\right )^{i \pi a b d^2} \int \exp \left (\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2+\frac {1}{2} i a^2 \pi d^2\right ) x^{i a b d^2 n \pi }dx\right )\)

\(\Big \downarrow \) 2706

\(\displaystyle x \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \left (\frac {i x \left (c x^n\right )^{-1/n} \int \exp \left (-\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2-\frac {1}{2} i a^2 \pi d^2+\frac {\left (1-i a b d^2 n \pi \right ) \log \left (c x^n\right )}{n}\right )d\log \left (c x^n\right )}{2 n}-\frac {i x \left (c x^n\right )^{-1/n} \int \exp \left (\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2+\frac {1}{2} i a^2 \pi d^2+\frac {\left (i a b n \pi d^2+1\right ) \log \left (c x^n\right )}{n}\right )d\log \left (c x^n\right )}{2 n}\right )\)

\(\Big \downarrow \) 2664

\(\displaystyle x \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \left (\frac {i x \left (c x^n\right )^{-1/n} e^{-\frac {2 a b n+\frac {i}{\pi d^2}}{2 b^2 n^2}} \int \exp \left (\frac {i \left (-i b^2 \pi \log \left (c x^n\right ) d^2-i a b \pi d^2+\frac {1}{n}\right )^2}{2 b^2 d^2 \pi }\right )d\log \left (c x^n\right )}{2 n}-\frac {i x \left (c x^n\right )^{-1/n} e^{-\frac {2 a b n-\frac {i}{\pi d^2}}{2 b^2 n^2}} \int \exp \left (-\frac {i \left (i b^2 \pi \log \left (c x^n\right ) d^2+i a b \pi d^2+\frac {1}{n}\right )^2}{2 b^2 d^2 \pi }\right )d\log \left (c x^n\right )}{2 n}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle x \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \left (-\frac {i x \left (c x^n\right )^{-1/n} e^{-\frac {2 a b n-\frac {i}{\pi d^2}}{2 b^2 n^2}} \int \exp \left (-\frac {i \left (i b^2 \pi \log \left (c x^n\right ) d^2+i a b \pi d^2+\frac {1}{n}\right )^2}{2 b^2 d^2 \pi }\right )d\log \left (c x^n\right )}{2 n}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) x \left (c x^n\right )^{-1/n} e^{-\frac {2 a b n+\frac {i}{\pi d^2}}{2 b^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {1}{n}\right )}{\sqrt {\pi } b d}\right )}{b d n}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle x \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \left (-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) x \left (c x^n\right )^{-1/n} e^{-\frac {2 a b n-\frac {i}{\pi d^2}}{2 b^2 n^2}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {1}{n}\right )}{\sqrt {\pi } b d}\right )}{b d n}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) x \left (c x^n\right )^{-1/n} e^{-\frac {2 a b n+\frac {i}{\pi d^2}}{2 b^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {1}{n}\right )}{\sqrt {\pi } b d}\right )}{b d n}\right )\)

Input:

Int[FresnelS[d*(a + b*Log[c*x^n])],x]
 

Output:

-(b*d*n*(((-1/4 + I/4)*x*Erf[((1/2 + I/2)*(n^(-1) + I*a*b*d^2*Pi + I*b^2*d 
^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(b*d*E^((2*a*b*n - I/(d^2*Pi))/(2*b^2* 
n^2))*n*(c*x^n)^n^(-1)) - ((1/4 - I/4)*x*Erfi[((1/2 + I/2)*(n^(-1) - I*a*b 
*d^2*Pi - I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(b*d*E^((2*a*b*n + I/ 
(d^2*Pi))/(2*b^2*n^2))*n*(c*x^n)^n^(-1)))) + x*FresnelS[d*(a + b*Log[c*x^n 
])]
 

