Optimal. Leaf size=227 \[ -\frac {a^4 C(a+b x)}{4 b^4}+\frac {a^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^4}+\frac {3 a^2 S(a+b x)}{2 \pi b^4}-\frac {3 a^2 (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 \pi b^4}+\frac {3 C(a+b x)}{4 \pi ^2 b^4}+\frac {a (a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^4}-\frac {(a+b x)^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{4 \pi b^4}+\frac {2 a \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi ^2 b^4}-\frac {3 (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{4 \pi ^2 b^4}+\frac {1}{4} x^4 C(a+b x) \]
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Rubi [A] time = 0.19, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6429, 3434, 3352, 3380, 2637, 3386, 3351, 3296, 2638, 3385} \[ -\frac {a^4 \text {FresnelC}(a+b x)}{4 b^4}+\frac {3 a^2 S(a+b x)}{2 \pi b^4}+\frac {a^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^4}-\frac {3 a^2 (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 \pi b^4}+\frac {3 \text {FresnelC}(a+b x)}{4 \pi ^2 b^4}+\frac {a (a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^4}-\frac {(a+b x)^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{4 \pi b^4}+\frac {2 a \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi ^2 b^4}-\frac {3 (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{4 \pi ^2 b^4}+\frac {1}{4} x^4 \text {FresnelC}(a+b x) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 3351
Rule 3352
Rule 3380
Rule 3385
Rule 3386
Rule 3434
Rule 6429
Rubi steps
\begin {align*} \int x^3 C(a+b x) \, dx &=\frac {1}{4} x^4 C(a+b x)-\frac {1}{4} b \int x^4 \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \, dx\\ &=\frac {1}{4} x^4 C(a+b x)-\frac {\operatorname {Subst}\left (\int \left (a^4 \cos \left (\frac {\pi x^2}{2}\right )-4 a^3 x \cos \left (\frac {\pi x^2}{2}\right )+6 a^2 x^2 \cos \left (\frac {\pi x^2}{2}\right )-4 a x^3 \cos \left (\frac {\pi x^2}{2}\right )+x^4 \cos \left (\frac {\pi x^2}{2}\right )\right ) \, dx,x,a+b x\right )}{4 b^4}\\ &=\frac {1}{4} x^4 C(a+b x)-\frac {\operatorname {Subst}\left (\int x^4 \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{4 b^4}+\frac {a \operatorname {Subst}\left (\int x^3 \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^4}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int x^2 \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^4}+\frac {a^3 \operatorname {Subst}\left (\int x \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^4}-\frac {a^4 \operatorname {Subst}\left (\int \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{4 b^4}\\ &=-\frac {a^4 C(a+b x)}{4 b^4}+\frac {1}{4} x^4 C(a+b x)-\frac {3 a^2 (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 b^4 \pi }-\frac {(a+b x)^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi }+\frac {a \operatorname {Subst}\left (\int x \cos \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^4}+\frac {a^3 \operatorname {Subst}\left (\int \cos \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^4}+\frac {3 \operatorname {Subst}\left (\int x^2 \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{4 b^4 \pi }+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^4 \pi }\\ &=-\frac {3 (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi ^2}-\frac {a^4 C(a+b x)}{4 b^4}+\frac {1}{4} x^4 C(a+b x)+\frac {3 a^2 S(a+b x)}{2 b^4 \pi }+\frac {a^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^4 \pi }-\frac {3 a^2 (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 b^4 \pi }+\frac {a (a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^4 \pi }-\frac {(a+b x)^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi }+\frac {3 \operatorname {Subst}\left (\int \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{4 b^4 \pi ^2}-\frac {a \operatorname {Subst}\left (\int \sin \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{b^4 \pi }\\ &=\frac {2 a \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^4 \pi ^2}-\frac {3 (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi ^2}-\frac {a^4 C(a+b x)}{4 b^4}+\frac {3 C(a+b x)}{4 b^4 \pi ^2}+\frac {1}{4} x^4 C(a+b x)+\frac {3 a^2 S(a+b x)}{2 b^4 \pi }+\frac {a^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^4 \pi }-\frac {3 a^2 (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 b^4 \pi }+\frac {a (a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^4 \pi }-\frac {(a+b x)^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi }\\ \end {align*}
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Mathematica [A] time = 0.37, size = 166, normalized size = 0.73 \[ \frac {\left (-\pi ^2 a^4+\pi ^2 b^4 x^4+3\right ) C(a+b x)+\pi a^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )+6 \pi a^2 S(a+b x)-\pi a^2 b x \sin \left (\frac {1}{2} \pi (a+b x)^2\right )-\pi b^3 x^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )+\pi a b^2 x^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )+5 a \cos \left (\frac {1}{2} \pi (a+b x)^2\right )-3 b x \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{4 \pi ^2 b^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} {\rm fresnelc}\left (b x + a\right ), x\right ) \]
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 \[ \int x^{3} {\rm fresnelc}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 187, normalized size = 0.82 \[ \frac {\frac {\FresnelC \left (b x +a \right ) b^{4} x^{4}}{4}-\frac {\left (b x +a \right )^{3} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{4 \pi }+\frac {-\frac {3 \left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{4 \pi }+\frac {3 \FresnelC \left (b x +a \right )}{4 \pi }}{\pi }+\frac {a \left (b x +a \right )^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {2 a \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi ^{2}}-\frac {3 a^{2} \left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{2 \pi }+\frac {3 a^{2} \mathrm {S}\left (b x +a \right )}{2 \pi }+\frac {a^{3} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {a^{4} \FresnelC \left (b x +a \right )}{4}}{b^{4}} \]
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 \[ \int x^{3} {\rm fresnelc}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,\mathrm {FresnelC}\left (a+b\,x\right ) \,d x \]
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 \[ \int x^{3} C\left (a + b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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