3.2.10 \(\int x^7 \text {FresnelC}(b x) \, dx\) [110]

Optimal. Leaf size=124 \[ \frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}-\frac {105 \text {FresnelC}(b x)}{8 b^8 \pi ^4}+\frac {1}{8} x^8 \text {FresnelC}(b x)+\frac {35 x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {x^7 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi } \]

[Out]

105/8*x*cos(1/2*b^2*Pi*x^2)/b^7/Pi^4-7/8*x^5*cos(1/2*b^2*Pi*x^2)/b^3/Pi^2-105/8*FresnelC(b*x)/b^8/Pi^4+1/8*x^8
*FresnelC(b*x)+35/8*x^3*sin(1/2*b^2*Pi*x^2)/b^5/Pi^3-1/8*x^7*sin(1/2*b^2*Pi*x^2)/b/Pi

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Rubi [A]
time = 0.06, antiderivative size = 124, 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 = {6562, 3467, 3466, 3433} \begin {gather*} -\frac {105 \text {FresnelC}(b x)}{8 \pi ^4 b^8}-\frac {x^7 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi b}+\frac {105 x \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^4 b^7}+\frac {35 x^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^3 b^5}-\frac {7 x^5 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^2 b^3}+\frac {1}{8} x^8 \text {FresnelC}(b x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*FresnelC[b*x],x]

[Out]

(105*x*Cos[(b^2*Pi*x^2)/2])/(8*b^7*Pi^4) - (7*x^5*Cos[(b^2*Pi*x^2)/2])/(8*b^3*Pi^2) - (105*FresnelC[b*x])/(8*b
^8*Pi^4) + (x^8*FresnelC[b*x])/8 + (35*x^3*Sin[(b^2*Pi*x^2)/2])/(8*b^5*Pi^3) - (x^7*Sin[(b^2*Pi*x^2)/2])/(8*b*
Pi)

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[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 6562

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

Rubi steps

\begin {align*} \int x^7 C(b x) \, dx &=\frac {1}{8} x^8 C(b x)-\frac {1}{8} b \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {1}{8} x^8 C(b x)-\frac {x^7 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }+\frac {7 \int x^6 \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{8 b \pi }\\ &=-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {1}{8} x^8 C(b x)-\frac {x^7 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }+\frac {35 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{8 b^3 \pi ^2}\\ &=-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {1}{8} x^8 C(b x)+\frac {35 x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {x^7 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }-\frac {105 \int x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{8 b^5 \pi ^3}\\ &=\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {1}{8} x^8 C(b x)+\frac {35 x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {x^7 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }-\frac {105 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{8 b^7 \pi ^4}\\ &=\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}-\frac {105 C(b x)}{8 b^8 \pi ^4}+\frac {1}{8} x^8 C(b x)+\frac {35 x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {x^7 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 89, normalized size = 0.72 \begin {gather*} \frac {-7 b x \left (-15+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )+\left (-105+b^8 \pi ^4 x^8\right ) \text {FresnelC}(b x)+b^3 \pi x^3 \left (35-b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^8 \pi ^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7*FresnelC[b*x],x]

[Out]

(-7*b*x*(-15 + b^4*Pi^2*x^4)*Cos[(b^2*Pi*x^2)/2] + (-105 + b^8*Pi^4*x^8)*FresnelC[b*x] + b^3*Pi*x^3*(35 - b^4*
Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2])/(8*b^8*Pi^4)

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Maple [A]
time = 0.30, size = 123, normalized size = 0.99

method	result	size
meijerg	\(\frac {b \,x^{9} \hypergeom \left (\left [\frac {1}{4}, \frac {9}{4}\right ], \left [\frac {1}{2}, \frac {5}{4}, \frac {13}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{9}\)	\(26\)
derivativedivides	\(\frac {\frac {\FresnelC \left (b x \right ) b^{8} x^{8}}{8}-\frac {b^{7} x^{7} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 \pi }+\frac {-\frac {7 b^{5} x^{5} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 \pi }+\frac {7 \left (\frac {5 b^{3} x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {15 \left (-\frac {b x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\FresnelC \left (b x \right )}{\pi }\right )}{\pi }\right )}{8 \pi }}{\pi }}{b^{8}}\)	\(123\)
default	\(\frac {\frac {\FresnelC \left (b x \right ) b^{8} x^{8}}{8}-\frac {b^{7} x^{7} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 \pi }+\frac {-\frac {7 b^{5} x^{5} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 \pi }+\frac {7 \left (\frac {5 b^{3} x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {15 \left (-\frac {b x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\FresnelC \left (b x \right )}{\pi }\right )}{\pi }\right )}{8 \pi }}{\pi }}{b^{8}}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*FresnelC(b*x),x,method=_RETURNVERBOSE)

[Out]

