Integrand size = 8, antiderivative size = 74 \[ \int x^3 \operatorname {FresnelC}(b x) \, dx=-\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {3 \operatorname {FresnelC}(b x)}{4 b^4 \pi ^2}+\frac {1}{4} x^4 \operatorname {FresnelC}(b x)-\frac {x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi } \]
[Out]
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6562, 3467, 3466, 3433} \[ \int x^3 \operatorname {FresnelC}(b x) \, dx=\frac {3 \operatorname {FresnelC}(b x)}{4 \pi ^2 b^4}-\frac {x^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b}-\frac {3 x \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {1}{4} x^4 \operatorname {FresnelC}(b x) \]
[In]
[Out]
Rule 3433
Rule 3466
Rule 3467
Rule 6562
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \operatorname {FresnelC}(b x)-\frac {1}{4} b \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = \frac {1}{4} x^4 \operatorname {FresnelC}(b x)-\frac {x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {3 \int x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b \pi } \\ & = -\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {1}{4} x^4 \operatorname {FresnelC}(b x)-\frac {x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {3 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2} \\ & = -\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {3 \operatorname {FresnelC}(b x)}{4 b^4 \pi ^2}+\frac {1}{4} x^4 \operatorname {FresnelC}(b x)-\frac {x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi } \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int x^3 \operatorname {FresnelC}(b x) \, dx=-\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {3 \operatorname {FresnelC}(b x)}{4 b^4 \pi ^2}+\frac {1}{4} x^4 \operatorname {FresnelC}(b x)-\frac {x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi } \]
[In]
[Out]
Time = 0.38 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84
method | result | size |
meijerg | \(\frac {-\frac {3 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) b x}{4}-\frac {b^{3} x^{3} \pi \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}+\frac {\left (5 x^{4} \pi ^{2} b^{4}+15\right ) \operatorname {FresnelC}\left (b x \right )}{20}}{b^{4} \pi ^{2}}\) | \(62\) |
derivativedivides | \(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{4} x^{4}}{4}-\frac {b^{3} x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }+\frac {-\frac {3 b x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }+\frac {3 \,\operatorname {FresnelC}\left (b x \right )}{4 \pi }}{\pi }}{b^{4}}\) | \(70\) |
default | \(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{4} x^{4}}{4}-\frac {b^{3} x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }+\frac {-\frac {3 b x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }+\frac {3 \,\operatorname {FresnelC}\left (b x \right )}{4 \pi }}{\pi }}{b^{4}}\) | \(70\) |
parts | \(\frac {x^{4} \operatorname {FresnelC}\left (b x \right )}{4}-\frac {b \left (\frac {x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {3 \left (-\frac {x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {\operatorname {FresnelC}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b^{2} \pi }\right )}{4}\) | \(93\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.80 \[ \int x^3 \operatorname {FresnelC}(b x) \, dx=-\frac {\pi b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 3 \, b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{4} x^{4} + 3\right )} \operatorname {C}\left (b x\right )}{4 \, \pi ^{2} b^{4}} \]
[In]
[Out]
Time = 0.56 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.51 \[ \int x^3 \operatorname {FresnelC}(b x) \, dx=\frac {5 x^{4} C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{64 \Gamma \left (\frac {9}{4}\right )} - \frac {5 x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{64 \pi b \Gamma \left (\frac {9}{4}\right )} - \frac {15 x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{64 \pi ^{2} b^{3} \Gamma \left (\frac {9}{4}\right )} + \frac {15 C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{64 \pi ^{2} b^{4} \Gamma \left (\frac {9}{4}\right )} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.27 \[ \int x^3 \operatorname {FresnelC}(b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {C}\left (b x\right ) - \frac {\sqrt {\frac {1}{2}} {\left (4 \, \sqrt {\frac {1}{2}} \pi ^{2} b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 12 \, \sqrt {\frac {1}{2}} \pi b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \left (3 i - 3\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) - \left (3 i + 3\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right )\right )}}{8 \, \pi ^{3} b^{4}} \]
[In]
[Out]
\[ \int x^3 \operatorname {FresnelC}(b x) \, dx=\int { x^{3} \operatorname {C}\left (b x\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int x^3 \operatorname {FresnelC}(b x) \, dx=\int x^3\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]
[In]
[Out]