Integrand size = 14, antiderivative size = 194 \[ \int (c+d x)^2 \operatorname {FresnelC}(a+b x) \, dx=-\frac {2 d^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2}-\frac {(b c-a d)^3 \operatorname {FresnelC}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \operatorname {FresnelC}(a+b x)}{3 d}+\frac {d (b c-a d) \operatorname {FresnelS}(a+b x)}{b^3 \pi }-\frac {(b c-a d)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {d (b c-a d) (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {d^2 (a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi } \]
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Time = 0.16 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6564, 3515, 3433, 3461, 2717, 3467, 3432, 3377, 2718} \[ \int (c+d x)^2 \operatorname {FresnelC}(a+b x) \, dx=-\frac {(b c-a d)^3 \operatorname {FresnelC}(a+b x)}{3 b^3 d}+\frac {d (b c-a d) \operatorname {FresnelS}(a+b x)}{\pi b^3}-\frac {(b c-a d)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^3}-\frac {d (a+b x) (b c-a d) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^3}-\frac {d^2 (a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi b^3}-\frac {2 d^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi ^2 b^3}+\frac {(c+d x)^3 \operatorname {FresnelC}(a+b x)}{3 d} \]
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Rule 2717
Rule 2718
Rule 3377
Rule 3432
Rule 3433
Rule 3461
Rule 3467
Rule 3515
Rule 6564
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^3 \operatorname {FresnelC}(a+b x)}{3 d}-\frac {b \int (c+d x)^3 \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \, dx}{3 d} \\ & = \frac {(c+d x)^3 \operatorname {FresnelC}(a+b x)}{3 d}-\frac {\text {Subst}\left (\int \left (b^3 c^3 \left (1-\frac {a d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{b^3 c^3}\right ) \cos \left (\frac {\pi x^2}{2}\right )+3 b^2 c^2 d \left (1+\frac {a d (-2 b c+a d)}{b^2 c^2}\right ) x \cos \left (\frac {\pi x^2}{2}\right )+3 b c d^2 \left (1-\frac {a d}{b c}\right ) x^2 \cos \left (\frac {\pi x^2}{2}\right )+d^3 x^3 \cos \left (\frac {\pi x^2}{2}\right )\right ) \, dx,x,a+b x\right )}{3 b^3 d} \\ & = \frac {(c+d x)^3 \operatorname {FresnelC}(a+b x)}{3 d}-\frac {d^2 \text {Subst}\left (\int x^3 \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{3 b^3}-\frac {(d (b c-a d)) \text {Subst}\left (\int x^2 \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3}-\frac {(b c-a d)^2 \text {Subst}\left (\int x \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3}-\frac {(b c-a d)^3 \text {Subst}\left (\int \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{3 b^3 d} \\ & = -\frac {(b c-a d)^3 \operatorname {FresnelC}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \operatorname {FresnelC}(a+b x)}{3 d}-\frac {d (b c-a d) (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {d^2 \text {Subst}\left (\int x \cos \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{6 b^3}-\frac {(b c-a d)^2 \text {Subst}\left (\int \cos \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^3}+\frac {(d (b c-a d)) \text {Subst}\left (\int \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3 \pi } \\ & = -\frac {(b c-a d)^3 \operatorname {FresnelC}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \operatorname {FresnelC}(a+b x)}{3 d}+\frac {d (b c-a d) \operatorname {FresnelS}(a+b x)}{b^3 \pi }-\frac {(b c-a d)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {d (b c-a d) (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {d^2 (a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi }+\frac {d^2 \text {Subst}\left (\int \sin \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{3 b^3 \pi } \\ & = -\frac {2 d^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2}-\frac {(b c-a d)^3 \operatorname {FresnelC}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \operatorname {FresnelC}(a+b x)}{3 d}+\frac {d (b c-a d) \operatorname {FresnelS}(a+b x)}{b^3 \pi }-\frac {(b c-a d)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {d (b c-a d) (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {d^2 (a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi } \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.22 \[ \int (c+d x)^2 \operatorname {FresnelC}(a+b x) \, dx=\frac {-2 d^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )+\pi ^2 \left (3 a b^2 c^2-3 a^2 b c d+a^3 d^2+b^3 x \left (3 c^2+3 c d x+d^2 x^2\right )\right ) \operatorname {FresnelC}(a+b x)+3 d (b c-a d) \pi \operatorname {FresnelS}(a+b x)-3 b^2 c^2 \pi \sin \left (\frac {1}{2} \pi (a+b x)^2\right )+3 a b c d \pi \sin \left (\frac {1}{2} \pi (a+b x)^2\right )-a^2 d^2 \pi \sin \left (\frac {1}{2} \pi (a+b x)^2\right )-3 b^2 c d \pi x \sin \left (\frac {1}{2} \pi (a+b x)^2\right )+a b d^2 \pi x \sin \left (\frac {1}{2} \pi (a+b x)^2\right )-b^2 d^2 \pi x^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2} \]
