Integrand size = 10, antiderivative size = 148 \[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=-\frac {2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2}+\frac {a^3 \operatorname {FresnelC}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \operatorname {FresnelC}(a+b x)-\frac {a \operatorname {FresnelS}(a+b x)}{b^3 \pi }-\frac {a^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }+\frac {a (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {(a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi } \]
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Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6564, 3515, 3433, 3461, 2717, 3467, 3432, 3377, 2718} \[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=\frac {a^3 \operatorname {FresnelC}(a+b x)}{3 b^3}-\frac {a^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^3}-\frac {a \operatorname {FresnelS}(a+b x)}{\pi b^3}+\frac {a (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^3}-\frac {(a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi b^3}-\frac {2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi ^2 b^3}+\frac {1}{3} x^3 \operatorname {FresnelC}(a+b x) \]
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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 {1}{3} x^3 \operatorname {FresnelC}(a+b x)-\frac {1}{3} b \int x^3 \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \, dx \\ & = \frac {1}{3} x^3 \operatorname {FresnelC}(a+b x)-\frac {\text {Subst}\left (\int \left (-a^3 \cos \left (\frac {\pi x^2}{2}\right )+3 a^2 x \cos \left (\frac {\pi x^2}{2}\right )-3 a x^2 \cos \left (\frac {\pi x^2}{2}\right )+x^3 \cos \left (\frac {\pi x^2}{2}\right )\right ) \, dx,x,a+b x\right )}{3 b^3} \\ & = \frac {1}{3} x^3 \operatorname {FresnelC}(a+b x)-\frac {\text {Subst}\left (\int x^3 \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{3 b^3}+\frac {a \text {Subst}\left (\int x^2 \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3}-\frac {a^2 \text {Subst}\left (\int x \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3}+\frac {a^3 \text {Subst}\left (\int \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{3 b^3} \\ & = \frac {a^3 \operatorname {FresnelC}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \operatorname {FresnelC}(a+b x)+\frac {a (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {\text {Subst}\left (\int x \cos \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{6 b^3}-\frac {a^2 \text {Subst}\left (\int \cos \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^3}-\frac {a \text {Subst}\left (\int \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3 \pi } \\ & = \frac {a^3 \operatorname {FresnelC}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \operatorname {FresnelC}(a+b x)-\frac {a \operatorname {FresnelS}(a+b x)}{b^3 \pi }-\frac {a^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }+\frac {a (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {(a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi }+\frac {\text {Subst}\left (\int \sin \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{3 b^3 \pi } \\ & = -\frac {2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2}+\frac {a^3 \operatorname {FresnelC}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \operatorname {FresnelC}(a+b x)-\frac {a \operatorname {FresnelS}(a+b x)}{b^3 \pi }-\frac {a^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }+\frac {a (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {(a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi } \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.78 \[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=-\frac {2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )-\pi ^2 \left (a^3+b^3 x^3\right ) \operatorname {FresnelC}(a+b x)+3 a \pi \operatorname {FresnelS}(a+b x)+a^2 \pi \sin \left (\frac {1}{2} \pi (a+b x)^2\right )-a b \pi x \sin \left (\frac {1}{2} \pi (a+b x)^2\right )+b^2 \pi x^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2} \]
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Time = 0.53 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {FresnelC}\left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \operatorname {FresnelC}\left (b x +a \right )}{3}-\frac {a^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {a \left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {a \,\operatorname {FresnelS}\left (b x +a \right )}{\pi }-\frac {\left (b x +a \right )^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{3 \pi }-\frac {2 \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{3 \pi ^{2}}}{b^{3}}\) | \(122\) |
default | \(\frac {\frac {\operatorname {FresnelC}\left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \operatorname {FresnelC}\left (b x +a \right )}{3}-\frac {a^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {a \left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {a \,\operatorname {FresnelS}\left (b x +a \right )}{\pi }-\frac {\left (b x +a \right )^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{3 \pi }-\frac {2 \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{3 \pi ^{2}}}{b^{3}}\) | \(122\) |
parts | \(\frac {x^{3} \operatorname {FresnelC}\left (b x +a \right )}{3}-\frac {b \left (\frac {x^{2} \sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {a \left (\frac {x \sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {a \left (\frac {\sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {\sqrt {\pi }\, a \,\operatorname {FresnelC}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )}{b}-\frac {\operatorname {FresnelS}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b}-\frac {2 \left (-\frac {\cos \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {\sqrt {\pi }\, a \,\operatorname {FresnelS}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )}{b^{2} \pi }\right )}{3}\) | \(286\) |
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Time = 0.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00 \[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=\frac {\pi ^{2} b^{4} x^{3} \operatorname {C}\left (b x + a\right ) + \pi ^{2} a^{3} \sqrt {b^{2}} \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 3 \, \pi a \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 2 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) - {\left (\pi b^{3} x^{2} - \pi a b^{2} x + \pi a^{2} b\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{3 \, \pi ^{2} b^{4}} \]
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\[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=\int x^{2} C\left (a + b x\right )\, dx \]
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Result contains complex when optimal does not.
Time = 0.81 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.86 \[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {C}\left (b x + a\right ) - \frac {{\left (12 \, {\left (-i \, \pi e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + i \, \pi e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a^{3} + 4 \, {\left (3 \, {\left (-i \, \pi e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + i \, \pi e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a^{2} + 2 \, \Gamma \left (2, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + 2 \, \Gamma \left (2, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} b x + 8 \, a {\left (\Gamma \left (2, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + \Gamma \left (2, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} + \sqrt {2 \, \pi b^{2} x^{2} + 4 \, \pi a b x + 2 \, \pi a^{2}} {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \pi ^{\frac {3}{2}} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )} - \left (i + 1\right ) \, \sqrt {2} \pi ^{\frac {3}{2}} {\left (\operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )}\right )} a^{3} + 6 \, {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} a\right )}\right )} b}{24 \, {\left (\pi ^{2} b^{5} x + \pi ^{2} a b^{4}\right )}} \]
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\[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=\int { x^{2} \operatorname {C}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=\int x^2\,\mathrm {FresnelC}\left (a+b\,x\right ) \,d x \]
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