Defintions of rubi rules used

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 2664

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[F^(a - b^2/ 
(4*c))   Int[F^((b + 2*c*x)^2/(4*c)), x], x] /; FreeQ[{F, a, b, c}, x]
 

rule 2706

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]^2*(b_.))*(f_.))*(( 
g_.) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[(g + h*x)^(m + 1)/(h*n*(c*(d + 
e*x)^n)^((m + 1)/n))   Subst[Int[E^(a*f*Log[F] + ((m + 1)*x)/n + b*f*Log[F] 
*x^2), x], x, Log[c*(d + e*x)^n]], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, 
m, n}, x] && EqQ[e*g - d*h, 0]
 

rule 2710

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]*(b_.))^2*(f_.)), x 
_Symbol] :> Simp[((c*(d + e*x)^n)^(2*a*b*f*Log[F])/(d + e*x)^(2*a*b*f*n*Log 
[F]))*Int[(d + e*x)^(2*a*b*f*n*Log[F])*F^(a^2*f + b^2*f*Log[c*(d + e*x)^n]^ 
2), x], x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] &&  !IntegerQ[2*a*b*f*Log[ 
F]]
 

rule 5126

Int[Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^2*(d_.)], x_Symbol] :> Simp[I 
/2   Int[E^((-I)*d*(a + b*Log[c*x^n])^2), x], x] - Simp[I/2   Int[E^(I*d*(a 
 + b*Log[c*x^n])^2), x], x] /; FreeQ[{a, b, c, d, n}, x]
 

rule 7022

Int[FresnelS[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Sim 
p[x*FresnelS[d*(a + b*Log[c*x^n])], x] - Simp[b*d*n   Int[Sin[(Pi/2)*(d*(a 
+ b*Log[c*x^n]))^2], x], x] /; FreeQ[{a, b, c, d, n}, x]
 

Maple [F]

\[\int \operatorname {FresnelS}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]

Input:

int(FresnelS(d*(a+b*ln(c*x^n))),x)
 

Output:

int(FresnelS(d*(a+b*ln(c*x^n))),x)
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (176) = 352\).

Time = 0.10 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.08 \[ \int \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {1}{2} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \frac {1}{2} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{2} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{2} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + x \operatorname {S}\left (b d \log \left (c x^{n}\right ) + a d\right ) \] Input:

integrate(fresnel_sin(d*(a+b*log(c*x^n))),x, algorithm="fricas")
 

Output:

-1/2*I*pi*sqrt(b^2*d^2*n^2)*e^(-log(c)/n - a/(b*n) - 1/2*I/(pi*b^2*d^2*n^2 
))*fresnel_cos((pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n 
 + I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) + 1/2*I*pi*sqrt(b^2*d^2*n^2)*e^( 
-log(c)/n - a/(b*n) + 1/2*I/(pi*b^2*d^2*n^2))*fresnel_cos((pi*b^2*d^2*n^2* 
log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n - I)*sqrt(b^2*d^2*n^2)/(pi*b^2 
*d^2*n^2)) - 1/2*pi*sqrt(b^2*d^2*n^2)*e^(-log(c)/n - a/(b*n) - 1/2*I/(pi*b 
^2*d^2*n^2))*fresnel_sin((pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi 
*a*b*d^2*n + I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - 1/2*pi*sqrt(b^2*d^2* 
n^2)*e^(-log(c)/n - a/(b*n) + 1/2*I/(pi*b^2*d^2*n^2))*fresnel_sin((pi*b^2* 
d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n - I)*sqrt(b^2*d^2*n^2) 
/(pi*b^2*d^2*n^2)) + x*fresnel_sin(b*d*log(c*x^n) + a*d)
 

Sympy [F]

\[ \int \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int S\left (d \left (a + b \log {\left (c x^{n} \right )}\right )\right )\, dx \] Input:

integrate(fresnels(d*(a+b*ln(c*x**n))),x)
 

Output:

Integral(fresnels(d*(a + b*log(c*x**n))), x)
 

Maxima [F]

\[ \int \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \operatorname {S}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \] Input:

integrate(fresnel_sin(d*(a+b*log(c*x^n))),x, algorithm="maxima")
 

Output:

integrate(fresnel_sin((b*log(c*x^n) + a)*d), x)
                                                                                    
                                                                                    
 

Giac [F]

\[ \int \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \operatorname {S}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \] Input:

integrate(fresnel_sin(d*(a+b*log(c*x^n))),x, algorithm="giac")
 

Output:

integrate(fresnel_sin((b*log(c*x^n) + a)*d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {FresnelS}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \] Input:

int(FresnelS(d*(a + b*log(c*x^n))),x)
 

Output:

int(FresnelS(d*(a + b*log(c*x^n))), x)
 

Reduce [F]

\[ \int \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {FresnelS}\left (d \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )\right )d x \] Input:

int(FresnelS(d*(a+b*log(c*x^n))),x)
 

Output:

int(FresnelS(d*(a+b*log(c*x^n))),x)