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Time = 0.72 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {FresnelC}\left (b x +a \right ) \left (a d -b c -d \left (b x +a \right )\right )^{3}}{3 b^{2} d}+\frac {-\frac {d^{3} \left (b x +a \right )^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {2 d^{3} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi ^{2}}+\frac {3 \left (a d -b c \right ) d^{2} \left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {3 \left (a d -b c \right ) d^{2} \operatorname {FresnelS}\left (b x +a \right )}{\pi }-\frac {3 \left (a d -b c \right )^{2} d \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\left (a d -b c \right )^{3} \operatorname {FresnelC}\left (b x +a \right )}{3 b^{2} d}}{b}\) | \(190\) |
default | \(\frac {-\frac {\operatorname {FresnelC}\left (b x +a \right ) \left (a d -b c -d \left (b x +a \right )\right )^{3}}{3 b^{2} d}+\frac {-\frac {d^{3} \left (b x +a \right )^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {2 d^{3} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi ^{2}}+\frac {3 \left (a d -b c \right ) d^{2} \left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {3 \left (a d -b c \right ) d^{2} \operatorname {FresnelS}\left (b x +a \right )}{\pi }-\frac {3 \left (a d -b c \right )^{2} d \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\left (a d -b c \right )^{3} \operatorname {FresnelC}\left (b x +a \right )}{3 b^{2} d}}{b}\) | \(190\) |
parts | \(\frac {\operatorname {FresnelC}\left (b x +a \right ) d^{2} x^{3}}{3}+\operatorname {FresnelC}\left (b x +a \right ) d c \,x^{2}+\operatorname {FresnelC}\left (b x +a \right ) c^{2} x +\frac {\operatorname {FresnelC}\left (b x +a \right ) c^{3}}{3 d}-\frac {b \left (\frac {d^{3} x^{2} \sin \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^{3} a \left (\frac {x \sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {a \left (\frac {\sin \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 {FresnelC}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )}{b}-\frac {\operatorname {FresnelS}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b}-\frac {2 d^{3} \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^{2} \pi }+\frac {3 c \,d^{2} x \sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {3 c \,d^{2} a \left (\frac {\sin \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 {FresnelC}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )}{b}-\frac {3 c \,d^{2} \operatorname {FresnelS}\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 {3 c^{2} d \sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {3 c^{2} d \sqrt {\pi }\, a \,\operatorname {FresnelC}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}+\frac {\sqrt {\pi }\, c^{3} \operatorname {FresnelC}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{\sqrt {b^{2} \pi }}\right )}{3 d}\) | \(620\) |
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Time = 0.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.28 \[ \int (c+d x)^2 \operatorname {FresnelC}(a+b x) \, dx=\frac {\pi ^{2} {\left (3 \, a b^{2} c^{2} - 3 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 2 \, b d^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) + 3 \, \pi {\left (b c d - a d^{2}\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + {\left (\pi ^{2} b^{4} d^{2} x^{3} + 3 \, \pi ^{2} b^{4} c d x^{2} + 3 \, \pi ^{2} b^{4} c^{2} x\right )} \operatorname {C}\left (b x + a\right ) - {\left (\pi b^{3} d^{2} x^{2} + \pi {\left (3 \, b^{3} c d - a b^{2} d^{2}\right )} x + \pi {\left (3 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2}\right )}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{3 \, \pi ^{2} b^{4}} \]
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\[ \int (c+d x)^2 \operatorname {FresnelC}(a+b x) \, dx=\int \left (c + d x\right )^{2} C\left (a + b x\right )\, dx \]
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\[ \int (c+d x)^2 \operatorname {FresnelC}(a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \operatorname {C}\left (b x + a\right ) \,d x } \]
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\[ \int (c+d x)^2 \operatorname {FresnelC}(a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \operatorname {C}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int (c+d x)^2 \operatorname {FresnelC}(a+b x) \, dx=\int \mathrm {FresnelC}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2 \,d x \]
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