3.140 \(\int x^7 C(b x)^2 \, dx\)

Optimal. Leaf size=253 \[ -\frac {105 C(b x)^2}{8 \pi ^4 b^8}-\frac {105 x^2}{16 \pi ^4 b^6}-\frac {x^7 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b}+\frac {7 x^6}{48 \pi ^2 b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{16 \pi ^2 b^2}-\frac {10 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^8}+\frac {105 x C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^4 b^7}+\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{16 \pi ^4 b^6}+\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^3 b^5}+\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{8 \pi ^3 b^4}-\frac {7 x^5 C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {1}{8} x^8 C(b x)^2 \]

[Out]

-105/16*x^2/b^6/Pi^4+7/48*x^6/b^2/Pi^2+55/16*x^2*cos(b^2*Pi*x^2)/b^6/Pi^4-1/16*x^6*cos(b^2*Pi*x^2)/b^2/Pi^2+10
5/4*x*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/b^7/Pi^4-7/4*x^5*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/b^3/Pi^2-105/8*Fres
nelC(b*x)^2/b^8/Pi^4+1/8*x^8*FresnelC(b*x)^2+35/4*x^3*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/b^5/Pi^3-1/4*x^7*Fresn
elC(b*x)*sin(1/2*b^2*Pi*x^2)/b/Pi-10*sin(b^2*Pi*x^2)/b^8/Pi^5+5/8*x^4*sin(b^2*Pi*x^2)/b^4/Pi^3

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Rubi [A]  time = 0.42, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6431, 6455, 6463, 6441, 30, 3380, 2634, 3379, 3296, 2637, 3309} \[ -\frac {x^7 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b}+\frac {35 x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^3 b^5}-\frac {7 x^5 \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {105 x \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^4 b^7}-\frac {105 \text {FresnelC}(b x)^2}{8 \pi ^4 b^8}+\frac {7 x^6}{48 \pi ^2 b^2}-\frac {105 x^2}{16 \pi ^4 b^6}+\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{8 \pi ^3 b^4}-\frac {10 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^8}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{16 \pi ^2 b^2}+\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{16 \pi ^4 b^6}+\frac {1}{8} x^8 \text {FresnelC}(b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[x^7*FresnelC[b*x]^2,x]

[Out]

(-105*x^2)/(16*b^6*Pi^4) + (7*x^6)/(48*b^2*Pi^2) + (55*x^2*Cos[b^2*Pi*x^2])/(16*b^6*Pi^4) - (x^6*Cos[b^2*Pi*x^
2])/(16*b^2*Pi^2) + (105*x*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(4*b^7*Pi^4) - (7*x^5*Cos[(b^2*Pi*x^2)/2]*Fresne
lC[b*x])/(4*b^3*Pi^2) - (105*FresnelC[b*x]^2)/(8*b^8*Pi^4) + (x^8*FresnelC[b*x]^2)/8 + (35*x^3*FresnelC[b*x]*S
in[(b^2*Pi*x^2)/2])/(4*b^5*Pi^3) - (x^7*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(4*b*Pi) - (10*Sin[b^2*Pi*x^2])/(b^
8*Pi^5) + (5*x^4*Sin[b^2*Pi*x^2])/(8*b^4*Pi^3)

Rule 30

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

Rule 2634

Int[sin[(c_.) + ((d_.)*(x_))/2]^2, x_Symbol] :> Simp[x/2, x] - Simp[Sin[2*c + d*x]/(2*d), x] /; FreeQ[{c, d},
x]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3309

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + ((f_.)*(x_))/2]^2, x_Symbol] :> Dist[1/2, Int[(c + d*x)^m, x], x] -
Dist[1/2, Int[(c + d*x)^m*Cos[2*e + f*x], x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 3379

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 3380

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[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 6431

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

Rule 6441

Int[Cos[(d_.)*(x_)^2]*FresnelC[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(Pi*b)/(2*d), Subst[Int[x^n, x], x, Fresne
lC[b*x]], x] /; FreeQ[{b, d, n}, x] && EqQ[d^2, (Pi^2*b^4)/4]

