Optimal. Leaf size=96 \[ -\frac {a^2 S(a+b x)}{2 b^2}-\frac {C(a+b x)}{2 \pi b^2}-\frac {a \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^2}+\frac {(a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 \pi b^2}+\frac {1}{2} x^2 S(a+b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6428, 3433, 3351, 3379, 2638, 3385, 3352} \[ -\frac {a^2 S(a+b x)}{2 b^2}-\frac {\text {FresnelC}(a+b x)}{2 \pi b^2}-\frac {a \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^2}+\frac {(a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 \pi b^2}+\frac {1}{2} x^2 S(a+b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2638
Rule 3351
Rule 3352
Rule 3379
Rule 3385
Rule 3433
Rule 6428
Rubi steps
\begin {align*} \int x S(a+b x) \, dx &=\frac {1}{2} x^2 S(a+b x)-\frac {1}{2} b \int x^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right ) \, dx\\ &=\frac {1}{2} x^2 S(a+b x)-\frac {\operatorname {Subst}\left (\int \left (a^2 \sin \left (\frac {\pi x^2}{2}\right )-2 a x \sin \left (\frac {\pi x^2}{2}\right )+x^2 \sin \left (\frac {\pi x^2}{2}\right )\right ) \, dx,x,a+b x\right )}{2 b^2}\\ &=\frac {1}{2} x^2 S(a+b x)-\frac {\operatorname {Subst}\left (\int x^2 \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2}+\frac {a \operatorname {Subst}\left (\int x \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^2}-\frac {a^2 \operatorname {Subst}\left (\int \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2}\\ &=\frac {(a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }-\frac {a^2 S(a+b x)}{2 b^2}+\frac {1}{2} x^2 S(a+b x)+\frac {a \operatorname {Subst}\left (\int \sin \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^2}-\frac {\operatorname {Subst}\left (\int \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2 \pi }\\ &=-\frac {a \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^2 \pi }+\frac {(a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }-\frac {C(a+b x)}{2 b^2 \pi }-\frac {a^2 S(a+b x)}{2 b^2}+\frac {1}{2} x^2 S(a+b x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 51, normalized size = 0.53 \[ -\frac {C(a+b x)+(a-b x) \left (\pi (a+b x) S(a+b x)+\cos \left (\frac {1}{2} \pi (a+b x)^2\right )\right )}{2 \pi b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x {\rm fresnels}\left (b x + a\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm fresnels}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 80, normalized size = 0.83 \[ \frac {\mathrm {S}\left (b x +a \right ) \left (\frac {\left (b x +a \right )^{2}}{2}-a \left (b x +a \right )\right )+\frac {\left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{2 \pi }-\frac {\FresnelC \left (b x +a \right )}{2 \pi }-\frac {a \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm fresnels}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {FresnelS}\left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x S\left (a + b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________