3.1.16 \(\int \frac {S(b x)}{x^8} \, dx\) [16]

Optimal. Leaf size=102 \[ -\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^4}-\frac {1}{672} b^7 \pi ^3 \text {CosIntegral}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {S(b x)}{7 x^7}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{336 x^2} \]

[Out]

-1/672*b^7*Pi^3*Ci(1/2*b^2*Pi*x^2)-1/168*b^3*Pi*cos(1/2*b^2*Pi*x^2)/x^4-1/7*FresnelS(b*x)/x^7-1/42*b*sin(1/2*b
^2*Pi*x^2)/x^6+1/336*b^5*Pi^2*sin(1/2*b^2*Pi*x^2)/x^2

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Rubi [A]
time = 0.08, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6561, 3460, 3378, 3383} \begin {gather*} -\frac {b \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{42 x^6}-\frac {1}{672} \pi ^3 b^7 \text {CosIntegral}\left (\frac {1}{2} \pi b^2 x^2\right )+\frac {\pi ^2 b^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{336 x^2}-\frac {\pi b^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{168 x^4}-\frac {S(b x)}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[FresnelS[b*x]/x^8,x]

[Out]

-1/168*(b^3*Pi*Cos[(b^2*Pi*x^2)/2])/x^4 - (b^7*Pi^3*CosIntegral[(b^2*Pi*x^2)/2])/672 - FresnelS[b*x]/(7*x^7) -
(b*Sin[(b^2*Pi*x^2)/2])/(42*x^6) + (b^5*Pi^2*Sin[(b^2*Pi*x^2)/2])/(336*x^2)

Rule 3378

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

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3460

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 6561

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

Rubi steps

\begin {align*} \int \frac {S(b x)}{x^8} \, dx &=-\frac {S(b x)}{7 x^7}+\frac {1}{7} b \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^7} \, dx\\ &=-\frac {S(b x)}{7 x^7}+\frac {1}{14} b \text {Subst}\left (\int \frac {\sin \left (\frac {1}{2} b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )\\ &=-\frac {S(b x)}{7 x^7}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}+\frac {1}{84} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\cos \left (\frac {1}{2} b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )\\ &=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^4}-\frac {S(b x)}{7 x^7}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}-\frac {1}{336} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {1}{2} b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )\\ &=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^4}-\frac {S(b x)}{7 x^7}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{336 x^2}-\frac {1}{672} \left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {1}{2} b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^4}-\frac {1}{672} b^7 \pi ^3 \text {Ci}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {S(b x)}{7 x^7}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{336 x^2}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 85, normalized size = 0.83 \begin {gather*} \frac {1}{672} \left (-\frac {4 b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}-b^7 \pi ^3 \text {CosIntegral}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {96 S(b x)}{x^7}+\frac {2 b \left (-8+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[FresnelS[b*x]/x^8,x]

[Out]

((-4*b^3*Pi*Cos[(b^2*Pi*x^2)/2])/x^4 - b^7*Pi^3*CosIntegral[(b^2*Pi*x^2)/2] - (96*FresnelS[b*x])/x^7 + (2*b*(-
8 + b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2])/x^6)/672

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Maple [A]
time = 0.42, size = 93, normalized size = 0.91

method	result	size
meijerg	\(\frac {\pi ^{\frac {7}{2}} b^{7} \left (\frac {\pi ^{\frac {3}{2}} x^{4} b^{4} \hypergeom \left (\left [1, 1, \frac {11}{4}\right ], \left [2, 3, \frac {7}{2}, \frac {15}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{165}-\frac {16 \left (-\frac {89}{21}+2 \gamma -2 \ln \left (2\right )+4 \ln \left (x \right )+2 \ln \left (\pi \right )+4 \ln \left (b \right )\right )}{21 \sqrt {\pi }}-\frac {128}{3 \pi ^{\frac {5}{2}} x^{4} b^{4}}\right )}{1024}\)	\(79\)
derivativedivides	\(b^{7} \left (-\frac {\mathrm {S}\left (b x \right )}{7 b^{7} x^{7}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{42 b^{6} x^{6}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 b^{4} x^{4}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b^{2} x^{2}}+\frac {\pi \cosineIntegral \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{4}\right )}{42}\right )\)	\(93\)
default	\(b^{7} \left (-\frac {\mathrm {S}\left (b x \right )}{7 b^{7} x^{7}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{42 b^{6} x^{6}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 b^{4} x^{4}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b^{2} x^{2}}+\frac {\pi \cosineIntegral \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{4}\right )}{42}\right )\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(b*x)/x^8,x,method=_RETURNVERBOSE)

