3.1.59 \(\int x^2 \cos (a+b x) \text {Si}(a+b x) \, dx\) [59]

Optimal. Leaf size=218 \[ \frac {a x}{2 b^2}-\frac {x^2}{4 b}+\frac {\cos (2 a+2 b x)}{2 b^3}-\frac {\text {CosIntegral}(2 a+2 b x)}{b^3}+\frac {a^2 \text {CosIntegral}(2 a+2 b x)}{2 b^3}+\frac {\log (a+b x)}{b^3}-\frac {a^2 \log (a+b x)}{2 b^3}-\frac {a \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {x \cos (a+b x) \sin (a+b x)}{2 b^2}-\frac {\sin ^2(a+b x)}{4 b^3}+\frac {2 x \cos (a+b x) \text {Si}(a+b x)}{b^2}-\frac {2 \sin (a+b x) \text {Si}(a+b x)}{b^3}+\frac {x^2 \sin (a+b x) \text {Si}(a+b x)}{b}+\frac {a \text {Si}(2 a+2 b x)}{b^3} \]

[Out]

1/2*a*x/b^2-1/4*x^2/b-Ci(2*b*x+2*a)/b^3+1/2*a^2*Ci(2*b*x+2*a)/b^3+1/2*cos(2*b*x+2*a)/b^3+ln(b*x+a)/b^3-1/2*a^2
*ln(b*x+a)/b^3+2*x*cos(b*x+a)*Si(b*x+a)/b^2+a*Si(2*b*x+2*a)/b^3-1/2*a*cos(b*x+a)*sin(b*x+a)/b^3+1/2*x*cos(b*x+
a)*sin(b*x+a)/b^2-2*Si(b*x+a)*sin(b*x+a)/b^3+x^2*Si(b*x+a)*sin(b*x+a)/b-1/4*sin(b*x+a)^2/b^3

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Rubi [A]
time = 0.48, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6654, 6874, 2715, 8, 3391, 30, 3393, 3383, 6648, 4669, 6873, 2718, 3380, 6652} \begin {gather*} \frac {a^2 \text {CosIntegral}(2 a+2 b x)}{2 b^3}-\frac {a^2 \log (a+b x)}{2 b^3}-\frac {\text {CosIntegral}(2 a+2 b x)}{b^3}+\frac {a \text {Si}(2 a+2 b x)}{b^3}-\frac {2 \text {Si}(a+b x) \sin (a+b x)}{b^3}+\frac {\log (a+b x)}{b^3}-\frac {\sin ^2(a+b x)}{4 b^3}+\frac {\cos (2 a+2 b x)}{2 b^3}-\frac {a \sin (a+b x) \cos (a+b x)}{2 b^3}+\frac {2 x \text {Si}(a+b x) \cos (a+b x)}{b^2}+\frac {a x}{2 b^2}+\frac {x \sin (a+b x) \cos (a+b x)}{2 b^2}+\frac {x^2 \text {Si}(a+b x) \sin (a+b x)}{b}-\frac {x^2}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Cos[a + b*x]*SinIntegral[a + b*x],x]

[Out]

(a*x)/(2*b^2) - x^2/(4*b) + Cos[2*a + 2*b*x]/(2*b^3) - CosIntegral[2*a + 2*b*x]/b^3 + (a^2*CosIntegral[2*a + 2
*b*x])/(2*b^3) + Log[a + b*x]/b^3 - (a^2*Log[a + b*x])/(2*b^3) - (a*Cos[a + b*x]*Sin[a + b*x])/(2*b^3) + (x*Co
s[a + b*x]*Sin[a + b*x])/(2*b^2) - Sin[a + b*x]^2/(4*b^3) + (2*x*Cos[a + b*x]*SinIntegral[a + b*x])/b^2 - (2*S
in[a + b*x]*SinIntegral[a + b*x])/b^3 + (x^2*Sin[a + b*x]*SinIntegral[a + b*x])/b + (a*SinIntegral[2*a + 2*b*x
])/b^3

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

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

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2718

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

Rule 3380

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]

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 3391

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^
2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[b*(c + d*x)*Cos[e + f*x
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 4669

Int[Cos[w_]^(p_.)*(u_.)*Sin[v_]^(p_.), x_Symbol] :> Dist[1/2^p, Int[u*Sin[2*v]^p, x], x] /; EqQ[w, v] && Integ
erQ[p]

Rule 6648

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

Rule 6652

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

Rule 6654

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

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int x^2 \cos (a+b x) \text {Si}(a+b x) \, dx &=\frac {x^2 \sin (a+b x) \text {Si}(a+b x)}{b}-\frac {2 \int x \sin (a+b x) \text {Si}(a+b x) \, dx}{b}-\int \frac {x^2 \sin ^2(a+b x)}{a+b x} \, dx\\ &=\frac {2 x \cos (a+b x) \text {Si}(a+b x)}{b^2}+\frac {x^2 \sin (a+b x) \text {Si}(a+b x)}{b}-\frac {2 \int \cos (a+b x) \text {Si}(a+b x) \, dx}{b^2}-\frac {2 \int \frac {x \cos (a+b x) \sin (a+b x)}{a+b x} \, dx}{b}-\int \left (-\frac {a \sin ^2(a+b x)}{b^2}+\frac {x \sin ^2(a+b x)}{b}+\frac {a^2 \sin ^2(a+b x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {2 x \cos (a+b x) \text {Si}(a+b x)}{b^2}-\frac {2 \sin (a+b x) \text {Si}(a+b x)}{b^3}+\frac {x^2 \sin (a+b x) \text {Si}(a+b x)}{b}+\frac {2 \int \frac {\sin ^2(a+b x)}{a+b x} \, dx}{b^2}+\frac {a \int \sin ^2(a+b x) \, dx}{b^2}-\frac {a^2 \int \frac {\sin ^2(a+b x)}{a+b x} \, dx}{b^2}-\frac {\int x \sin ^2(a+b x) \, dx}{b}-\frac {\int \frac {x \sin (2 (a+b x))}{a+b x} \, dx}{b}\\ &=-\frac {a \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {x \cos (a+b x) \sin (a+b x)}{2 b^2}-\frac {\sin ^2(a+b x)}{4 b^3}+\frac {2 x \cos (a+b x) \text {Si}(a+b x)}{b^2}-\frac {2 \sin (a+b x) \text {Si}(a+b x)}{b^3}+\frac {x^2 \sin (a+b x) \text {Si}(a+b x)}{b}+\frac {2 \int \left (\frac {1}{2 (a+b x)}-\frac {\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b^2}+\frac {a \int 1 \, dx}{2 b^2}-\frac {a^2 \int \left (\frac {1}{2 (a+b x)}-\frac {\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b^2}-\frac {\int x \, dx}{2 b}-\frac {\int \frac {x \sin (2 a+2 b x)}{a+b x} \, dx}{b}\\ &=\frac {a x}{2 b^2}-\frac {x^2}{4 b}+\frac {\log (a+b x)}{b^3}-\frac {a^2 \log (a+b x)}{2 b^3}-\frac {a \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {x \cos (a+b x) \sin (a+b x)}{2 b^2}-\frac {\sin ^2(a+b x)}{4 b^3}+\frac {2 x \cos (a+b x) \text {Si}(a+b x)}{b^2}-\frac {2 \sin (a+b x) \text {Si}(a+b x)}{b^3}+\frac {x^2 \sin (a+b x) \text {Si}(a+b x)}{b}-\frac {\int \frac {\cos (2 a+2 b x)}{a+b x} \, dx}{b^2}+\frac {a^2 \int \frac {\cos (2 a+2 b x)}{a+b x} \, dx}{2 b^2}-\frac {\int \left (\frac {\sin (2 a+2 b x)}{b}+\frac {a \sin (2 a+2 b x)}{b (-a-b x)}\right ) \, dx}{b}\\ &=\frac {a x}{2 b^2}-\frac {x^2}{4 b}-\frac {\text {Ci}(2 a+2 b x)}{b^3}+\frac {a^2 \text {Ci}(2 a+2 b x)}{2 b^3}+\frac {\log (a+b x)}{b^3}-\frac {a^2 \log (a+b x)}{2 b^3}-\frac {a \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {x \cos (a+b x) \sin (a+b x)}{2 b^2}-\frac {\sin ^2(a+b x)}{4 b^3}+\frac {2 x \cos (a+b x) \text {Si}(a+b x)}{b^2}-\frac {2 \sin (a+b x) \text {Si}(a+b x)}{b^3}+\frac {x^2 \sin (a+b x) \text {Si}(a+b x)}{b}-\frac {\int \sin (2 a+2 b x) \, dx}{b^2}-\frac {a \int \frac {\sin (2 a+2 b x)}{-a-b x} \, dx}{b^2}\\ &=\frac {a x}{2 b^2}-\frac {x^2}{4 b}+\frac {\cos (2 a+2 b x)}{2 b^3}-\frac {\text {Ci}(2 a+2 b x)}{b^3}+\frac {a^2 \text {Ci}(2 a+2 b x)}{2 b^3}+\frac {\log (a+b x)}{b^3}-\frac {a^2 \log (a+b x)}{2 b^3}-\frac {a \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {x \cos (a+b x) \sin (a+b x)}{2 b^2}-\frac {\sin ^2(a+b x)}{4 b^3}+\frac {2 x \cos (a+b x) \text {Si}(a+b x)}{b^2}-\frac {2 \sin (a+b x) \text {Si}(a+b x)}{b^3}+\frac {x^2 \sin (a+b x) \text {Si}(a+b x)}{b}+\frac {a \text {Si}(2 a+2 b x)}{b^3}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 134, normalized size = 0.61 \begin {gather*} \frac {4 a b x-2 b^2 x^2+5 \cos (2 (a+b x))+4 \left (-2+a^2\right ) \text {CosIntegral}(2 (a+b x))+8 \log (a+b x)-4 a^2 \log (a+b x)-2 a \sin (2 (a+b x))+2 b x \sin (2 (a+b x))+8 \left (2 b x \cos (a+b x)+\left (-2+b^2 x^2\right ) \sin (a+b x)\right ) \text {Si}(a+b x)+8 a \text {Si}(2 (a+b x))}{8 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cos[a + b*x]*SinIntegral[a + b*x],x]

[Out]

(4*a*b*x - 2*b^2*x^2 + 5*Cos[2*(a + b*x)] + 4*(-2 + a^2)*CosIntegral[2*(a + b*x)] + 8*Log[a + b*x] - 4*a^2*Log
[a + b*x] - 2*a*Sin[2*(a + b*x)] + 2*b*x*Sin[2*(a + b*x)] + 8*(2*b*x*Cos[a + b*x] + (-2 + b^2*x^2)*Sin[a + b*x
])*SinIntegral[a + b*x] + 8*a*SinIntegral[2*(a + b*x)])/(8*b^3)

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Maple [A]
time = 0.66, size = 212, normalized size = 0.97

method	result	size
derivativedivides	\(\frac {\sinIntegral \left (b x +a \right ) \left (a^{2} \sin \left (b x +a \right )-2 a \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )+\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {a^{2} \ln \left (b x +a \right )}{2}+\frac {a^{2} \cosineIntegral \left (2 b x +2 a \right )}{2}-\sin \left (b x +a \right ) \cos \left (b x +a \right ) a +a \left (b x +a \right )-\left (b x +a \right ) \left (-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}+a \sinIntegral \left (2 b x +2 a \right )+\cos ^{2}\left (b x +a \right )+\ln \left (b x +a \right )-\cosineIntegral \left (2 b x +2 a \right )}{b^{3}}\)	\(212\)
default	\(\frac {\sinIntegral \left (b x +a \right ) \left (a^{2} \sin \left (b x +a \right )-2 a \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )+\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {a^{2} \ln \left (b x +a \right )}{2}+\frac {a^{2} \cosineIntegral \left (2 b x +2 a \right )}{2}-\sin \left (b x +a \right ) \cos \left (b x +a \right ) a +a \left (b x +a \right )-\left (b x +a \right ) \left (-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}+a \sinIntegral \left (2 b x +2 a \right )+\cos ^{2}\left (b x +a \right )+\ln \left (b x +a \right )-\cosineIntegral \left (2 b x +2 a \right )}{b^{3}}\)	\(212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cos(b*x+a)*Si(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/b^3*(Si(b*x+a)*(a^2*sin(b*x+a)-2*a*(cos(b*x+a)+(b*x+a)*sin(b*x+a))+(b*x+a)^2*sin(b*x+a)-2*sin(b*x+a)+2*(b*x+
a)*cos(b*x+a))-1/2*a^2*ln(b*x+a)+1/2*a^2*Ci(2*b*x+2*a)-sin(b*x+a)*cos(b*x+a)*a+a*(b*x+a)-(b*x+a)*(-1/2*sin(b*x
+a)*cos(b*x+a)+1/2*b*x+1/2*a)+1/4*(b*x+a)^2-1/4*sin(b*x+a)^2+a*Si(2*b*x+2*a)+cos(b*x+a)^2+ln(b*x+a)-Ci(2*b*x+2
*a))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(b*x+a)*sin_integral(b*x+a),x, algorithm="maxima")

