Optimal. Leaf size=101 \[ \frac {\cos (c) \text {FresnelC}(b x) S(b x)}{2 b}+\frac {1}{8} i b x^2 \cos (c) \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )-\frac {1}{8} i b x^2 \cos (c) \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )+\frac {\text {FresnelC}(b x)^2 \sin (c)}{2 b} \]
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Rubi [A]
time = 0.04, 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 = {6584, 6576, 30,
6582} \begin {gather*} \frac {1}{8} i b x^2 \cos (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 \cos (c) \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )+\frac {\cos (c) \text {FresnelC}(b x) S(b x)}{2 b}+\frac {\sin (c) \text {FresnelC}(b x)^2}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 6576
Rule 6582
Rule 6584
Rubi steps
\begin {align*} \int C(b x) \sin \left (c+\frac {1}{2} b^2 \pi x^2\right ) \, dx &=\cos (c) \int C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx+\sin (c) \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=\frac {\cos (c) C(b x) S(b x)}{2 b}+\frac {1}{8} i b x^2 \cos (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 \cos (c) \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )+\frac {\sin (c) \text {Subst}(\int x \, dx,x,C(b x))}{b}\\ &=\frac {\cos (c) C(b x) S(b x)}{2 b}+\frac {1}{8} i b x^2 \cos (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 \cos (c) \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )+\frac {C(b x)^2 \sin (c)}{2 b}\\ \end {align*}
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Mathematica [F]
time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int \text {FresnelC}(b x) \sin \left (c+\frac {1}{2} b^2 \pi x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \FresnelC \left (b x \right ) \sin \left (c +\frac {b^{2} \pi \,x^{2}}{2}\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sin {\left (\frac {\pi b^{2} x^{2}}{2} + c \right )} C\left (b x\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sin \left (\frac {\Pi \,b^2\,x^2}{2}+c\right )\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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