Optimal. Leaf size=143 \[ \frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {C(b x) S(b x)}{2 \pi b^2}+\frac {x S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^2}+\frac {1}{2} x^2 S(b x)^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6430, 6454, 6446, 3379, 2638} \[ \frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {\text {FresnelC}(b x) S(b x)}{2 \pi b^2}+\frac {x S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^2}+\frac {1}{2} x^2 S(b x)^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2638
Rule 3379
Rule 6430
Rule 6446
Rule 6454
Rubi steps
\begin {align*} \int x S(b x)^2 \, dx &=\frac {1}{2} x^2 S(b x)^2-b \int x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b \pi }+\frac {1}{2} x^2 S(b x)^2-\frac {\int x \sin \left (b^2 \pi x^2\right ) \, dx}{2 \pi }-\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b \pi }\\ &=\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b \pi }-\frac {C(b x) S(b x)}{2 b^2 \pi }+\frac {1}{2} x^2 S(b x)^2+\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {\operatorname {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 \pi }\\ &=\frac {\cos \left (b^2 \pi x^2\right )}{4 b^2 \pi ^2}+\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b \pi }-\frac {C(b x) S(b x)}{2 b^2 \pi }+\frac {1}{2} x^2 S(b x)^2+\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int x S(b x)^2 \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x {\rm fresnels}\left (b x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm fresnels}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int x \mathrm {S}\left (b x \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm fresnels}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x S^{2}\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________