\(\int \frac {\operatorname {FresnelS}(d (a+b \log (c x^n)))}{x^2} \, dx\) [58]

Optimal result

Integrand size = 17, antiderivative size = 217 \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) e^{\frac {2 a b n+\frac {i}{d^2 \pi }}{2 b^2 n^2}} \left (c x^n\right )^{\frac {1}{n}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {1}{n}-i a b d^2 \pi -i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )}{x}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) e^{\frac {2 a b n-\frac {i}{d^2 \pi }}{2 b^2 n^2}} \left (c x^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {1}{n}+i a b d^2 \pi +i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )}{x}-\frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \] Output:

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

Mathematica [A] (verified)

Time = 2.53 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.90 \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=-\frac {\sqrt [4]{-1} \sqrt {2} e^{\frac {2 a b n-\frac {i}{d^2 \pi }}{2 b^2 n^2}} \left (c x^n\right )^{\frac {1}{n}} \left (i \text {erfi}\left (\frac {(-1)^{3/4} \left (-i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {2 \pi }}\right )+e^{\frac {i}{b^2 d^2 n^2 \pi }} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )\right )+4 \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{4 x} \] Input:

Integrate[FresnelS[d*(a + b*Log[c*x^n])]/x^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {7025, 5128, 2712, 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.

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

\(\Big \downarrow \) 7025

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

\(\Big \downarrow \) 5128

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

\(\Big \downarrow \) 2712

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

\(\Big \downarrow \) 2706

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

\(\Big \downarrow \) 2664

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

Int[FresnelS[d*(a + b*Log[c*x^n])]/x^2,x]
 

Output:

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

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 2712

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

rule 5128

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

rule 7025

Int[FresnelS[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_. 
), x_Symbol] :> Simp[(e*x)^(m + 1)*(FresnelS[d*(a + b*Log[c*x^n])]/(e*(m + 
1))), x] - Simp[b*d*(n/(m + 1))   Int[(e*x)^m*Sin[(Pi/2)*(d*(a + b*Log[c*x^ 
n]))^2], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
 

Maple [F]

\[\int \frac {\operatorname {FresnelS}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}d x\]

Input:

int(FresnelS(d*(a+b*ln(c*x^n)))/x^2,x)
 

Output:

int(FresnelS(d*(a+b*ln(c*x^n)))/x^2,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. 444 vs. \(2 (177) = 354\).

Time = 0.10 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.05 \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {-i \, \pi \sqrt {b^{2} d^{2} n^{2}} x e^{\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + i \, \pi \sqrt {b^{2} d^{2} n^{2}} x e^{\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \pi \sqrt {b^{2} d^{2} n^{2}} x e^{\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \pi \sqrt {b^{2} d^{2} n^{2}} x e^{\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - 2 \, \operatorname {S}\left (b d \log \left (c x^{n}\right ) + a d\right )}{2 \, x} \] Input:

integrate(fresnel_sin(d*(a+b*log(c*x^n)))/x^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {S\left (a d + b d \log {\left (c x^{n} \right )}\right )}{x^{2}}\, dx \] Input:

integrate(fresnels(d*(a+b*ln(c*x**n)))/x**2,x)
 

Output:

Integral(fresnels(a*d + b*d*log(c*x**n))/x**2, x)
 

Maxima [F]

\[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\operatorname {S}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \] Input:

integrate(fresnel_sin(d*(a+b*log(c*x^n)))/x^2,x, algorithm="maxima")
 

Output:

integrate(fresnel_sin((b*log(c*x^n) + a)*d)/x^2, x)
                                                                                    
                                                                                    
 

Giac [F]

\[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\operatorname {S}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \] Input:

integrate(fresnel_sin(d*(a+b*log(c*x^n)))/x^2,x, algorithm="giac")
 

Output:

integrate(fresnel_sin((b*log(c*x^n) + a)*d)/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\mathrm {FresnelS}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \] Input:

int(FresnelS(d*(a + b*log(c*x^n)))/x^2,x)
 

Output:

int(FresnelS(d*(a + b*log(c*x^n)))/x^2, x)
 

Reduce [F]

\[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\mathrm {FresnelS}\left (d \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )\right )}{x^{2}}d x \] Input:

int(FresnelS(d*(a+b*log(c*x^n)))/x^2,x)
 

Output:

int(FresnelS(d*(a+b*log(c*x^n)))/x^2,x)