1/b^8*(1/8*FresnelC(b*x)*b^8*x^8-1/8/Pi*b^7*x^7*sin(1/2*b^2*Pi*x^2)+7/8/Pi*(-1/Pi*b^5*x^5*cos(1/2*b^2*Pi*x^2)+
5/Pi*(1/Pi*b^3*x^3*sin(1/2*b^2*Pi*x^2)-3/Pi*(-1/Pi*b*x*cos(1/2*b^2*Pi*x^2)+1/Pi*FresnelC(b*x)))))

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Maxima [C] Result contains complex when optimal does not.
time = 0.49, size = 126, normalized size = 1.02 \begin {gather*} \frac {1}{8} \, x^{8} \operatorname {C}\left (b x\right ) - \frac {\sqrt {\frac {1}{2}} {\left (-\left (105 i - 105\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) + \left (105 i + 105\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right ) + 28 \, {\left (\sqrt {\frac {1}{2}} \pi ^{3} b^{5} x^{5} - 15 \, \sqrt {\frac {1}{2}} \pi b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 4 \, {\left (\sqrt {\frac {1}{2}} \pi ^{4} b^{7} x^{7} - 35 \, \sqrt {\frac {1}{2}} \pi ^{2} b^{3} x^{3}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )}}{16 \, \pi ^{5} b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*fresnel_cos(b*x),x, algorithm="maxima")

[Out]

1/8*x^8*fresnel_cos(b*x) - 1/16*sqrt(1/2)*(-(105*I - 105)*(1/4)^(1/4)*pi*erf(sqrt(1/2*I*pi)*b*x) + (105*I + 10
5)*(1/4)^(1/4)*pi*erf(sqrt(-1/2*I*pi)*b*x) + 28*(sqrt(1/2)*pi^3*b^5*x^5 - 15*sqrt(1/2)*pi*b*x)*cos(1/2*pi*b^2*
x^2) + 4*(sqrt(1/2)*pi^4*b^7*x^7 - 35*sqrt(1/2)*pi^2*b^3*x^3)*sin(1/2*pi*b^2*x^2))/(pi^5*b^8)

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Fricas [A]
time = 0.36, size = 85, normalized size = 0.69 \begin {gather*} -\frac {7 \, {\left (\pi ^{2} b^{5} x^{5} - 15 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{4} b^{8} x^{8} - 105\right )} \operatorname {C}\left (b x\right ) + {\left (\pi ^{3} b^{7} x^{7} - 35 \, \pi b^{3} x^{3}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{8 \, \pi ^{4} b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*fresnel_cos(b*x),x, algorithm="fricas")

[Out]

-1/8*(7*(pi^2*b^5*x^5 - 15*b*x)*cos(1/2*pi*b^2*x^2) - (pi^4*b^8*x^8 - 105)*fresnel_cos(b*x) + (pi^3*b^7*x^7 -
35*pi*b^3*x^3)*sin(1/2*pi*b^2*x^2))/(pi^4*b^8)

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Sympy [A]
time = 1.25, size = 184, normalized size = 1.48 \begin {gather*} \frac {45 x^{8} C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{512 \Gamma \left (\frac {13}{4}\right )} - \frac {45 x^{7} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{512 \pi b \Gamma \left (\frac {13}{4}\right )} - \frac {315 x^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{512 \pi ^{2} b^{3} \Gamma \left (\frac {13}{4}\right )} + \frac {1575 x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{512 \pi ^{3} b^{5} \Gamma \left (\frac {13}{4}\right )} + \frac {4725 x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{512 \pi ^{4} b^{7} \Gamma \left (\frac {13}{4}\right )} - \frac {4725 C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{512 \pi ^{4} b^{8} \Gamma \left (\frac {13}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*fresnelc(b*x),x)

[Out]

45*x**8*fresnelc(b*x)*gamma(1/4)/(512*gamma(13/4)) - 45*x**7*sin(pi*b**2*x**2/2)*gamma(1/4)/(512*pi*b*gamma(13
/4)) - 315*x**5*cos(pi*b**2*x**2/2)*gamma(1/4)/(512*pi**2*b**3*gamma(13/4)) + 1575*x**3*sin(pi*b**2*x**2/2)*ga
mma(1/4)/(512*pi**3*b**5*gamma(13/4)) + 4725*x*cos(pi*b**2*x**2/2)*gamma(1/4)/(512*pi**4*b**7*gamma(13/4)) - 4
725*fresnelc(b*x)*gamma(1/4)/(512*pi**4*b**8*gamma(13/4))

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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(x^7*fresnel_cos(b*x),x, algorithm="giac")

[Out]

integrate(x^7*fresnel_cos(b*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^7\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*FresnelC(b*x),x)

[Out]

int(x^7*FresnelC(b*x), x)

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