3.50 \(\int (c+d x) S(a+b x)^2 \, dx\)

Optimal. Leaf size=279 \[ \frac{i d (a+b x)^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-\frac{1}{2} i \pi (a+b x)^2\right )}{8 \pi b^2}-\frac{i d (a+b x)^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},\frac{1}{2} i \pi (a+b x)^2\right )}{8 \pi b^2}+\frac{(a+b x) (b c-a d) S(a+b x)^2}{b^2}-\frac{(b c-a d) S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} \pi b^2}+\frac{2 (b c-a d) S(a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^2}-\frac{d \text{FresnelC}(a+b x) S(a+b x)}{2 \pi b^2}+\frac{d (a+b x)^2 S(a+b x)^2}{2 b^2}+\frac{d (a+b x) S(a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^2}+\frac{d \cos \left (\pi (a+b x)^2\right )}{4 \pi ^2 b^2} \]

[Out]

(d*Cos[Pi*(a + b*x)^2])/(4*b^2*Pi^2) + (2*(b*c - a*d)*Cos[(Pi*(a + b*x)^2)/2]*FresnelS[a + b*x])/(b^2*Pi) + (d
*(a + b*x)*Cos[(Pi*(a + b*x)^2)/2]*FresnelS[a + b*x])/(b^2*Pi) - (d*FresnelC[a + b*x]*FresnelS[a + b*x])/(2*b^
2*Pi) + ((b*c - a*d)*(a + b*x)*FresnelS[a + b*x]^2)/b^2 + (d*(a + b*x)^2*FresnelS[a + b*x]^2)/(2*b^2) - ((b*c
- a*d)*FresnelS[Sqrt[2]*(a + b*x)])/(Sqrt[2]*b^2*Pi) + ((I/8)*d*(a + b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}
, (-I/2)*Pi*(a + b*x)^2])/(b^2*Pi) - ((I/8)*d*(a + b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*Pi*(a + b*
x)^2])/(b^2*Pi)

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Rubi [A]  time = 0.196207, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6432, 6420, 6452, 3351, 6430, 6454, 6446, 3379, 2638} \[ \frac{i d (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i \pi (a+b x)^2\right )}{8 \pi b^2}-\frac{i d (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i \pi (a+b x)^2\right )}{8 \pi b^2}+\frac{(a+b x) (b c-a d) S(a+b x)^2}{b^2}-\frac{(b c-a d) S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} \pi b^2}+\frac{2 (b c-a d) S(a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^2}-\frac{d \text{FresnelC}(a+b x) S(a+b x)}{2 \pi b^2}+\frac{d (a+b x)^2 S(a+b x)^2}{2 b^2}+\frac{d (a+b x) S(a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^2}+\frac{d \cos \left (\pi (a+b x)^2\right )}{4 \pi ^2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)*FresnelS[a + b*x]^2,x]

[Out]

(d*Cos[Pi*(a + b*x)^2])/(4*b^2*Pi^2) + (2*(b*c - a*d)*Cos[(Pi*(a + b*x)^2)/2]*FresnelS[a + b*x])/(b^2*Pi) + (d
*(a + b*x)*Cos[(Pi*(a + b*x)^2)/2]*FresnelS[a + b*x])/(b^2*Pi) - (d*FresnelC[a + b*x]*FresnelS[a + b*x])/(2*b^
2*Pi) + ((b*c - a*d)*(a + b*x)*FresnelS[a + b*x]^2)/b^2 + (d*(a + b*x)^2*FresnelS[a + b*x]^2)/(2*b^2) - ((b*c
- a*d)*FresnelS[Sqrt[2]*(a + b*x)])/(Sqrt[2]*b^2*Pi) + ((I/8)*d*(a + b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}
, (-I/2)*Pi*(a + b*x)^2])/(b^2*Pi) - ((I/8)*d*(a + b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*Pi*(a + b*
x)^2])/(b^2*Pi)

Rule 6432

Int[FresnelS[(a_) + (b_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/b^(m + 1), Subst[Int[ExpandI
ntegrand[FresnelS[x]^2, (b*c - a*d + d*x)^m, x], x], x, a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]

Rule 6420

Int[FresnelS[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[((a + b*x)*FresnelS[a + b*x]^2)/b, x] - Dist[2, Int[(a +
 b*x)*Sin[(Pi*(a + b*x)^2)/2]*FresnelS[a + b*x], x], x] /; FreeQ[{a, b}, x]

