Optimal. Leaf size=101 \[ -\frac {1}{8} i b x^2 \sin (c) \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )+\frac {1}{8} i b x^2 \sin (c) \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )-\frac {\sin (c) C(b x) S(b x)}{2 b}+\frac {\cos (c) C(b x)^2}{2 b} \]
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Rubi [A] time = 0.05, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {6443, 6441, 30, 6447} \[ -\frac {1}{8} i b x^2 \sin (c) \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )+\frac {1}{8} i b x^2 \sin (c) \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )-\frac {\sin (c) \text {FresnelC}(b x) S(b x)}{2 b}+\frac {\cos (c) \text {FresnelC}(b x)^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 6441
Rule 6443
Rule 6447
Rubi steps
\begin {align*} \int \cos \left (c+\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx &=\cos (c) \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx-\sin (c) \int C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac {C(b x) S(b x) \sin (c)}{2 b}-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right ) \sin (c)+\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right ) \sin (c)+\frac {\cos (c) \operatorname {Subst}(\int x \, dx,x,C(b x))}{b}\\ &=\frac {\cos (c) C(b x)^2}{2 b}-\frac {C(b x) S(b x) \sin (c)}{2 b}-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right ) \sin (c)+\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right ) \sin (c)\\ \end {align*}
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Mathematica [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \cos \left (c+\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + c\right ) {\rm fresnelc}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + c\right ) {\rm fresnelc}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \cos \left (c +\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + c\right ) {\rm fresnelc}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (\frac {\Pi \,b^2\,x^2}{2}+c\right )\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\left (\frac {\pi b^{2} x^{2}}{2} + c \right )} C\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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