Integrand size = 10, antiderivative size = 127 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^5} \, dx=-\frac {b^2}{24 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{6 x}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x)^2}{4 x^4}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x^3}+\frac {1}{12} b^4 \pi \text {Si}\left (b^2 \pi x^2\right ) \]
[Out]
Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6565, 6591, 6599, 6575, 30, 3456, 3461, 3378, 3380} \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^5} \, dx=-\frac {1}{12} \pi ^2 b^4 \operatorname {FresnelS}(b x)^2-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^3}-\frac {b^2}{24 x^2}+\frac {b^2 \cos \left (\pi b^2 x^2\right )}{24 x^2}+\frac {1}{12} \pi b^4 \text {Si}\left (b^2 \pi x^2\right )-\frac {\pi b^3 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x}-\frac {\operatorname {FresnelS}(b x)^2}{4 x^4} \]
[In]
[Out]
Rule 30
Rule 3378
Rule 3380
Rule 3456
Rule 3461
Rule 6565
Rule 6575
Rule 6591
Rule 6599
Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelS}(b x)^2}{4 x^4}+\frac {1}{2} b \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \\ & = -\frac {b^2}{24 x^2}-\frac {\operatorname {FresnelS}(b x)^2}{4 x^4}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x^3}-\frac {1}{12} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3} \, dx+\frac {1}{6} \left (b^3 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2} \, dx \\ & = -\frac {b^2}{24 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{6 x}-\frac {\operatorname {FresnelS}(b x)^2}{4 x^4}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x^3}-\frac {1}{24} b^2 \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{12} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x} \, dx-\frac {1}{6} \left (b^5 \pi ^2\right ) \int \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = -\frac {b^2}{24 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{6 x}-\frac {\operatorname {FresnelS}(b x)^2}{4 x^4}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x^3}+\frac {1}{24} b^4 \pi \text {Si}\left (b^2 \pi x^2\right )+\frac {1}{24} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{6} \left (b^4 \pi ^2\right ) \text {Subst}(\int x \, dx,x,\operatorname {FresnelS}(b x)) \\ & = -\frac {b^2}{24 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{6 x}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x)^2}{4 x^4}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x^3}+\frac {1}{12} b^4 \pi \text {Si}\left (b^2 \pi x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^5} \, dx=-\frac {b^2}{24 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{6 x}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x)^2}{4 x^4}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x^3}+\frac {1}{12} b^4 \pi \text {Si}\left (b^2 \pi x^2\right ) \]
[In]
[Out]
\[\int \frac {\operatorname {FresnelS}\left (b x \right )^{2}}{x^{5}}d x\]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.87 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^5} \, dx=\frac {\pi b^{4} x^{4} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 2 \, \pi b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - b^{2} x^{2} - 2 \, b x \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{4} x^{4} + 3\right )} \operatorname {S}\left (b x\right )^{2}}{12 \, x^{4}} \]
[In]
[Out]
\[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^5} \, dx=\int \frac {S^{2}\left (b x\right )}{x^{5}}\, dx \]
[In]
[Out]
\[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^5} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{5}} \,d x } \]
[In]
[Out]
\[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^5} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{5}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^5} \, dx=\int \frac {{\mathrm {FresnelS}\left (b\,x\right )}^2}{x^5} \,d x \]
[In]
[Out]