Optimal. Leaf size=69 \[ \frac {(a+b x) C(a+b x)^2}{b}-\frac {2 C(a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b}+\frac {S\left (\sqrt {2} (a+b x)\right )}{\sqrt {2} \pi b} \]
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Rubi [A] time = 0.15, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6421, 6453, 3351} \[ \frac {(a+b x) \text {FresnelC}(a+b x)^2}{b}-\frac {2 \text {FresnelC}(a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b}+\frac {S\left (\sqrt {2} (a+b x)\right )}{\sqrt {2} \pi b} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 6421
Rule 6453
Rubi steps
\begin {align*} \int C(a+b x)^2 \, dx &=\frac {(a+b x) C(a+b x)^2}{b}-2 \int (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) C(a+b x) \, dx\\ &=\frac {(a+b x) C(a+b x)^2}{b}-\frac {2 \operatorname {Subst}\left (\int x \cos \left (\frac {\pi x^2}{2}\right ) C(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) C(a+b x)^2}{b}-\frac {2 C(a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b \pi }+\frac {\operatorname {Subst}\left (\int \sin \left (\pi x^2\right ) \, dx,x,a+b x\right )}{b \pi }\\ &=\frac {(a+b x) C(a+b x)^2}{b}+\frac {S\left (\sqrt {2} (a+b x)\right )}{\sqrt {2} b \pi }-\frac {2 C(a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b \pi }\\ \end {align*}
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Mathematica [A] time = 0.02, size = 66, normalized size = 0.96 \[ \frac {2 \pi (a+b x) C(a+b x)^2-4 C(a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )+\sqrt {2} S\left (\sqrt {2} (a+b x)\right )}{2 \pi b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\rm fresnelc}\left (b x + a\right )^{2}, x\right ) \]
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 \[ \int {\rm fresnelc}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 60, normalized size = 0.87 \[ \frac {\left (b x +a \right ) \FresnelC \left (b x +a \right )^{2}-\frac {2 \FresnelC \left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {\sqrt {2}\, \mathrm {S}\left (\left (b x +a \right ) \sqrt {2}\right )}{2 \pi }}{b} \]
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 \[ \int {\rm fresnelc}\left (b x + a\right )^{2}\,{d x} \]
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 \[ \int {\mathrm {FresnelC}\left (a+b\,x\right )}^2 \,d x \]
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 \[ \int C^{2}\left (a + b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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