Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=-\frac {b^2}{120 x^4}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}+\frac {1}{72} b^6 \pi ^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{45 x^3}-\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^5}-\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{72 x^2}-\frac {1}{45} b^5 \pi ^2 \text {Int}\left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2},x\right ) \]
[Out]
Not integrable
Time = 0.15 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}+\frac {1}{3} b \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6} \, dx \\ & = -\frac {b^2}{120 x^4}-\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^5}-\frac {1}{30} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^5} \, dx+\frac {1}{15} \left (b^3 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^4} \, dx \\ & = -\frac {b^2}{120 x^4}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{45 x^3}-\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^5}-\frac {1}{60} b^2 \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{90} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx \\ & = -\frac {b^2}{120 x^4}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{45 x^3}-\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^5}+\frac {1}{180} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{120} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx \\ & = -\frac {b^2}{120 x^4}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{45 x^3}-\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^5}-\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{72 x^2}-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{180} \left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac {1}{120} \left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right ) \\ & = -\frac {b^2}{120 x^4}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}+\frac {1}{72} b^6 \pi ^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{45 x^3}-\frac {\operatorname {FresnelS}(b x)^2}{6 x^6}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^5}-\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{72 x^2}-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx \\ \end{align*}
Not integrable
Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx \]
[In]
[Out]
Not integrable
Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00
\[\int \frac {\operatorname {FresnelS}\left (b x \right )^{2}}{x^{7}}d x\]
[In]
[Out]
Not integrable
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{7}} \,d x } \]
[In]
[Out]
Not integrable
Time = 1.33 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int \frac {S^{2}\left (b x\right )}{x^{7}}\, dx \]
[In]
[Out]
Not integrable
Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{7}} \,d x } \]
[In]
[Out]
Not integrable
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{7}} \,d x } \]
[In]
[Out]
Not integrable
Time = 4.85 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^7} \, dx=\int \frac {{\mathrm {FresnelS}\left (b\,x\right )}^2}{x^7} \,d x \]
[In]
[Out]