Integrand size = 17, antiderivative size = 228 \[ \int \frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) e^{\frac {2 i+2 a b d^2 n \pi }{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{2/n} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2}{n}-i a b d^2 \pi -i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )}{x^2}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) e^{-\frac {2 \left (i-a b d^2 n \pi \right )}{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{2/n} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2}{n}+i a b d^2 \pi +i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )}{x^2}-\frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]
(1/8+1/8*I)*exp((2*I+2*a*b*d^2*n*Pi)/b^2/d^2/n^2/Pi)*(c*x^n)^(2/n)*erf((1/ 2+1/2*I)*(2/n-I*a*b*d^2*Pi-I*b^2*d^2*Pi*ln(c*x^n))/b/d/Pi^(1/2))/x^2-(1/8+ 1/8*I)*(c*x^n)^(2/n)*erfi((1/2+1/2*I)*(2/n+I*a*b*d^2*Pi+I*b^2*d^2*Pi*ln(c* x^n))/b/d/Pi^(1/2))/exp(2*(I-a*b*d^2*n*Pi)/b^2/d^2/n^2/Pi)/x^2-1/2*Fresnel C(d*(a+b*ln(c*x^n)))/x^2
Time = 2.61 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.87 \[ \int \frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {\sqrt [4]{-1} e^{\frac {2 \left (\frac {a n}{b}-\frac {i}{b^2 d^2 \pi }+n \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{n^2}} \left (\text {erfi}\left (\frac {(-1)^{3/4} \left (-2 i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {2 \pi }}\right )+i e^{\frac {4 i}{b^2 d^2 n^2 \pi }} \text {erfi}\left (\frac {\sqrt [4]{-1} \left (2 i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {2 \pi }}\right )\right )}{4 \sqrt {2}}-\frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]
-1/4*((-1)^(1/4)*E^((2*((a*n)/b - I/(b^2*d^2*Pi) + n*(-(n*Log[x]) + Log[c* x^n])))/n^2)*(Erfi[((-1)^(3/4)*(-2*I + a*b*d^2*n*Pi + b^2*d^2*n*Pi*Log[c*x ^n]))/(b*d*n*Sqrt[2*Pi])] + I*E^((4*I)/(b^2*d^2*n^2*Pi))*Erfi[((-1)^(1/4)* (2*I + a*b*d^2*n*Pi + b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[2*Pi])]))/Sqrt [2] - FresnelC[d*(a + b*Log[c*x^n])]/(2*x^2)
Time = 0.81 (sec) , antiderivative size = 254, 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.
| \(\displaystyle \int \frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 7026 |
| \(\displaystyle \frac {1}{2} b d n \int \frac {\cos \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right )}{x^3}dx-\frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\) |
| \(\Big \downarrow \) 5129 |
| \(\displaystyle -\frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} b d n \left (\frac {1}{2} \int \frac {e^{-\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2}}{x^3}dx+\frac {1}{2} \int \frac {e^{\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2}}{x^3}dx\right )\) |
| \(\Big \downarrow \) 2712 |
| \(\displaystyle -\frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} b d n \left (\frac {1}{2} x^{i \pi a b d^2 n} \left (c x^n\right )^{-i \pi a b d^2} \int \exp \left (-\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2-\frac {1}{2} i a^2 \pi d^2\right ) x^{-i a b n \pi d^2-3}dx+\frac {1}{2} x^{-i \pi a b d^2 n} \left (c x^n\right )^{i \pi a b d^2} \int \exp \left (\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2+\frac {1}{2} i a^2 \pi d^2\right ) x^{i a b d^2 n \pi -3}dx\right )\) |
| \(\Big \downarrow \) 2706 |
| \(\displaystyle -\frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} b d n \left (\frac {\left (c x^n\right )^{2/n} \int \exp \left (-\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2-\frac {1}{2} i a^2 \pi d^2-\frac {\left (i a b n \pi d^2+2\right ) \log \left (c x^n\right )}{n}\right )d\log \left (c x^n\right )}{2 n x^2}+\frac {\left (c x^n\right )^{2/n} \int \exp \left (\frac {1}{2} i b^2 \pi \log ^2\left (c x^n\right ) d^2+\frac {1}{2} i a^2 \pi d^2-\frac {\left (2-i a b d^2 n \pi \right ) \log \left (c x^n\right )}{n}\right )d\log \left (c x^n\right )}{2 n