Integrand size = 14, antiderivative size = 279 \[ \int (c+d x) \operatorname {FresnelS}(a+b x)^2 \, dx=\frac {d \cos \left (\pi (a+b x)^2\right )}{4 b^2 \pi ^2}+\frac {2 (b c-a d) \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \operatorname {FresnelS}(a+b x)}{b^2 \pi }+\frac {d (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \operatorname {FresnelS}(a+b x)}{b^2 \pi }-\frac {d \operatorname {FresnelC}(a+b x) \operatorname {FresnelS}(a+b x)}{2 b^2 \pi }+\frac {(b c-a d) (a+b x) \operatorname {FresnelS}(a+b x)^2}{b^2}+\frac {d (a+b x)^2 \operatorname {FresnelS}(a+b x)^2}{2 b^2}-\frac {(b c-a d) \operatorname {FresnelS}\left (\sqrt {2} (a+b x)\right )}{\sqrt {2} b^2 \pi }+\frac {i d (a+b x)^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi }-\frac {i d (a+b x)^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi } \]
[Out]
Time = 0.15 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6567, 6555, 6587, 3432, 6565, 6589, 6581, 3460, 2718} \[ \int (c+d x) \operatorname {FresnelS}(a+b x)^2 \, dx=\frac {i d (a+b x)^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i \pi (a+b x)^2\right )}{8 \pi b^2}-\frac {i d (a+b x)^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i \pi (a+b x)^2\right )}{8 \pi b^2}+\frac {(a+b x) (b c-a d) \operatorname {FresnelS}(a+b x)^2}{b^2}-\frac {(b c-a d) \operatorname {FresnelS}\left (\sqrt {2} (a+b x)\right )}{\sqrt {2} \pi b^2}+\frac {2 (b c-a d) \operatorname {FresnelS}(a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^2}-\frac {d \operatorname {FresnelC}(a+b x) \operatorname {FresnelS}(a+b x)}{2 \pi b^2}+\frac {d (a+b x)^2 \operatorname {FresnelS}(a+b x)^2}{2 b^2}+\frac {d (a+b x) \operatorname {FresnelS}(a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^2}+\frac {d \cos \left (\pi (a+b x)^2\right )}{4 \pi ^2 b^2} \]
[In]
[Out]
Rule 2718
Rule 3432
Rule 3460
Rule 6555
Rule 6565
Rule 6567
Rule 6581
Rule 6587
Rule 6589
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (b c \left (1-\frac {a d}{b c}\right ) \operatorname {FresnelS}(x)^2+d x \operatorname {FresnelS}(x)^2\right ) \, dx,x,a+b x\right )}{b^2} \\ & = \frac {d \text {Subst}\left (\int x \operatorname {FresnelS}(x)^2 \, dx,x,a+b x\right )}{b^2}+\frac {(b c-a d) \text {Subst}\left (\int \operatorname {FresnelS}(x)^2 \, dx,x,a+b x\right )}{b^2} \\ & = \frac {(b c-a d) (a+b x) \operatorname {FresnelS}(a+b x)^2}{b^2}+\frac {d (a+b x)^2 \operatorname {FresnelS}(a+b x)^2}{2 b^2}-\frac {d \text {Subst}\left (\int x^2 \operatorname {FresnelS}(x) \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^2}-\frac {(2 (b c-a d)) \text {Subst}\left (\int x \operatorname {FresnelS}(x) \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^2} \\ & = \frac {2 (b c-a d) \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \operatorname {FresnelS}(a+b x)}{b^2 \pi }+\frac {d (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \operatorname {FresnelS}(a+b x)}{b^2 \pi }+\frac {(b c-a d) (a+b x) \operatorname {FresnelS}(a+b x)^2}{b^2}+\frac {d (a+b x)^2 \operatorname {FresnelS}(a+b x)^2}{2 b^2}-\frac {d \text {Subst}\left (\int x \sin \left (\pi x^2\right ) \, dx,x,a+b x\right )}{2 b^2 \pi }-\frac {d \text {Subst}\left (\int \cos \left (\frac {\pi x^2}{2}\right ) \operatorname {FresnelS}(x) \, dx,x,a+b x\right )}{b^2 \pi }-\frac {(b c-a d) \text {Subst}\left (\int \sin \left (\pi x^2\right ) \, dx,x,a+b x\right )}{b^2 \pi } \\ & = \frac {2 (b c-a d) \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \operatorname {FresnelS}(a+b x)}{b^2 \pi }+\frac {d (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \operatorname {FresnelS}(a+b x)}{b^2 \pi }-\frac {d \operatorname {FresnelC}(a+b x) \operatorname {FresnelS}(a+b x)}{2 b^2 \pi }+\frac {(b c-a d) (a+b x) \operatorname {FresnelS}(a+b x)^2}{b^2}+\frac {d (a+b x)^2 \operatorname {FresnelS}(a+b x)^2}{2 b^2}-\frac {(b c-a d) \operatorname {FresnelS}\left (\sqrt {2} (a+b x)\right )}{\sqrt {2} b^2 \pi }+\frac {i d (a+b x)^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi }-\frac {i d (a+b x)^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi }-\frac {d \text {Subst}\left (\int \sin (\pi x) \, dx,x,(a+b x)^2\right )}{4 b^2 \pi } \\ & = \frac {d \cos \left (\pi (a+b x)^2\right )}{4 b^2 \pi ^2}+\frac {2 (b c-a d) \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \operatorname {FresnelS}(a+b x)}{b^2 \pi }+\frac {d (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \operatorname {FresnelS}(a+b x)}{b^2 \pi }-\frac {d \operatorname {FresnelC}(a+b x) \operatorname {FresnelS}(a+b x)}{2 b^2 \pi }+\frac {(b c-a d) (a+b x) \operatorname {FresnelS}(a+b x)^2}{b^2}+\frac {d (a+b x)^2 \operatorname {FresnelS}(a+b x)^2}{2 b^2}-\frac {(b c-a d) \operatorname {FresnelS}\left (\sqrt {2} (a+b x)\right )}{\sqrt {2} b^2 \pi }+\frac {i d (a+b x)^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi }-\frac {i d (a+b x)^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi } \\ \end{align*}
\[ \int (c+d x) \operatorname {FresnelS}(a+b x)^2 \, dx=\int (c+d x) \operatorname {FresnelS}(a+b x)^2 \, dx \]
[In]
[Out]
\[\int \left (d x +c \right ) \operatorname {FresnelS}\left (b x +a \right )^{2}d x\]
[In]
[Out]
\[ \int (c+d x) \operatorname {FresnelS}(a+b x)^2 \, dx=\int { {\left (d x + c\right )} \operatorname {S}\left (b x + a\right )^{2} \,d x } \]
[In]
[Out]
\[ \int (c+d x) \operatorname {FresnelS}(a+b x)^2 \, dx=\int \left (c + d x\right ) S^{2}\left (a + b x\right )\, dx \]
[In]
[Out]
\[ \int (c+d x) \operatorname {FresnelS}(a+b x)^2 \, dx=\int { {\left (d x + c\right )} \operatorname {S}\left (b x + a\right )^{2} \,d x } \]
[In]
[Out]
\[ \int (c+d x) \operatorname {FresnelS}(a+b x)^2 \, dx=\int { {\left (d x + c\right )} \operatorname {S}\left (b x + a\right )^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int (c+d x) \operatorname {FresnelS}(a+b x)^2 \, dx=\int {\mathrm {FresnelS}\left (a+b\,x\right )}^2\,\left (c+d\,x\right ) \,d x \]
[In]
[Out]