Integrand size = 17, antiderivative size = 231 \[ \int x^2 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\left (\frac {1}{12}+\frac {i}{12}\right ) e^{-\frac {3 a}{b n}+\frac {9 i}{2 b^2 d^2 n^2 \pi }} x^3 \left (c x^n\right )^{-3/n} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {3}{n}+i a b d^2 \pi +i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )-\left (\frac {1}{12}+\frac {i}{12}\right ) e^{-\frac {3 a}{b n}-\frac {9 i}{2 b^2 d^2 n^2 \pi }} x^3 \left (c x^n\right )^{-3/n} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {3}{n}-i a b d^2 \pi -i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )+\frac {1}{3} x^3 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
(1/12+1/12*I)*exp(-3*a/b/n+9/2*I/b^2/d^2/n^2/Pi)*x^3*erf((1/2+1/2*I)*(3/n+ I*a*b*d^2*Pi+I*b^2*d^2*Pi*ln(c*x^n))/b/d/Pi^(1/2))/((c*x^n)^(3/n))-(1/12+1 /12*I)*exp(-3*a/b/n-9/2*I/b^2/d^2/n^2/Pi)*x^3*erfi((1/2+1/2*I)*(3/n-I*a*b* d^2*Pi-I*b^2*d^2*Pi*ln(c*x^n))/b/d/Pi^(1/2))/((c*x^n)^(3/n))+1/3*x^3*Fresn elC(d*(a+b*ln(c*x^n)))
Time = 4.61 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.38 \[ \int x^2 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{12} x^3 \left (4 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\sqrt [4]{-1} \sqrt {2} e^{\frac {1}{2} \left (-\frac {6 a}{b n}-\frac {9 i}{b^2 d^2 n^2 \pi }-i a^2 d^2 \pi +2 i a b d^2 \pi \left (n \log (x)-\log \left (c x^n\right )\right )-i b^2 d^2 \pi \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right )} \left (c x^n\right )^{-3/n} \left (i e^{\frac {9 i}{b^2 d^2 n^2 \pi }} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-3 i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )+\text {erfi}\left (\frac {(-1)^{3/4} \left (3 i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {2 \pi }}\right )\right ) \left (\cos \left (\frac {1}{2} d^2 \pi \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2\right )+i \sin \left (\frac {1}{2} d^2 \pi \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2\right )\right )\right ) \]
(x^3*(4*FresnelC[d*(a + b*Log[c*x^n])] + ((-1)^(1/4)*Sqrt[2]*E^(((-6*a)/(b *n) - (9*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)*(I*E^((9*I)/( b^2*d^2*n^2*Pi))*Erfi[((1/2 + I/2)*(-3*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)*(3*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*Sin[(d^2*Pi*(a - b*n*Log[x] + b*Log[c*x^n])^2)/2])) /(c*x^n)^(3/n)))/12
Time = 0.86 (sec) , antiderivative size = 257, 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 = {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.
| \(\displaystyle \int x^2 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
| \(\Big \downarrow \) 7026 |
| \(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} b d n \int x^2 \cos \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right )dx\) |
| \(\Big \downarrow \) 5129 |
| \(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} b d n \left (\frac {1}{2} \int e^{-\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} x^2dx+\frac {1}{2} \int e^{\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} x^2dx\right )\) |
| \(\Big \downarrow \) 2712 |
| \(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} b d n \left (\frac {1}{2} x^{i \pi a b d^2 n} \left (c x^n\right )^{-i \pi a b d^2} \int \exp \left (-\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2-\frac {1}{2} i a^2 \pi d^2\right ) x^{2-i a b d^2 n \pi }dx+\frac {1}{2} x^{-i \pi a b d^2 n} \left (c x^n\right )^{i \pi a b d^2} \int \exp \left (\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2+\frac {1}{2} i a^2 \pi d^2\right ) x^{i a b n \pi d^2+2}dx\right )\) |
| \(\Big \downarrow \) 2706 |
