Integrand size = 19, antiderivative size = 280 \[ \int (e x)^m \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) e^{\frac {i (1+m) \left (1+m+2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+m+i a b d^2 n \pi +i b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )}{1+m}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) e^{-\frac {i (1+m) \left (1+m-2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+m-i a b d^2 n \pi -i b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )}{1+m}+\frac {(e x)^{1+m} \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)} \]
(1/4+1/4*I)*exp(1/2*I*(1+m)*(1+m+2*I*a*b*d^2*n*Pi)/b^2/d^2/n^2/Pi)*x*(e*x) ^m*erf((1/2+1/2*I)*(1+m+I*a*b*d^2*n*Pi+I*b^2*d^2*n*Pi*ln(c*x^n))/b/d/n/Pi^ (1/2))/(1+m)/((c*x^n)^((1+m)/n))-(1/4+1/4*I)*x*(e*x)^m*erfi((1/2+1/2*I)*(1 +m-I*a*b*d^2*n*Pi-I*b^2*d^2*n*Pi*ln(c*x^n))/b/d/n/Pi^(1/2))/exp(1/2*I*(1+m )*(1+m-2*I*a*b*d^2*n*Pi)/b^2/d^2/n^2/Pi)/(1+m)/((c*x^n)^((1+m)/n))+(e*x)^( 1+m)*FresnelC(d*(a+b*ln(c*x^n)))/e/(1+m)
Time = 3.66 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.87 \[ \int (e x)^m \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^m \left ((-1)^{3/4} \sqrt {2} e^{-\frac {(1+m) \left (i+i m+2 a b d^2 n \pi +2 b^2 d^2 n \pi \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{2 b^2 d^2 n^2 \pi }} x^{-m} \left (\text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i+i m+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )-e^{\frac {i (1+m)^2}{b^2 d^2 n^2 \pi }} \text {erfi}\left (\frac {(-1)^{3/4} \left (1+m+i a b d^2 n \pi +i b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {2 \pi }}\right )\right )+4 x \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{4 (1+m)} \]
((e*x)^m*(((-1)^(3/4)*Sqrt[2]*(Erf[((1/2 + I/2)*(I + I*m + a*b*d^2*n*Pi + b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[Pi])] - E^((I*(1 + m)^2)/(b^2*d^2*n^ 2*Pi))*Erfi[((-1)^(3/4)*(1 + m + I*a*b*d^2*n*Pi + I*b^2*d^2*n*Pi*Log[c*x^n ]))/(b*d*n*Sqrt[2*Pi])]))/(E^(((1 + m)*(I + I*m + 2*a*b*d^2*n*Pi + 2*b^2*d ^2*n*Pi*(-(n*Log[x]) + Log[c*x^n])))/(2*b^2*d^2*n^2*Pi))*x^m) + 4*x*Fresne lC[d*(a + b*Log[c*x^n])]))/(4*(1 + m))
Time = 0.98 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.22, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {7026, 5129, 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.