x^2}\right )\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle -\frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} b d n \left (\frac {\left (c x^n\right )^{2/n} e^{\frac {2 \pi a b d^2 n+2 i}{\pi b^2 d^2 n^2}} \int \exp \left (-\frac {i \left (-i b^2 \pi \log \left (c x^n\right ) d^2-i a b \pi d^2+\frac {2}{n}\right )^2}{2 b^2 d^2 \pi }\right )d\log \left (c x^n\right )}{2 n x^2}+\frac {\left (c x^n\right )^{2/n} e^{-\frac {2 \left (-\pi a b d^2 n+i\right )}{\pi b^2 d^2 n^2}} \int \exp \left (\frac {i \left (i b^2 \pi \log \left (c x^n\right ) d^2+i a b \pi d^2+\frac {2}{n}\right )^2}{2 b^2 d^2 \pi }\right )d\log \left (c x^n\right )}{2 n x^2}\right )\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} b d n \left (\frac {\left (c x^n\right )^{2/n} e^{\frac {2 \pi a b d^2 n+2 i}{\pi b^2 d^2 n^2}} \int \exp \left (-\frac {i \left (-i b^2 \pi \log \left (c x^n\right ) d^2-i a b \pi d^2+\frac {2}{n}\right )^2}{2 b^2 d^2 \pi }\right )d\log \left (c x^n\right )}{2 n x^2}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (c x^n\right )^{2/n} e^{-\frac {2 \left (-\pi a b d^2 n+i\right )}{\pi b^2 d^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {2}{n}\right )}{\sqrt {\pi } b d}\right )}{b d n x^2}\right )\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle -\frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} b d n \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (c x^n\right )^{2/n} e^{\frac {2 \pi a b d^2 n+2 i}{\pi b^2 d^2 n^2}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {2}{n}\right )}{\sqrt {\pi } b d}\right )}{b d n x^2}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (c x^n\right )^{2/n} e^{-\frac {2 \left (-\pi a b d^2 n+i\right )}{\pi b^2 d^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {2}{n}\right )}{\sqrt {\pi } b d}\right )}{b d n x^2}\right )\) |
(b*d*n*(((1/4 + I/4)*E^((2*I + 2*a*b*d^2*n*Pi)/(b^2*d^2*n^2*Pi))*(c*x^n)^( 2/n)*Erf[((1/2 + I/2)*(2/n - I*a*b*d^2*Pi - I*b^2*d^2*Pi*Log[c*x^n]))/(b*d *Sqrt[Pi])])/(b*d*n*x^2) - ((1/4 + I/4)*(c*x^n)^(2/n)*Erfi[((1/2 + I/2)*(2 /n + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(b*d*E^((2* (I - a*b*d^2*n*Pi))/(b^2*d^2*n^2*Pi))*n*x^2)))/2 - FresnelC[d*(a + b*Log[c *x^n])]/(2*x^2)
3.2.68.3.1 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]
\[\int \frac {\operatorname {FresnelC}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (183) = 366\).
Time = 0.28 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.02 \[ \int \frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {\pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + i \, \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - i \, \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - 2 \, \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right )}{4 \, x^{2}} \]
1/4*(pi*sqrt(b^2*d^2*n^2)*x^2*e^(2*log(c)/n + 2*a/(b*n) + 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 + 2*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) + pi*sqrt(b^2*d^2*n^2)*x^2* e^(2*log(c)/n + 2*a/(b*n) - 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 - 2*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^2*e^(2*log(c)/n + 2*a/(b*n) + 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*lo g(c) + pi*a*b*d^2*n + 2*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^2*e^(2*log(c)/n + 2*a/(b*n) - 2*I/(pi*b^2*d^2*n^2))*fresne l_sin((pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n - 2*I)*s qrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - 2*fresnel_cos(b*d*log(c*x^n) + a*d))/ x^2
\[ \int \frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {C\left (a d + b d \log {\left (c x^{n} \right )}\right )}{x^{3}}\, dx \]
\[ \int \frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
\[ \int \frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathrm {FresnelC}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \]