Optimal. Leaf size=122 \[ -\frac {(b c-a d)^2 C(a+b x)}{2 b^2 d}-\frac {(b c-a d) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^2}+\frac {d S(a+b x)}{2 \pi b^2}-\frac {d (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 \pi b^2}+\frac {(c+d x)^2 C(a+b x)}{2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6429, 3434, 3352, 3380, 2637, 3386, 3351} \[ -\frac {(b c-a d)^2 \text {FresnelC}(a+b x)}{2 b^2 d}-\frac {(b c-a d) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^2}+\frac {d S(a+b x)}{2 \pi b^2}-\frac {d (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 \pi b^2}+\frac {(c+d x)^2 \text {FresnelC}(a+b x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3351
Rule 3352
Rule 3380
Rule 3386
Rule 3434
Rule 6429
Rubi steps
\begin {align*} \int (c+d x) C(a+b x) \, dx &=\frac {(c+d x)^2 C(a+b x)}{2 d}-\frac {b \int (c+d x)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \, dx}{2 d}\\ &=\frac {(c+d x)^2 C(a+b x)}{2 d}-\frac {\operatorname {Subst}\left (\int \left (b^2 c^2 \left (1+\frac {a d (-2 b c+a d)}{b^2 c^2}\right ) \cos \left (\frac {\pi x^2}{2}\right )+2 b c d \left (1-\frac {a d}{b c}\right ) x \cos \left (\frac {\pi x^2}{2}\right )+d^2 x^2 \cos \left (\frac {\pi x^2}{2}\right )\right ) \, dx,x,a+b x\right )}{2 b^2 d}\\ &=\frac {(c+d x)^2 C(a+b x)}{2 d}-\frac {d \operatorname {Subst}\left (\int x^2 \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2}-\frac {(b c-a d) \operatorname {Subst}\left (\int x \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^2}-\frac {(b c-a d)^2 \operatorname {Subst}\left (\int \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2 d}\\ &=-\frac {(b c-a d)^2 C(a+b x)}{2 b^2 d}+\frac {(c+d x)^2 C(a+b x)}{2 d}-\frac {d (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }-\frac {(b c-a d) \operatorname {Subst}\left (\int \cos \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^2}+\frac {d \operatorname {Subst}\left (\int \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2 \pi }\\ &=-\frac {(b c-a d)^2 C(a+b x)}{2 b^2 d}+\frac {(c+d x)^2 C(a+b x)}{2 d}+\frac {d S(a+b x)}{2 b^2 \pi }-\frac {(b c-a d) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^2 \pi }-\frac {d (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }\\ \end {align*}
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Mathematica [A] time = 0.28, size = 74, normalized size = 0.61 \[ \frac {-\pi (a+b x) C(a+b x) (a d-b (2 c+d x))+\sin \left (\frac {1}{2} \pi (a+b x)^2\right ) (a d-2 b c-b d x)+d S(a+b x)}{2 \pi b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x + c\right )} {\rm fresnelc}\left (b x + a\right ), x\right ) \]
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 \[ \int {\left (d x + c\right )} {\rm fresnelc}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 107, normalized size = 0.88 \[ \frac {\frac {\FresnelC \left (b x +a \right ) \left (\frac {\left (b x +a \right )^{2} d}{2}-a d \left (b x +a \right )+b c \left (b x +a \right )\right )}{b}-\frac {\frac {d \left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {d \,\mathrm {S}\left (b x +a \right )}{\pi }+\frac {\left (-2 a d +2 b c \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }}{2 b}}{b} \]
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 \[ \int {\left (d x + c\right )} {\rm fresnelc}\left (b x + a\right )\,{d x} \]
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 \[ \int \mathrm {FresnelC}\left (a+b\,x\right )\,\left (c+d\,x\right ) \,d x \]
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 \[ \int \left (c + d x\right ) C\left (a + b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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