| \(\displaystyle \int (e x)^m \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
| \(\Big \downarrow \) 7026 |
| \(\displaystyle \frac {(e x)^{m+1} \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b d n \int (e x)^m \cos \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right )dx}{m+1}\) |
| \(\Big \downarrow \) 5129 |
| \(\displaystyle \frac {(e x)^{m+1} \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b d n \left (\frac {1}{2} \int e^{-\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} (e x)^mdx+\frac {1}{2} \int e^{\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} (e x)^mdx\right )}{m+1}\) |
| \(\Big \downarrow \) 2712 |
| \(\displaystyle \frac {(e x)^{m+1} \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b d n \left (\frac {1}{2} (e x)^m \left (c x^n\right )^{-i \pi a b d^2} x^{-m+i \pi a b d^2 n} \int \exp \left (-\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2-\frac {1}{2} i a^2 \pi d^2\right ) x^{m-i a b d^2 n \pi }dx+\frac {1}{2} (e x)^m \left (c x^n\right )^{i \pi a b d^2} x^{-m-i \pi a b d^2 n} \int \exp \left (\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2+\frac {1}{2} i a^2 \pi d^2\right ) x^{i a b n \pi d^2+m}dx\right )}{m+1}\) |
| \(\Big \downarrow \) 2706 |
| \(\displaystyle \frac {(e x)^{m+1} \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b d n \left (\frac {x (e x)^m \left (c x^n\right )^{-\frac {-i \pi a b d^2 n+m+1}{n}-i \pi a b d^2} \int \exp \left (-\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2-\frac {1}{2} i a^2 \pi d^2+\frac {\left (-i a b n \pi d^2+m+1\right ) \log \left (c x^n\right )}{n}\right )d\log \left (c x^n\right )}{2 n}+\frac {x (e x)^m \left (c x^n\right )^{i \pi a b d^2-\frac {i \pi a b d^2 n+m+1}{n}} \int \exp \left (\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2+\frac {1}{2} i a^2 \pi d^2+\frac {\left (i a b n \pi d^2+m+1\right ) \log \left (c x^n\right )}{n}\right )d\log \left (c x^n\right )}{2 n}\right )}{m+1}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle \frac {(e x)^{m+1} \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b d n \left (\frac {x (e x)^m \exp \left (-\frac {i (m+1) \left (-2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \left (c x^n\right )^{-\frac {-i \pi a b d^2 n+m+1}{n}-i \pi a b d^2} \int \exp \left (\frac {i \left (-i b^2 n \pi \log \left (c x^n\right ) d^2-i a b n \pi d^2+m+1\right )^2}{2 b^2 d^2 n^2 \pi }\right )d\log \left (c x^n\right )}{2 n}+\frac {x (e x)^m \exp \left (\frac {i (m+1) \left (2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \left (c x^n\right )^{i \pi a b d^2-\frac {i \pi a b d^2 n+m+1}{n}} \int \exp \left (-\frac {i \left (i b^2 n \pi \log \left (c x^n\right ) d^2+i a b n \pi d^2+m+1\right )^2}{2 b^2 d^2 n^2 \pi }\right )d\log \left (c x^n\right )}{2 n}\right )}{m+1}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {(e x)^{m+1} \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b d n \left (\frac {x (e x)^m \exp \left (\frac {i (m+1) \left (2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \left (c x^n\right )^{i \pi a b d^2-\frac {i \pi a b d^2 n+m+1}{n}} \int \exp \left (-\frac {i \left (i b^2 n \pi \log \left (c x^n\right ) d^2+i a b n \pi d^2+m+1\right )^2}{2 b^2 d^2 n^2 \pi }\right )d\log \left (c x^n\right )}{2 n}+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) x (e x)^m \exp \left (-\frac {i (m+1) \left (-2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \left (c x^n\right )^{-\frac {-i \pi a b d^2 n+m+1}{n}-i \pi a b d^2} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2 n-i \pi b^2 d^2 n \log \left (c x^n\right )+m+1\right )}{\sqrt {\pi } b d n}\right )}{b d n}\right )}{m+1}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {(e x)^{m+1} \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b d n \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) x (e x)^m \exp \left (-\frac {i (m+1) \left (-2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \left (c x^n\right )^{-\frac {-i \pi a b d^2 n+m+1}{n}-i \pi a b d^2} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2 n-i \pi b^2 d^2 n \log \left (c x^n\right )+m+1\right )}{\sqrt {\pi } b d n}\right )}{b d n}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) x (e x)^m \exp \left (\frac {i (m+1) \left (2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \left (c x^n\right )^{i \pi a b d^2-\frac {i \pi a b d^2 n+m+1}{n}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2 n+i \pi b^2 d^2 n \log \left (c x^n\right )+m+1\right )}{\sqrt {\pi } b d n}\right )}{b d n}\right )}{m+1}\) |