Rule 6455

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

Rule 6463

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

Rubi steps

\begin {align*} \int x^7 C(b x)^2 \, dx &=\frac {1}{8} x^8 C(b x)^2-\frac {1}{4} b \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=\frac {1}{8} x^8 C(b x)^2-\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {\int x^7 \sin \left (b^2 \pi x^2\right ) \, dx}{8 \pi }+\frac {7 \int x^6 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b \pi }\\ &=-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 b^3 \pi ^2}+\frac {1}{8} x^8 C(b x)^2-\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {35 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{4 b^3 \pi ^2}+\frac {7 \int x^5 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^2 \pi ^2}+\frac {\operatorname {Subst}\left (\int x^3 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 \pi }\\ &=-\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 b^3 \pi ^2}+\frac {1}{8} x^8 C(b x)^2+\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^5 \pi ^3}-\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }-\frac {105 \int x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^5 \pi ^3}-\frac {35 \int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{8 b^4 \pi ^3}+\frac {3 \operatorname {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 b^2 \pi ^2}+\frac {7 \operatorname {Subst}\left (\int x^2 \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^2 \pi ^2}\\ &=-\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}+\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 b^7 \pi ^4}-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 b^3 \pi ^2}+\frac {1}{8} x^8 C(b x)^2+\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^5 \pi ^3}-\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {3 x^4 \sin \left (b^2 \pi x^2\right )}{16 b^4 \pi ^3}-\frac {105 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{4 b^7 \pi ^4}-\frac {105 \int x \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^6 \pi ^4}-\frac {3 \operatorname {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^4 \pi ^3}-\frac {35 \operatorname {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 b^4 \pi ^3}+\frac {7 \operatorname {Subst}\left (\int x^2 \, dx,x,x^2\right )}{16 b^2 \pi ^2}+\frac {7 \operatorname {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 b^2 \pi ^2}\\ &=\frac {7 x^6}{48 b^2 \pi ^2}+\frac {41 x^2 \cos \left (b^2 \pi x^2\right )}{16 b^6 \pi ^4}-\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}+\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 b^7 \pi ^4}-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 b^3 \pi ^2}+\frac {1}{8} x^8 C(b x)^2+\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^5 \pi ^3}-\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{8 b^4 \pi ^3}-\frac {105 \operatorname {Subst}(\int x \, dx,x,C(b x))}{4 b^8 \pi ^4}-\frac {3 \operatorname {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^6 \pi ^4}-\frac {35 \operatorname {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 b^6 \pi ^4}-\frac {105 \operatorname {Subst}\left (\int \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^6 \pi ^4}-\frac {7 \operatorname {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^4 \pi ^3}\\ &=-\frac {105 x^2}{16 b^6 \pi ^4}+\frac {7 x^6}{48 b^2 \pi ^2}+\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{16 b^6 \pi ^4}-\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}+\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 b^7 \pi ^4}-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 b^3 \pi ^2}-\frac {105 C(b x)^2}{8 b^8 \pi ^4}+\frac {1}{8} x^8 C(b x)^2+\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^5 \pi ^3}-\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }-\frac {73 \sin \left (b^2 \pi x^2\right )}{8 b^8 \pi ^5}+\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{8 b^4 \pi ^3}-\frac {7 \operatorname {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^6 \pi ^4}\\ &=-\frac {105 x^2}{16 b^6 \pi ^4}+\frac {7 x^6}{48 b^2 \pi ^2}+\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{16 b^6 \pi ^4}-\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}+\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 b^7 \pi ^4}-\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 b^3 \pi ^2}-\frac {105 C(b x)^2}{8 b^8 \pi ^4}+\frac {1}{8} x^8 C(b x)^2+\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^5 \pi ^3}-\frac {x^7 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }-\frac {10 \sin \left (b^2 \pi x^2\right )}{b^8 \pi ^5}+\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{8 b^4 \pi ^3}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 253, normalized size = 1.00 \[ -\frac {105 C(b x)^2}{8 \pi ^4 b^8}-\frac {105 x^2}{16 \pi ^4 b^6}-\frac {x^7 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b}+\frac {7 x^6}{48 \pi ^2 b^2}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{16 \pi ^2 b^2}-\frac {10 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^8}+\frac {105 x C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^4 b^7}+\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{16 \pi ^4 b^6}+\frac {35 x^3 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^3 b^5}+\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{8 \pi ^3 b^4}-\frac {7 x^5 C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {1}{8} x^8 C(b x)^2 \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*FresnelC[b*x]^2,x]

[Out]

(-105*x^2)/(16*b^6*Pi^4) + (7*x^6)/(48*b^2*Pi^2) + (55*x^2*Cos[b^2*Pi*x^2])/(16*b^6*Pi^4) - (x^6*Cos[b^2*Pi*x^
2])/(16*b^2*Pi^2) + (105*x*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(4*b^7*Pi^4) - (7*x^5*Cos[(b^2*Pi*x^2)/2]*Fresne
lC[b*x])/(4*b^3*Pi^2) - (105*FresnelC[b*x]^2)/(8*b^8*Pi^4) + (x^8*FresnelC[b*x]^2)/8 + (35*x^3*FresnelC[b*x]*S
in[(b^2*Pi*x^2)/2])/(4*b^5*Pi^3) - (x^7*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(4*b*Pi) - (10*Sin[b^2*Pi*x^2])/(b^
8*Pi^5) + (5*x^4*Sin[b^2*Pi*x^2])/(8*b^4*Pi^3)

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fricas [F]  time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{7} {\rm fresnelc}\left (b x\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*fresnelc(b*x)^2,x, algorithm="fricas")

[Out]

integral(x^7*fresnelc(b*x)^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{7} {\rm fresnelc}\left (b x\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*fresnelc(b*x)^2,x, algorithm="giac")

[Out]

integrate(x^7*fresnelc(b*x)^2, x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[ \int x^{7} \FresnelC \left (b x \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*FresnelC(b*x)^2,x)

[Out]

int(x^7*FresnelC(b*x)^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{7} {\rm fresnelc}\left (b x\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*fresnelc(b*x)^2,x, algorithm="maxima")

[Out]

integrate(x^7*fresnelc(b*x)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^7\,{\mathrm {FresnelC}\left (b\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*FresnelC(b*x)^2,x)

[Out]

int(x^7*FresnelC(b*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{7} C^{2}\left (b x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*fresnelc(b*x)**2,x)

[Out]

Integral(x**7*fresnelc(b*x)**2, x)

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