Optimal. Leaf size=296 \[ -\frac{2 d^2 (b c-a d) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi ^2 b^4}+\frac{d^2 (a+b x)^2 (b c-a d) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^4}-\frac{3 d (b c-a d)^2 \text{FresnelC}(a+b x)}{2 \pi b^4}-\frac{(b c-a d)^4 S(a+b x)}{4 b^4 d}+\frac{(b c-a d)^3 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^4}+\frac{3 d (a+b x) (b c-a d)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 \pi b^4}+\frac{3 d^3 S(a+b x)}{4 \pi ^2 b^4}-\frac{3 d^3 (a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 \pi ^2 b^4}+\frac{d^3 (a+b x)^3 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 \pi b^4}+\frac{(c+d x)^4 S(a+b x)}{4 d} \]
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Rubi [A] time = 0.400987, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {6428, 3433, 3351, 3379, 2638, 3385, 3352, 3296, 2637, 3386} \[ -\frac{2 d^2 (b c-a d) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi ^2 b^4}+\frac{d^2 (a+b x)^2 (b c-a d) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^4}-\frac{3 d (b c-a d)^2 \text{FresnelC}(a+b x)}{2 \pi b^4}-\frac{(b c-a d)^4 S(a+b x)}{4 b^4 d}+\frac{(b c-a d)^3 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^4}+\frac{3 d (a+b x) (b c-a d)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 \pi b^4}+\frac{3 d^3 S(a+b x)}{4 \pi ^2 b^4}-\frac{3 d^3 (a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 \pi ^2 b^4}+\frac{d^3 (a+b x)^3 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 \pi b^4}+\frac{(c+d x)^4 S(a+b x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 6428
Rule 3433
Rule 3351
Rule 3379
Rule 2638
Rule 3385
Rule 3352
Rule 3296
Rule 2637
Rule 3386
Rubi steps
\begin{align*} \int (c+d x)^3 S(a+b x) \, dx &=\frac{(c+d x)^4 S(a+b x)}{4 d}-\frac{b \int (c+d x)^4 \sin \left (\frac{1}{2} \pi (a+b x)^2\right ) \, dx}{4 d}\\ &=\frac{(c+d x)^4 S(a+b x)}{4 d}-\frac{\operatorname{Subst}\left (\int \left (b^4 c^4 \left (1+\frac{a d \left (-4 b^3 c^3+6 a b^2 c^2 d-4 a^2 b c d^2+a^3 d^3\right )}{b^4 c^4}\right ) \sin \left (\frac{\pi x^2}{2}\right )+4 b^3 c^3 d \left (1-\frac{a d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{b^3 c^3}\right ) x \sin \left (\frac{\pi x^2}{2}\right )+6 b^2 c^2 d^2 \left (1+\frac{a d (-2 b c+a d)}{b^2 c^2}\right ) x^2 \sin \left (\frac{\pi x^2}{2}\right )+4 b c d^3 \left (1-\frac{a d}{b c}\right ) x^3 \sin \left (\frac{\pi x^2}{2}\right )+d^4 x^4 \sin \left (\frac{\pi x^2}{2}\right )\right ) \, dx,x,a+b x\right )}{4 b^4 d}\\ &=\frac{(c+d x)^4 S(a+b x)}{4 d}-\frac{d^3 \operatorname{Subst}\left (\int x^4 \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{4 b^4}-\frac{\left (d^2 (b c-a d)\right ) \operatorname{Subst}\left (\int x^3 \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^4}-\frac{\left (3 d (b c-a d)^2\right ) \operatorname{Subst}\left (\int x^2 \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^4}-\frac{(b c-a d)^3 \operatorname{Subst}\left (\int x \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^4}-\frac{(b c-a d)^4 \operatorname{Subst}\left (\int \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{4 b^4 d}\\ &=\frac{3 d (b c-a d)^2 (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 b^4 \pi }+\frac{d^3 (a+b x)^3 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi }-\frac{(b c-a d)^4 S(a+b x)}{4 b^4 d}+\frac{(c+d x)^4 S(a+b x)}{4 d}-\frac{\left (d^2 (b c-a d)\right ) \operatorname{Subst}\left (\int x \sin \left (\frac{\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^4}-\frac{(b c-a d)^3 \operatorname{Subst}\left (\int \sin \left (\frac{\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^4}-\frac{\left (3 d^3\right ) \operatorname{Subst}\left (\int x^2 \cos \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{4 b^4 \pi }-\frac{\left (3 d (b c-a d)^2\right ) \operatorname{Subst}\left (\int \cos \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^4 \pi }\\ &=\frac{(b c-a d)^3 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^4 \pi }+\frac{3 d (b c-a d)^2 (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 b^4 \pi }+\frac{d^2 (b c-a d) (a+b x)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^4 \pi }+\frac{d^3 (a+b x)^3 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi }-\frac{3 d (b c-a d)^2 C(a+b x)}{2 b^4 \pi }-\frac{(b c-a d)^4 S(a+b x)}{4 b^4 d}+\frac{(c+d x)^4 S(a+b x)}{4 d}-\frac{3 d^3 (a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi ^2}+\frac{\left (3 d^3\right ) \operatorname{Subst}\left (\int \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{4 b^4 \pi ^2}-\frac{\left (d^2 (b c-a d)\right ) \operatorname{Subst}\left (\int \cos \left (\frac{\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{b^4 \pi }\\ &=\frac{(b c-a d)^3 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^4 \pi }+\frac{3 d (b c-a d)^2 (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 b^4 \pi }+\frac{d^2 (b c-a d) (a+b x)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^4 \pi }+\frac{d^3 (a+b x)^3 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi }-\frac{3 d (b c-a d)^2 C(a+b x)}{2 b^4 \pi }-\frac{(b c-a d)^4 S(a+b x)}{4 b^4 d}+\frac{3 d^3 S(a+b x)}{4 b^4 \pi ^2}+\frac{(c+d x)^4 S(a+b x)}{4 d}-\frac{2 d^2 (b c-a d) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^4 \pi ^2}-\frac{3 d^3 (a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi ^2}\\ \end{align*}
Mathematica [A] time = 0.813954, size = 424, normalized size = 1.43 \[ \frac{S(a+b x) \left (6 \pi ^2 b^2 c^2 d \left (b^2 x^2-a^2\right )+4 \pi ^2 b c d^2 \left (a^3+b^3 x^3\right )+d^3 \left (-\pi ^2 a^4+\pi ^2 b^4 x^4+3\right )+4 \pi ^2 b^3 c^3 (a+b x)\right )+4 \pi a^2 b c d^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )+\pi a^2 b d^3 x \cos \left (\frac{1}{2} \pi (a+b x)^2\right )-\pi a^3 d^3 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )+6 \pi b^3 c^2 d x \cos \left (\frac{1}{2} \pi (a+b x)^2\right )-6 \pi a b^2 c^2 d \cos \left (\frac{1}{2} \pi (a+b x)^2\right )+4 \pi b^3 c^3 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )+4 \pi b^3 c d^2 x^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )-4 \pi a b^2 c d^2 x \cos \left (\frac{1}{2} \pi (a+b x)^2\right )+\pi b^3 d^3 x^3 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )-\pi a b^2 d^3 x^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right )-8 b c d^2 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )-6 \pi d (b c-a d)^2 \text{FresnelC}(a+b x)-3 b d^3 x \sin \left (\frac{1}{2} \pi (a+b x)^2\right )+5 a d^3 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 \pi ^2 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 400, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ({\frac{{\it FresnelS} \left ( bx+a \right ) \left ( d \left ( bx+a \right ) -ad+bc \right ) ^{4}}{4\,d{b}^{3}}}-{\frac{1}{4\,d{b}^{3}} \left ( -{\frac{{d}^{4} \left ( bx+a \right ) ^{3}}{\pi }\cos \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) }+3\,{\frac{{d}^{4}}{\pi } \left ({\frac{ \left ( bx+a \right ) \sin \left ( 1/2\,\pi \, \left ( bx+a \right ) ^{2} \right ) }{\pi }}-{\frac{{\it FresnelS} \left ( bx+a \right ) }{\pi }} \right ) }-{\frac{ \left ( -4\,a{d}^{4}+4\,bc{d}^{3} \right ) \left ( bx+a \right ) ^{2}}{\pi }\cos \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) }+2\,{\frac{ \left ( -4\,a{d}^{4}+4\,bc{d}^{3} \right ) \sin \left ( 1/2\,\pi \, \left ( bx+a \right ) ^{2} \right ) }{{\pi }^{2}}}-{\frac{ \left ( 6\,{a}^{2}{d}^{4}-12\,abc{d}^{3}+6\,{b}^{2}{c}^{2}{d}^{2} \right ) \left ( bx+a \right ) }{\pi }\cos \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) }+{\frac{ \left ( 6\,{a}^{2}{d}^{4}-12\,abc{d}^{3}+6\,{b}^{2}{c}^{2}{d}^{2} \right ){\it FresnelC} \left ( bx+a \right ) }{\pi }}-{\frac{-4\,{a}^{3}{d}^{4}+12\,{a}^{2}bc{d}^{3}-12\,a{b}^{2}{c}^{2}{d}^{2}+4\,{b}^{3}{c}^{3}d}{\pi }\cos \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) }+{a}^{4}{d}^{4}{\it FresnelS} \left ( bx+a \right ) -4\,{a}^{3}bc{d}^{3}{\it FresnelS} \left ( bx+a \right ) +6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}{\it FresnelS} \left ( bx+a \right ) -4\,a{b}^{3}{c}^{3}d{\it FresnelS} \left ( bx+a \right ) +{b}^{4}{c}^{4}{\it FresnelS} \left ( bx+a \right ) \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{3}{\rm fresnels}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}{\rm fresnels}\left (b x + a\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right )^{3} S\left (a + b x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{3}{\rm fresnels}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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