[Out]

b^7*(-1/7*FresnelS(b*x)/b^7/x^7-1/42/b^6/x^6*sin(1/2*b^2*Pi*x^2)+1/42*Pi*(-1/4/b^4/x^4*cos(1/2*b^2*Pi*x^2)-1/4
*Pi*(-1/2/b^2/x^2*sin(1/2*b^2*Pi*x^2)+1/4*Pi*Ci(1/2*b^2*Pi*x^2))))

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Maxima [C] Result contains complex when optimal does not.
time = 0.32, size = 46, normalized size = 0.45 \begin {gather*} -\frac {1}{224} \, {\left (\pi ^{3} \Gamma \left (-3, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + \pi ^{3} \Gamma \left (-3, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{7} - \frac {\operatorname {S}\left (b x\right )}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_sin(b*x)/x^8,x, algorithm="maxima")

[Out]

-1/224*(pi^3*gamma(-3, 1/2*I*pi*b^2*x^2) + pi^3*gamma(-3, -1/2*I*pi*b^2*x^2))*b^7 - 1/7*fresnel_sin(b*x)/x^7

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Fricas [A]
time = 0.37, size = 98, normalized size = 0.96 \begin {gather*} -\frac {\pi ^{3} b^{7} x^{7} \operatorname {Ci}\left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \pi ^{3} b^{7} x^{7} \operatorname {Ci}\left (-\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 8 \, \pi b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 4 \, {\left (\pi ^{2} b^{5} x^{5} - 8 \, b x\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 192 \, \operatorname {S}\left (b x\right )}{1344 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_sin(b*x)/x^8,x, algorithm="fricas")

[Out]

-1/1344*(pi^3*b^7*x^7*cos_integral(1/2*pi*b^2*x^2) + pi^3*b^7*x^7*cos_integral(-1/2*pi*b^2*x^2) + 8*pi*b^3*x^3
*cos(1/2*pi*b^2*x^2) - 4*(pi^2*b^5*x^5 - 8*b*x)*sin(1/2*pi*b^2*x^2) + 192*fresnel_sin(b*x))/x^7

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Sympy [A]
time = 2.96, size = 68, normalized size = 0.67 \begin {gather*} \frac {\pi ^{5} b^{11} x^{4} \Gamma \left (\frac {11}{4}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {11}{4} \\ 2, 3, \frac {7}{2}, \frac {15}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{61440 \Gamma \left (\frac {15}{4}\right )} - \frac {\pi ^{3} b^{7} \log {\left (b^{4} x^{4} \right )}}{1344} - \frac {\pi b^{3}}{24 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)/x**8,x)

[Out]

pi**5*b**11*x**4*gamma(11/4)*hyper((1, 1, 11/4), (2, 3, 7/2, 15/4), -pi**2*b**4*x**4/16)/(61440*gamma(15/4)) -
pi**3*b**7*log(b**4*x**4)/1344 - pi*b**3/(24*x**4)

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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(fresnel_sin(b*x)/x^8,x, algorithm="giac")

[Out]

integrate(fresnel_sin(b*x)/x^8, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {FresnelS}\left (b\,x\right )}{x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(b*x)/x^8,x)

[Out]

int(FresnelS(b*x)/x^8, x)

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