-((b*d*n*(((-1/4 - I/4)*E^(((I/2)*(1 + m)*(1 + m + (2*I)*a*b*d^2*n*Pi))/(b ^2*d^2*n^2*Pi))*x*(e*x)^m*(c*x^n)^(I*a*b*d^2*Pi - (1 + m + I*a*b*d^2*n*Pi) /n)*Erf[((1/2 + I/2)*(1 + m + I*a*b*d^2*n*Pi + I*b^2*d^2*n*Pi*Log[c*x^n])) /(b*d*n*Sqrt[Pi])])/(b*d*n) + ((1/4 + I/4)*x*(e*x)^m*(c*x^n)^((-I)*a*b*d^2 *Pi - (1 + m - I*a*b*d^2*n*Pi)/n)*Erfi[((1/2 + I/2)*(1 + m - I*a*b*d^2*n*P i - I*b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[Pi])])/(b*d*E^(((I/2)*(1 + m)* (1 + m - (2*I)*a*b*d^2*n*Pi))/(b^2*d^2*n^2*Pi))*n)))/(1 + m)) + ((e*x)^(1 + m)*FresnelC[d*(a + b*Log[c*x^n])])/(e*(1 + m))
3.2.69.3.1 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_.))*(( 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]
Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^2*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[1/2 Int[(e*x)^m/E^(I*d*(a + b*Log[c*x^n])^2), x], x] + Simp[1/2 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]
Int[FresnelC[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_. ), x_Symbol] :> Simp[(e*x)^(m + 1)*(FresnelC[d*(a + b*Log[c*x^n])]/(e*(m + 1))), x] - Simp[b*d*(n/(m + 1)) Int[(e*x)^m*Cos[(Pi/2)*(d*(a + b*Log[c*x^ n]))^2], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
\[\int \left (e x \right )^{m} \operatorname {FresnelC}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 689 vs. \(2 (310) = 620\).
Time = 0.31 (sec) , antiderivative size = 689, normalized size of antiderivative = 2.46 \[ \int (e x)^m \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {\pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m \log \left (e\right ) - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} - \frac {i \, m}{\pi b^{2} d^{2} n^{2}} - \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 \, m + 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}} e^{\left (m \log \left (e\right ) - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} + \frac {i \, m}{\pi b^{2} d^{2} n^{2}} + \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 \, m - 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}} e^{\left (m \log \left (e\right ) - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} - \frac {i \, m}{\pi b^{2} d^{2} n^{2}} - \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 \, m + 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}} e^{\left (m \log \left (e\right ) - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} + \frac {i \, m}{\pi b^{2} d^{2} n^{2}} + \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 \, m - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - 2 \, x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right )}{2 \, {\left (m + 1\right )}} \]
-1/2*(pi*sqrt(b^2*d^2*n^2)*e^(m*log(e) - m*log(c)/n - a*m/(b*n) - log(c)/n - a/(b*n) - 1/2*I*m^2/(pi*b^2*d^2*n^2) - I*m/(pi*b^2*d^2*n^2) - 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*m + I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) + pi*sqrt(b^2* d^2*n^2)*e^(m*log(e) - m*log(c)/n - a*m/(b*n) - log(c)/n - a/(b*n) + 1/2*I *m^2/(pi*b^2*d^2*n^2) + I*m/(pi*b^2*d^2*n^2) + 1/2*I/(pi*b^2*d^2*n^2))*fre snel_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*m - I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - I*pi*sqrt(b^2*d^2*n^2)*e^(m*lo g(e) - m*log(c)/n - a*m/(b*n) - log(c)/n - a/(b*n) - 1/2*I*m^2/(pi*b^2*d^2 *n^2) - I*m/(pi*b^2*d^2*n^2) - 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*m + I)*sqrt(b^2*d ^2*n^2)/(pi*b^2*d^2*n^2)) + I*pi*sqrt(b^2*d^2*n^2)*e^(m*log(e) - m*log(c)/ n - a*m/(b*n) - log(c)/n - a/(b*n) + 1/2*I*m^2/(pi*b^2*d^2*n^2) + I*m/(pi* b^2*d^2*n^2) + 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*m - I)*sqrt(b^2*d^2*n^2)/(pi*b^2* d^2*n^2)) - 2*x*e^(m*log(e) + m*log(x))*fresnel_cos(b*d*log(c*x^n) + a*d)) /(m + 1)
\[ \int (e x)^m \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (e x\right )^{m} C\left (a d + b d \log {\left (c x^{n} \right )}\right )\, dx \]
\[ \int (e x)^m \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int (e x)^m \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
Timed out. \[ \int (e x)^m \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {FresnelC}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \]