3.1.92 \(\int x^7 \cos (\frac {1}{2} b^2 \pi x^2) S(b x) \, dx\) [92]

Optimal. Leaf size=217 \[ \frac {4 x^3}{b^5 \pi ^3}-\frac {x^7}{14 b \pi }+\frac {17 x^3 \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^8 \pi ^4}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac {531 S\left (\sqrt {2} b x\right )}{16 \sqrt {2} b^8 \pi ^4}-\frac {24 x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^6 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {147 x \sin \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}+\frac {x^5 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \]

[Out]

4*x^3/b^5/Pi^3-1/14*x^7/b/Pi+17/8*x^3*cos(b^2*Pi*x^2)/b^5/Pi^3-48*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b^8/Pi^4+6
*x^4*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b^4/Pi^2-24*x^2*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/b^6/Pi^3+x^6*FresnelS
(b*x)*sin(1/2*b^2*Pi*x^2)/b^2/Pi-147/16*x*sin(b^2*Pi*x^2)/b^7/Pi^4+1/4*x^5*sin(b^2*Pi*x^2)/b^3/Pi^2+531/32*Fre
snelS(b*x*2^(1/2))/b^8/Pi^4*2^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6597, 3472, 30, 3467, 3466, 3432, 6589, 6587} \begin {gather*} \frac {531 S\left (\sqrt {2} b x\right )}{16 \sqrt {2} \pi ^4 b^8}+\frac {4 x^3}{\pi ^3 b^5}+\frac {x^6 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {48 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}-\frac {147 x \sin \left (\pi b^2 x^2\right )}{16 \pi ^4 b^7}-\frac {24 x^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac {17 x^3 \cos \left (\pi b^2 x^2\right )}{8 \pi ^3 b^5}+\frac {6 x^4 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^5 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac {x^7}{14 \pi b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x],x]

[Out]

(4*x^3)/(b^5*Pi^3) - x^7/(14*b*Pi) + (17*x^3*Cos[b^2*Pi*x^2])/(8*b^5*Pi^3) - (48*Cos[(b^2*Pi*x^2)/2]*FresnelS[
b*x])/(b^8*Pi^4) + (6*x^4*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^4*Pi^2) + (531*FresnelS[Sqrt[2]*b*x])/(16*Sqrt
[2]*b^8*Pi^4) - (24*x^2*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^6*Pi^3) + (x^6*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2]
)/(b^2*Pi) - (147*x*Sin[b^2*Pi*x^2])/(16*b^7*Pi^4) + (x^5*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3466

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(-e^(n - 1))*(e*x)^(m - n + 1)*(Cos[c +
d*x^n]/(d*n)), x] + Dist[e^n*((m - n + 1)/(d*n)), Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}
, x] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3467

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(Sin[c + d*
x^n]/(d*n)), x] - Dist[e^n*((m - n + 1)/(d*n)), Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x
] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3472

Int[(x_)^(m_.)*Sin[(a_.) + ((b_.)*(x_)^(n_))/2]^2, x_Symbol] :> Dist[1/2, Int[x^m, x], x] - Dist[1/2, Int[x^m*
Cos[2*a + b*x^n], x], x] /; FreeQ[{a, b, m, n}, x]

Rule 6587

Int[FresnelS[(b_.)*(x_)]*(x_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[(-Cos[d*x^2])*(FresnelS[b*x]/(2*d)), x] + D
ist[1/(2*b*Pi), Int[Sin[2*d*x^2], x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4]

Rule 6589

Int[FresnelS[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[(-x^(m - 1))*Cos[d*x^2]*(FresnelS[b*x]
/(2*d)), x] + (Dist[(m - 1)/(2*d), Int[x^(m - 2)*Cos[d*x^2]*FresnelS[b*x], x], x] + Dist[1/(2*b*Pi), Int[x^(m
- 1)*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IGtQ[m, 1]

Rule 6597

Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m - 1)*Sin[d*x^2]*(FresnelS[b*x]/(2
*d)), x] + (-Dist[1/(Pi*b), Int[x^(m - 1)*Sin[d*x^2]^2, x], x] - Dist[(m - 1)/(2*d), Int[x^(m - 2)*Sin[d*x^2]*
FresnelS[b*x], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IGtQ[m, 1]

Rubi steps

\begin {align*} \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx &=\frac {x^6 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {6 \int x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac {\int x^6 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac {x^6 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {24 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^4 \pi ^2}-\frac {3 \int x^4 \sin \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}-\frac {\int x^6 \, dx}{2 b \pi }+\frac {\int x^6 \cos \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac {x^7}{14 b \pi }+\frac {3 x^3 \cos \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac {24 x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^6 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x^5 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {48 \int x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^6 \pi ^3}-\frac {9 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{2 b^5 \pi ^3}+\frac {24 \int x^2 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac {5 \int x^4 \sin \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}\\ &=-\frac {x^7}{14 b \pi }+\frac {17 x^3 \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^8 \pi ^4}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac {24 x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^6 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {9 x \sin \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^5 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {9 \int \sin \left (b^2 \pi x^2\right ) \, dx}{4 b^7 \pi ^4}+\frac {24 \int \sin \left (b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4}-\frac {15 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{8 b^5 \pi ^3}+\frac {12 \int x^2 \, dx}{b^5 \pi ^3}-\frac {12 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}\\ &=\frac {4 x^3}{b^5 \pi ^3}-\frac {x^7}{14 b \pi }+\frac {17 x^3 \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^8 \pi ^4}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac {9 S\left (\sqrt {2} b x\right )}{4 \sqrt {2} b^8 \pi ^4}+\frac {12 \sqrt {2} S\left (\sqrt {2} b x\right )}{b^8 \pi ^4}-\frac {24 x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^6 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {147 x \sin \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}+\frac {x^5 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {15 \int \sin \left (b^2 \pi x^2\right ) \, dx}{16 b^7 \pi ^4}+\frac {6 \int \sin \left (b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4}\\ &=\frac {4 x^3}{b^5 \pi ^3}-\frac {x^7}{14 b \pi }+\frac {17 x^3 \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^8 \pi ^4}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac {51 S\left (\sqrt {2} b x\right )}{16 \sqrt {2} b^8 \pi ^4}+\frac {15 \sqrt {2} S\left (\sqrt {2} b x\right )}{b^8 \pi ^4}-\frac {24 x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^6 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {147 x \sin \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}+\frac {x^5 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 163, normalized size = 0.75 \begin {gather*} \frac {896 b^3 \pi x^3-16 b^7 \pi ^3 x^7+476 b^3 \pi x^3 \cos \left (b^2 \pi x^2\right )+3717 \sqrt {2} S\left (\sqrt {2} b x\right )+224 S(b x) \left (6 \left (-8+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )+b^2 \pi x^2 \left (-24+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )-2058 b x \sin \left (b^2 \pi x^2\right )+56 b^5 \pi ^2 x^5 \sin \left (b^2 \pi x^2\right )}{224 b^8 \pi ^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x],x]

[Out]

(896*b^3*Pi*x^3 - 16*b^7*Pi^3*x^7 + 476*b^3*Pi*x^3*Cos[b^2*Pi*x^2] + 3717*Sqrt[2]*FresnelS[Sqrt[2]*b*x] + 224*
FresnelS[b*x]*(6*(-8 + b^4*Pi^2*x^4)*Cos[(b^2*Pi*x^2)/2] + b^2*Pi*x^2*(-24 + b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2]
) - 2058*b*x*Sin[b^2*Pi*x^2] + 56*b^5*Pi^2*x^5*Sin[b^2*Pi*x^2])/(224*b^8*Pi^4)

