Integrand size = 19, antiderivative size = 17 \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\frac {\operatorname {FresnelC}(b x)^{1+n}}{b (1+n)} \]
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Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6576, 30} \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\frac {\operatorname {FresnelC}(b x)^{n+1}}{b (n+1)} \]
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Rule 30
Rule 6576
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^n \, dx,x,\operatorname {FresnelC}(b x)\right )}{b} \\ & = \frac {\operatorname {FresnelC}(b x)^{1+n}}{b (1+n)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\frac {\operatorname {FresnelC}(b x)^{1+n}}{b (1+n)} \]
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Time = 0.81 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {\operatorname {FresnelC}\left (b x \right )^{1+n}}{b \left (1+n \right )}\) | \(18\) |
default | \(\frac {\operatorname {FresnelC}\left (b x \right )^{1+n}}{b \left (1+n \right )}\) | \(18\) |
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Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\frac {\operatorname {C}\left (b x\right )^{n} \operatorname {C}\left (b x\right )}{b n + b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (12) = 24\).
Time = 0.62 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00 \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: b = 0 \wedge n = -1 \\0^{n} x & \text {for}\: b = 0 \\\frac {\log {\left (C\left (b x\right ) \right )}}{b} & \text {for}\: n = -1 \\\frac {C\left (b x\right ) C^{n}\left (b x\right )}{b n + b} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\frac {\operatorname {C}\left (b x\right )^{n + 1}}{b {\left (n + 1\right )}} \]
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\[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\int { \operatorname {C}\left (b x\right )^{n} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \]
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Timed out. \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\int {\mathrm {FresnelC}\left (b\,x\right )}^n\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
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