3.13 \(\int \text{Si}(b x)^2 \, dx\)

Optimal. Leaf size=32 \[ x \text{Si}(b x)^2-\frac{\text{Si}(2 b x)}{b}+\frac{2 \text{Si}(b x) \cos (b x)}{b} \]

[Out]

(2*Cos[b*x]*SinIntegral[b*x])/b + x*SinIntegral[b*x]^2 - SinIntegral[2*b*x]/b

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Rubi [A]  time = 0.0508099, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6505, 6511, 12, 4406, 3299} \[ x \text{Si}(b x)^2-\frac{\text{Si}(2 b x)}{b}+\frac{2 \text{Si}(b x) \cos (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[SinIntegral[b*x]^2,x]

[Out]

(2*Cos[b*x]*SinIntegral[b*x])/b + x*SinIntegral[b*x]^2 - SinIntegral[2*b*x]/b

Rule 6505

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

Rule 6511

Int[Sin[(a_.) + (b_.)*(x_)]*SinIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[(Cos[a + b*x]*SinIntegral[c +
d*x])/b, x] + Dist[d/b, Int[(Cos[a + b*x]*Sin[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 3299

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

Rubi steps

\begin{align*} \int \text{Si}(b x)^2 \, dx &=x \text{Si}(b x)^2-2 \int \sin (b x) \text{Si}(b x) \, dx\\ &=\frac{2 \cos (b x) \text{Si}(b x)}{b}+x \text{Si}(b x)^2-2 \int \frac{\cos (b x) \sin (b x)}{b x} \, dx\\ &=\frac{2 \cos (b x) \text{Si}(b x)}{b}+x \text{Si}(b x)^2-\frac{2 \int \frac{\cos (b x) \sin (b x)}{x} \, dx}{b}\\ &=\frac{2 \cos (b x) \text{Si}(b x)}{b}+x \text{Si}(b x)^2-\frac{2 \int \frac{\sin (2 b x)}{2 x} \, dx}{b}\\ &=\frac{2 \cos (b x) \text{Si}(b x)}{b}+x \text{Si}(b x)^2-\frac{\int \frac{\sin (2 b x)}{x} \, dx}{b}\\ &=\frac{2 \cos (b x) \text{Si}(b x)}{b}+x \text{Si}(b x)^2-\frac{\text{Si}(2 b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0075411, size = 32, normalized size = 1. \[ x \text{Si}(b x)^2-\frac{\text{Si}(2 b x)}{b}+\frac{2 \text{Si}(b x) \cos (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[SinIntegral[b*x]^2,x]

[Out]

(2*Cos[b*x]*SinIntegral[b*x])/b + x*SinIntegral[b*x]^2 - SinIntegral[2*b*x]/b

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Maple [A]  time = 0.05, size = 32, normalized size = 1. \begin{align*}{\frac{bx \left ({\it Si} \left ( bx \right ) \right ) ^{2}+2\,\cos \left ( bx \right ){\it Si} \left ( bx \right ) -{\it Si} \left ( 2\,bx \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Si(b*x)^2,x)

[Out]

1/b*(b*x*Si(b*x)^2+2*cos(b*x)*Si(b*x)-Si(2*b*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Si(b*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sin_integral(b*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Si(b*x)**2,x)

[Out]

Integral(Si(b*x)**2, x)

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Giac [C]  time = 1.33996, size = 66, normalized size = 2.06 \begin{align*} x \operatorname{Si}\left (b x\right )^{2} + \frac{2 \, \cos \left (b x\right ) \operatorname{Si}\left (b x\right )}{b} - \frac{\Im \left ( \operatorname{Ci}\left (2 \, b x\right ) \right ) - \Im \left ( \operatorname{Ci}\left (-2 \, b x\right ) \right ) + 2 \, \operatorname{Si}\left (2 \, b x\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sin_integral(b*x)^2 + 2*cos(b*x)*sin_integral(b*x)/b - 1/2*(imag_part(cos_integral(2*b*x)) - imag_part(cos_i
ntegral(-2*b*x)) + 2*sin_integral(2*b*x))/b