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Maple [A]
time = 0.77, size = 321, normalized size = 1.48

method	result	size
default	\(\frac {\frac {\mathrm {S}\left (b x \right ) \left (\frac {b^{6} x^{6} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {6 \left (-\frac {b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}}{\pi }\right )}{\pi }\right )}{b^{7}}-\frac {\frac {\frac {1}{7} \pi ^{2} b^{7} x^{7}-8 b^{3} x^{3}}{2 \pi ^{3}}+\frac {-\frac {3 \pi \,b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {9 \pi \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-12 \sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{\pi ^{4}}-\frac {\frac {\pi \,b^{5} x^{5} \sin \left (b^{2} \pi \,x^{2}\right )}{2}-\frac {5 \pi \left (-\frac {b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\frac {3 b x \sin \left (b^{2} \pi \,x^{2}\right )}{4 \pi }-\frac {3 \sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{8 \pi }}{\pi }\right )}{2}-\frac {12 b x \sin \left (b^{2} \pi \,x^{2}\right )}{\pi }+\frac {6 \sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{\pi }}{2 \pi ^{3}}}{b^{7}}}{b}\)	\(321\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x,method=_RETURNVERBOSE)

[Out]

(FresnelS(b*x)/b^7*(1/Pi*b^6*x^6*sin(1/2*b^2*Pi*x^2)-6/Pi*(-1/Pi*b^4*x^4*cos(1/2*b^2*Pi*x^2)+4/Pi*(1/Pi*b^2*x^
2*sin(1/2*b^2*Pi*x^2)+2/Pi^2*cos(1/2*b^2*Pi*x^2))))-1/b^7*(1/2/Pi^3*(1/7*Pi^2*b^7*x^7-8*b^3*x^3)+3/Pi^4*(-1/2*
Pi*b^3*x^3*cos(b^2*Pi*x^2)+3/2*Pi*(1/2/Pi*b*x*sin(b^2*Pi*x^2)-1/4/Pi*2^(1/2)*FresnelS(b*x*2^(1/2)))-4*2^(1/2)*
FresnelS(b*x*2^(1/2)))-1/2/Pi^3*(1/2*Pi*b^5*x^5*sin(b^2*Pi*x^2)-5/2*Pi*(-1/2/Pi*b^3*x^3*cos(b^2*Pi*x^2)+3/2/Pi
*(1/2/Pi*b*x*sin(b^2*Pi*x^2)-1/4/Pi*2^(1/2)*FresnelS(b*x*2^(1/2))))-12/Pi*b*x*sin(b^2*Pi*x^2)+6/Pi*2^(1/2)*Fre
snelS(b*x*2^(1/2)))))/b

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="maxima")

[Out]

integrate(x^7*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)

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Fricas [A]
time = 0.36, size = 169, normalized size = 0.78 \begin {gather*} -\frac {16 \, \pi ^{3} b^{8} x^{7} - 952 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 420 \, \pi b^{4} x^{3} - 1344 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - 3717 \, \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 28 \, {\left ({\left (4 \, \pi ^{2} b^{6} x^{5} - 147 \, b^{2} x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 8 \, {\left (\pi ^{3} b^{7} x^{6} - 24 \, \pi b^{3} x^{2}\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{224 \, \pi ^{4} b^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="fricas")

[Out]

-1/224*(16*pi^3*b^8*x^7 - 952*pi*b^4*x^3*cos(1/2*pi*b^2*x^2)^2 - 420*pi*b^4*x^3 - 1344*(pi^2*b^5*x^4 - 8*b)*co
s(1/2*pi*b^2*x^2)*fresnel_sin(b*x) - 3717*sqrt(2)*sqrt(b^2)*fresnel_sin(sqrt(2)*sqrt(b^2)*x) - 28*((4*pi^2*b^6
*x^5 - 147*b^2*x)*cos(1/2*pi*b^2*x^2) + 8*(pi^3*b^7*x^6 - 24*pi*b^3*x^2)*fresnel_sin(b*x))*sin(1/2*pi*b^2*x^2)
)/(pi^4*b^9)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{7} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*cos(1/2*b**2*pi*x**2)*fresnels(b*x),x)

[Out]

Integral(x**7*cos(pi*b**2*x**2/2)*fresnels(b*x), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="giac")

[Out]

integrate(x^7*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^7\,\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*FresnelS(b*x)*cos((Pi*b^2*x^2)/2),x)

[Out]

int(x^7*FresnelS(b*x)*cos((Pi*b^2*x^2)/2), x)

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