Rule 6452

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

Rule 3351

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

Rule 6430

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

Rule 6454

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*b^4)/4] && IGtQ[m, 1]

Rule 6446

Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)], x_Symbol] :> Simp[(FresnelC[b*x]*FresnelS[b*x])/(2*b), x] + (-Simp
[(1*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -((I*b^2*Pi*x^2)/2)])/8, x] + Simp[(1*I*b*x^2*HypergeometricPF
Q[{1, 1}, {3/2, 2}, (1*I*b^2*Pi*x^2)/2])/8, x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4]

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 2638

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

Rubi steps

\begin{align*} \int (c+d x) S(a+b x)^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (b c \left (1-\frac{a d}{b c}\right ) S(x)^2+d x S(x)^2\right ) \, dx,x,a+b x\right )}{b^2}\\ &=\frac{d \operatorname{Subst}\left (\int x S(x)^2 \, dx,x,a+b x\right )}{b^2}+\frac{(b c-a d) \operatorname{Subst}\left (\int S(x)^2 \, dx,x,a+b x\right )}{b^2}\\ &=\frac{(b c-a d) (a+b x) S(a+b x)^2}{b^2}+\frac{d (a+b x)^2 S(a+b x)^2}{2 b^2}-\frac{d \operatorname{Subst}\left (\int x^2 S(x) \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^2}-\frac{(2 (b c-a d)) \operatorname{Subst}\left (\int x S(x) \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^2}\\ &=\frac{2 (b c-a d) \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^2 \pi }+\frac{d (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^2 \pi }+\frac{(b c-a d) (a+b x) S(a+b x)^2}{b^2}+\frac{d (a+b x)^2 S(a+b x)^2}{2 b^2}-\frac{d \operatorname{Subst}\left (\int x \sin \left (\pi x^2\right ) \, dx,x,a+b x\right )}{2 b^2 \pi }-\frac{d \operatorname{Subst}\left (\int \cos \left (\frac{\pi x^2}{2}\right ) S(x) \, dx,x,a+b x\right )}{b^2 \pi }-\frac{(b c-a d) \operatorname{Subst}\left (\int \sin \left (\pi x^2\right ) \, dx,x,a+b x\right )}{b^2 \pi }\\ &=\frac{2 (b c-a d) \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^2 \pi }+\frac{d (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^2 \pi }-\frac{d C(a+b x) S(a+b x)}{2 b^2 \pi }+\frac{(b c-a d) (a+b x) S(a+b x)^2}{b^2}+\frac{d (a+b x)^2 S(a+b x)^2}{2 b^2}-\frac{(b c-a d) S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} b^2 \pi }+\frac{i d (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi }-\frac{i d (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi }-\frac{d \operatorname{Subst}\left (\int \sin (\pi x) \, dx,x,(a+b x)^2\right )}{4 b^2 \pi }\\ &=\frac{d \cos \left (\pi (a+b x)^2\right )}{4 b^2 \pi ^2}+\frac{2 (b c-a d) \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^2 \pi }+\frac{d (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^2 \pi }-\frac{d C(a+b x) S(a+b x)}{2 b^2 \pi }+\frac{(b c-a d) (a+b x) S(a+b x)^2}{b^2}+\frac{d (a+b x)^2 S(a+b x)^2}{2 b^2}-\frac{(b c-a d) S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} b^2 \pi }+\frac{i d (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi }-\frac{i d (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi }\\ \end{align*}

Mathematica [F]  time = 0.585015, size = 0, normalized size = 0. \[ \int (c+d x) S(a+b x)^2 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c + d*x)*FresnelS[a + b*x]^2,x]

[Out]

Integrate[(c + d*x)*FresnelS[a + b*x]^2, x]

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) \left ({\it FresnelS} \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)*FresnelS(b*x+a)^2,x)

[Out]

int((d*x+c)*FresnelS(b*x+a)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}{\rm fresnels}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*fresnels(b*x+a)^2,x, algorithm="maxima")

[Out]

integrate((d*x + c)*fresnels(b*x + a)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d x + c\right )}{\rm fresnels}\left (b x + a\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*fresnels(b*x+a)^2,x, algorithm="fricas")

[Out]

integral((d*x + c)*fresnels(b*x + a)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right ) S^{2}\left (a + b x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*fresnels(b*x+a)**2,x)

[Out]

Integral((c + d*x)*fresnels(a + b*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}{\rm fresnels}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*fresnels(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*x + c)*fresnels(b*x + a)^2, x)