| \(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} b d n \left (\frac {x^3 \left (c x^n\right )^{-3/n} \int \exp \left (-\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2-\frac {1}{2} i a^2 \pi d^2+\frac {\left (3-i a b d^2 n \pi \right ) \log \left (c x^n\right )}{n}\right )d\log \left (c x^n\right )}{2 n}+\frac {x^3 \left (c x^n\right )^{-3/n} \int \exp \left (\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2+\frac {1}{2} i a^2 \pi d^2+\frac {\left (i a b n \pi d^2+3\right ) \log \left (c x^n\right )}{n}\right )d\log \left (c x^n\right )}{2 n}\right )\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} b d n \left (\frac {x^3 \left (c x^n\right )^{-3/n} e^{-\frac {3 a}{b n}-\frac {9 i}{2 \pi b^2 d^2 n^2}} \int \exp \left (\frac {i \left (-i b^2 \pi \log \left (c x^n\right ) d^2-i a b \pi d^2+\frac {3}{n}\right )^2}{2 b^2 d^2 \pi }\right )d\log \left (c x^n\right )}{2 n}+\frac {x^3 \left (c x^n\right )^{-3/n} e^{-\frac {3 a}{b n}+\frac {9 i}{2 \pi b^2 d^2 n^2}} \int \exp \left (-\frac {i \left (i b^2 \pi \log \left (c x^n\right ) d^2+i a b \pi d^2+\frac {3}{n}\right )^2}{2 b^2 d^2 \pi }\right )d\log \left (c x^n\right )}{2 n}\right )\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} b d n \left (\frac {x^3 \left (c x^n\right )^{-3/n} e^{-\frac {3 a}{b n}+\frac {9 i}{2 \pi b^2 d^2 n^2}} \int \exp \left (-\frac {i \left (i b^2 \pi \log \left (c x^n\right ) d^2+i a b \pi d^2+\frac {3}{n}\right )^2}{2 b^2 d^2 \pi }\right )d\log \left (c x^n\right )}{2 n}+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) x^3 \left (c x^n\right )^{-3/n} e^{-\frac {3 a}{b n}-\frac {9 i}{2 \pi b^2 d^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {3}{n}\right )}{\sqrt {\pi } b d}\right )}{b d n}\right )\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} b d n \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) x^3 \left (c x^n\right )^{-3/n} e^{-\frac {3 a}{b n}-\frac {9 i}{2 \pi b^2 d^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {3}{n}\right )}{\sqrt {\pi } b d}\right )}{b d n}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) x^3 \left (c x^n\right )^{-3/n} e^{-\frac {3 a}{b n}+\frac {9 i}{2 \pi b^2 d^2 n^2}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {3}{n}\right )}{\sqrt {\pi } b d}\right )}{b d n}\right )\) |
-1/3*(b*d*n*(((-1/4 - I/4)*E^((-3*a)/(b*n) + ((9*I)/2)/(b^2*d^2*n^2*Pi))*x ^3*Erf[((1/2 + I/2)*(3/n + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*S qrt[Pi])])/(b*d*n*(c*x^n)^(3/n)) + ((1/4 + I/4)*E^((-3*a)/(b*n) - ((9*I)/2 )/(b^2*d^2*n^2*Pi))*x^3*Erfi[((1/2 + I/2)*(3/n - I*a*b*d^2*Pi - I*b^2*d^2* Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(b*d*n*(c*x^n)^(3/n)))) + (x^3*FresnelC[d *(a + b*Log[c*x^n])])/3
3.2.63.3.1 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]
\[\int x^{2} \operatorname {FresnelC}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (187) = 374\).
Time = 0.29 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.94 \[ \int x^2 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{3} \, x^{3} \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \frac {1}{6} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} - \frac {9 i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{6} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} + \frac {9 i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \frac {1}{6} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} - \frac {9 i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{6} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} + \frac {9 i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) \]
1/3*x^3*fresnel_cos(b*d*log(c*x^n) + a*d) - 1/6*pi*sqrt(b^2*d^2*n^2)*e^(-3 *log(c)/n - 3*a/(b*n) - 9/2*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 + 3*I)*sqrt(b^2*d^2*n^2)/(pi *b^2*d^2*n^2)) - 1/6*pi*sqrt(b^2*d^2*n^2)*e^(-3*log(c)/n - 3*a/(b*n) + 9/2 *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 - 3*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) + 1/6*I*pi*s qrt(b^2*d^2*n^2)*e^(-3*log(c)/n - 3*a/(b*n) - 9/2*I/(pi*b^2*d^2*n^2))*fres nel_sin((pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n + 3*I) *sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - 1/6*I*pi*sqrt(b^2*d^2*n^2)*e^(-3*lo g(c)/n - 3*a/(b*n) + 9/2*I/(pi*b^2*d^2*n^2))*fresnel_sin((pi*b^2*d^2*n^2*l og(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n - 3*I)*sqrt(b^2*d^2*n^2)/(pi*b^ 2*d^2*n^2))
\[ \int x^2 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^{2} C\left (a d + b d \log {\left (c x^{n} \right )}\right )\, dx \]
\[ \int x^2 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int x^2 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
Timed out. \[ \int x^2 \operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^2\,\mathrm {FresnelC}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]