Optimal. Leaf size=196 \[ -\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^2 b^3}+\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^2 b^3}-\frac {3 C(b x) S(b x)}{2 \pi ^2 b^5}-\frac {x^3 C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^3 b^5}+\frac {3 x C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^2 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {x^4}{8 \pi b} \]
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Rubi [A] time = 0.16, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6463, 6455, 6447, 3379, 2638, 3380, 3309, 30, 3296} \[ -\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^2 b^3}+\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^2 b^3}-\frac {3 \text {FresnelC}(b x) S(b x)}{2 \pi ^2 b^5}+\frac {3 x \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^3 \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {x^2 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^3 b^5}+\frac {x^4}{8 \pi b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2638
Rule 3296
Rule 3309
Rule 3379
Rule 3380
Rule 6447
Rule 6455
Rule 6463
Rubi steps
\begin {align*} \int x^4 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx &=-\frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac {3 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{b^2 \pi }+\frac {\int x^3 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=-\frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac {3 x C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {3 \int C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^4 \pi ^2}-\frac {3 \int x \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^3 \pi ^2}+\frac {\operatorname {Subst}\left (\int x \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b \pi }\\ &=-\frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }-\frac {3 C(b x) S(b x)}{2 b^5 \pi ^2}-\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {3 x C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {3 \operatorname {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}+\frac {\operatorname {Subst}\left (\int x \, dx,x,x^2\right )}{4 b \pi }+\frac {\operatorname {Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi }\\ &=\frac {x^4}{8 b \pi }+\frac {3 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}-\frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }-\frac {3 C(b x) S(b x)}{2 b^5 \pi ^2}-\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {3 x C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {x^2 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {\operatorname {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}\\ &=\frac {x^4}{8 b \pi }+\frac {\cos \left (b^2 \pi x^2\right )}{b^5 \pi ^3}-\frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }-\frac {3 C(b x) S(b x)}{2 b^5 \pi ^2}-\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {3 x C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {x^2 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end {align*}
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Mathematica [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int x^4 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{4} {\rm fresnelc}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ), x\right ) \]
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 \[ \int x^{4} {\rm fresnelc}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int x^{4} \FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )\, dx \]
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 \[ \int x^{4} {\rm fresnelc}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \]
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 \[ \int x^4\,\mathrm {FresnelC}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
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 \[ \int x^{4} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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