Integrand size = 13, antiderivative size = 214 \[ \int \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\left (\frac {1}{4}+\frac {i}{4}\right ) e^{-\frac {2 a b n-\frac {i}{d^2 \pi }}{2 b^2 n^2}} x \left (c x^n\right )^{-1/n} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {1}{n}+i a b d^2 \pi +i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )-\left (\frac {1}{4}+\frac {i}{4}\right ) e^{-\frac {2 a b n+\frac {i}{d^2 \pi }}{2 b^2 n^2}} x \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {1}{n}-i a b d^2 \pi -i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )+x \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
(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*FresnelC(d*(a+b*l n(c*x^n)))
Time = 4.68 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.47 \[ \int \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {\sqrt [4]{-1} e^{\frac {1}{2} \left (-\frac {2 a}{b n}-\frac {i}{b^2 d^2 n^2 \pi }-i a^2 d^2 \pi +2 i a b d^2 \pi \left (n \log (x)-\log \left (c x^n\right )\right )-i b^2 d^2 \pi \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right )} x \left (c x^n\right )^{-1/n} \left (i e^{\frac {i}{b^2 d^2 n^2 \pi }} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )+\text {erfi}\left (\frac {(-1)^{3/4} \left (i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {2 \pi }}\right )\right ) \left (\cos \left (\frac {1}{2} d^2 \pi \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2\right )+i \sin \left (\frac {1}{2} d^2 \pi \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2\right )\right )}{2 \sqrt {2}} \]
x*FresnelC[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*(I*E^(I/(b^2*d^2*n^2*Pi))*Erf i[((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])] + 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))
Time = 0.76 (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 = {7023, 5127, 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.
| \(\displaystyle \int \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
| \(\Big \downarrow \) 7023 |
| \(\displaystyle x \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \int \cos \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right )dx\) |
| \(\Big \downarrow \) 5127 |
| \(\displaystyle x \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \left (\frac {1}{2} \int e^{-\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2}dx+\frac {1}{2} \int e^{\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2}dx\right )\) |
| \(\Big \downarrow \) 2710 |
| \(\displaystyle x \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \left (\frac {1}{2} x^{i \pi a b d^2 n} \left (c x^n\right )^{-i \pi a b d^2} \int \exp \left (-\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2-\frac {1}{2} i a^2 \pi d^2\right ) x^{-i a b d^2 n \pi }dx+\frac {1}{2} x^{-i \pi a b d^2 n} \left (c x^n\right )^{i \pi a b d^2} \int \exp \left (\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2+\frac {1}{2} i a^2 \pi d^2\right ) x^{i a b d^2 n \pi }dx\right )\) |
| \(\Big \downarrow \) 2706 |
| \(\displaystyle x \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \left (\frac {x \left (c x^n\right )^{-1/n} \int \exp \left (-\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2-\frac {1}{2} i a^2 \pi d^2+\frac {\left (1-i a b d^2 n \pi \right ) \log \left (c x^n\right )}{n}\right )d\log \left (c x^n\right )}{2 n}+\frac {x \left (c x^n\right )^{-1/n} \int \exp \left (\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2+\frac {1}{2} i a^2 \pi d^2+\frac {\left (i a b n \pi d^2+1\right ) \log \left (c x^n\right )}{n}\right )d\log \left (c x^n\right )}{2 n}\right )\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle x \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \left (\frac {x \left (c x^n\right )^{-1/n} e^{-\frac {2 a b n+\frac {i}{\pi d^2}}{2 b^2 n^2}} \int \exp \left (\frac {i \left (-i b^2 \pi \log \left (c x^n\right ) d^2-i a b \pi d^2+\frac {1}{n}\right )^2}{2 b^2 d^2 \pi }\right )d\log \left (c x^n\right )}{2 n}+\frac {x \left (c x^n\right )^{-1/n} e^{-\frac {2 a b n-\frac {i}{\pi d^2}}{2 b^2 n^2}} \int \exp \left (-\frac {i \left (i b^2 \pi \log \left (c x^n\right ) d^2+i a b \pi d^2+\frac {1}{n}\right )^2}{2 b^2 d^2 \pi }\right )d\log \left (c x^n\right )}{2 n}\right )\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle x \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \left (\frac {x \left (c x^n\right )^{-1/n} e^{-\frac {2 a b n-\frac {i}{\pi d^2}}{2 b^2 n^2}} \int \exp \left (-\frac {i \left (i b^2 \pi \log \left (c x^n\right ) d^2+i a b \pi d^2+\frac {1}{n}\right )^2}{2 b^2 d^2 \pi }\right )d\log \left (c x^n\right )}{2 n}+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) x \left (c x^n\right )^{-1/n} e^{-\frac {2 a b n+\frac {i}{\pi d^2}}{2 b^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {1}{n}\right )}{\sqrt {\pi } b d}\right )}{b d n}\right )\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle x \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) x \left (c x^n\right )^{-1/n} e^{-\frac {2 a b n+\frac {i}{\pi d^2}}{2 b^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {1}{n}\right )}{\sqrt {\pi } b d}\right )}{b d n}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) x \left (c x^n\right )^{-1/n} e^{-\frac {2 a b n-\frac {i}{\pi d^2}}{2 b^2 n^2}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {1}{n}\right )}{\sqrt {\pi } b d}\right )}{b d n}\right )\) |
-(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*FresnelC[d*(a + b*Log[c*x^n ])]
3.2.65.3.1 Defintions of rubi rules used
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]
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]
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]
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]
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]]
Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^2*(d_.)], x_Symbol] :> Simp[1 /2 Int[E^((-I)*d*(a + b*Log[c*x^n])^2), x], x] + Simp[1/2 Int[E^(I*d*(a + b*Log[c*x^n])^2), x], x] /; FreeQ[{a, b, c, d, n}, x]
Int[FresnelC[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Sim p[x*FresnelC[d*(a + b*Log[c*x^n])], x] - Simp[b*d*n Int[Cos[(Pi/2)*(d*(a + b*Log[c*x^n]))^2], x], x] /; FreeQ[{a, b, c, d, n}, x]
\[\int \operatorname {FresnelC}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
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 = 0.28 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.08 \[ \int \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {1}{2} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{2} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \frac {1}{2} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{2} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + x \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right ) \]
-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_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_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_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*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_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_cos(b*d*log(c*x^n) + a*d)
\[ \int \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int C\left (d \left (a + b \log {\left (c x^{n} \right )}\right )\right )\, dx \]
\[ \int \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
Timed out. \[ \int \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {FresnelC}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]