Integrand size = 12, antiderivative size = 121 \[ \int (c+d x) \operatorname {FresnelS}(a+b x) \, dx=\frac {(b c-a d) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^2 \pi }+\frac {d (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }-\frac {d \operatorname {FresnelC}(a+b x)}{2 b^2 \pi }-\frac {(b c-a d)^2 \operatorname {FresnelS}(a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \operatorname {FresnelS}(a+b x)}{2 d} \]
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Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6563, 3514, 3432, 3460, 2718, 3466, 3433} \[ \int (c+d x) \operatorname {FresnelS}(a+b x) \, dx=-\frac {(b c-a d)^2 \operatorname {FresnelS}(a+b x)}{2 b^2 d}+\frac {(b c-a d) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^2}-\frac {d \operatorname {FresnelC}(a+b x)}{2 \pi b^2}+\frac {d (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 \pi b^2}+\frac {(c+d x)^2 \operatorname {FresnelS}(a+b x)}{2 d} \]
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Rule 2718
Rule 3432
Rule 3433
Rule 3460
Rule 3466
Rule 3514
Rule 6563
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^2 \operatorname {FresnelS}(a+b x)}{2 d}-\frac {b \int (c+d x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right ) \, dx}{2 d} \\ & = \frac {(c+d x)^2 \operatorname {FresnelS}(a+b x)}{2 d}-\frac {\text {Subst}\left (\int \left (b^2 c^2 \left (1+\frac {a d (-2 b c+a d)}{b^2 c^2}\right ) \sin \left (\frac {\pi x^2}{2}\right )+2 b c d \left (1-\frac {a d}{b c}\right ) x \sin \left (\frac {\pi x^2}{2}\right )+d^2 x^2 \sin \left (\frac {\pi x^2}{2}\right )\right ) \, dx,x,a+b x\right )}{2 b^2 d} \\ & = \frac {(c+d x)^2 \operatorname {FresnelS}(a+b x)}{2 d}-\frac {d \text {Subst}\left (\int x^2 \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2}-\frac {(b c-a d) \text {Subst}\left (\int x \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^2}-\frac {(b c-a d)^2 \text {Subst}\left (\int \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2 d} \\ & = \frac {d (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }-\frac {(b c-a d)^2 \operatorname {FresnelS}(a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \operatorname {FresnelS}(a+b x)}{2 d}-\frac {(b c-a d) \text {Subst}\left (\int \sin \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^2}-\frac {d \text {Subst}\left (\int \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2 \pi } \\ & = \frac {(b c-a d) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^2 \pi }+\frac {d (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }-\frac {d \operatorname {FresnelC}(a+b x)}{2 b^2 \pi }-\frac {(b c-a d)^2 \operatorname {FresnelS}(a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \operatorname {FresnelS}(a+b x)}{2 d} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.50 \[ \int (c+d x) \operatorname {FresnelS}(a+b x) \, dx=\frac {-d \operatorname {FresnelC}(a+b x)+(2 b c-a d+b d x) \left (\cos \left (\frac {1}{2} \pi (a+b x)^2\right )+\pi (a+b x) \operatorname {FresnelS}(a+b x)\right )}{2 b^2 \pi } \]
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Time = 0.54 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {FresnelS}\left (b x +a \right ) \left (d a \left (b x +a \right )-c b \left (b x +a \right )-\frac {d \left (b x +a \right )^{2}}{2}\right )}{b}+\frac {\frac {d \left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {d \,\operatorname {FresnelC}\left (b x +a \right )}{\pi }-\frac {\left (2 a d -2 b c \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }}{2 b}}{b}\) | \(109\) |
default | \(\frac {-\frac {\operatorname {FresnelS}\left (b x +a \right ) \left (d a \left (b x +a \right )-c b \left (b x +a \right )-\frac {d \left (b x +a \right )^{2}}{2}\right )}{b}+\frac {\frac {d \left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {d \,\operatorname {FresnelC}\left (b x +a \right )}{\pi }-\frac {\left (2 a d -2 b c \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }}{2 b}}{b}\) | \(109\) |
parts | \(\frac {\operatorname {FresnelS}\left (b x +a \right ) d \,x^{2}}{2}+\operatorname {FresnelS}\left (b x +a \right ) c x -\frac {b \left (-\frac {d x \cos \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {d a \left (-\frac {\cos \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {\sqrt {\pi }\, a \,\operatorname {FresnelS}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )}{b}+\frac {d \,\operatorname {FresnelC}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}-\frac {2 c \cos \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {2 c \sqrt {\pi }\, a \,\operatorname {FresnelS}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )}{2}\) | \(254\) |
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Time = 0.26 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.09 \[ \int (c+d x) \operatorname {FresnelS}(a+b x) \, dx=\frac {\pi {\left (2 \, a b c - a^{2} d\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - \sqrt {b^{2}} d \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + {\left (b^{2} d x + 2 \, b^{2} c - a b d\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) + {\left (\pi b^{3} d x^{2} + 2 \, \pi b^{3} c x\right )} \operatorname {S}\left (b x + a\right )}{2 \, \pi b^{3}} \]
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\[ \int (c+d x) \operatorname {FresnelS}(a+b x) \, dx=\int \left (c + d x\right ) S\left (a + b x\right )\, dx \]
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\[ \int (c+d x) \operatorname {FresnelS}(a+b x) \, dx=\int { {\left (d x + c\right )} \operatorname {S}\left (b x + a\right ) \,d x } \]
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\[ \int (c+d x) \operatorname {FresnelS}(a+b x) \, dx=\int { {\left (d x + c\right )} \operatorname {S}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int (c+d x) \operatorname {FresnelS}(a+b x) \, dx=\int \mathrm {FresnelS}\left (a+b\,x\right )\,\left (c+d\,x\right ) \,d x \]
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