[Out]

integrate(x^2*cos(b*x + a)*sin_integral(b*x + a), x)

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Fricas [A]
time = 0.37, size = 141, normalized size = 0.65 \begin {gather*} -\frac {b^{2} x^{2} - 8 \, b x \cos \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) - 2 \, a b x - 5 \, \cos \left (b x + a\right )^{2} - {\left (a^{2} - 2\right )} \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) - {\left (a^{2} - 2\right )} \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) + 2 \, {\left (a^{2} - 2\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (b x - a\right )} \cos \left (b x + a\right ) + 2 \, {\left (b^{2} x^{2} - 2\right )} \operatorname {Si}\left (b x + a\right )\right )} \sin \left (b x + a\right ) - 4 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right )}{4 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(b*x+a)*sin_integral(b*x+a),x, algorithm="fricas")

[Out]

-1/4*(b^2*x^2 - 8*b*x*cos(b*x + a)*sin_integral(b*x + a) - 2*a*b*x - 5*cos(b*x + a)^2 - (a^2 - 2)*cos_integral
(2*b*x + 2*a) - (a^2 - 2)*cos_integral(-2*b*x - 2*a) + 2*(a^2 - 2)*log(b*x + a) - 2*((b*x - a)*cos(b*x + a) +
2*(b^2*x^2 - 2)*sin_integral(b*x + a))*sin(b*x + a) - 4*a*sin_integral(2*b*x + 2*a))/b^3

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \cos {\left (a + b x \right )} \operatorname {Si}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cos(b*x+a)*Si(b*x+a),x)

[Out]

Integral(x**2*cos(a + b*x)*Si(a + b*x), x)

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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.43, size = 431, normalized size = 1.98 \begin {gather*} {\left (\frac {2 \, x \cos \left (b x + a\right )}{b^{2}} + \frac {{\left (b^{2} x^{2} - 2\right )} \sin \left (b x + a\right )}{b^{3}}\right )} \operatorname {Si}\left (b x + a\right ) - \frac {2 \, b^{2} x^{2} \tan \left (b x + a\right )^{2} - 4 \, a b x \tan \left (b x + a\right )^{2} + 4 \, a^{2} \log \left ({\left | b x + a \right |}\right ) \tan \left (b x + a\right )^{2} - 2 \, a^{2} \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} - 2 \, a^{2} \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} + 2 \, b^{2} x^{2} - 4 \, a \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} + 4 \, a \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} - 8 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )^{2} - 4 \, a b x + 4 \, a^{2} \log \left ({\left | b x + a \right |}\right ) - 2 \, a^{2} \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) - 2 \, a^{2} \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) - 4 \, b x \tan \left (b x + a\right ) - 8 \, \log \left ({\left | b x + a \right |}\right ) \tan \left (b x + a\right )^{2} + 4 \, \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} + 4 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} - 4 \, a \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) + 4 \, a \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) - 8 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) + 4 \, a \tan \left (b x + a\right ) + 5 \, \tan \left (b x + a\right )^{2} - 8 \, \log \left ({\left | b x + a \right |}\right ) + 4 \, \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) + 4 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) - 5}{8 \, {\left (b^{3} \tan \left (b x + a\right )^{2} + b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(b*x+a)*sin_integral(b*x+a),x, algorithm="giac")

[Out]

(2*x*cos(b*x + a)/b^2 + (b^2*x^2 - 2)*sin(b*x + a)/b^3)*sin_integral(b*x + a) - 1/8*(2*b^2*x^2*tan(b*x + a)^2
- 4*a*b*x*tan(b*x + a)^2 + 4*a^2*log(abs(b*x + a))*tan(b*x + a)^2 - 2*a^2*real_part(cos_integral(2*b*x + 2*a))
*tan(b*x + a)^2 - 2*a^2*real_part(cos_integral(-2*b*x - 2*a))*tan(b*x + a)^2 + 2*b^2*x^2 - 4*a*imag_part(cos_i
ntegral(2*b*x + 2*a))*tan(b*x + a)^2 + 4*a*imag_part(cos_integral(-2*b*x - 2*a))*tan(b*x + a)^2 - 8*a*sin_inte
gral(2*b*x + 2*a)*tan(b*x + a)^2 - 4*a*b*x + 4*a^2*log(abs(b*x + a)) - 2*a^2*real_part(cos_integral(2*b*x + 2*
a)) - 2*a^2*real_part(cos_integral(-2*b*x - 2*a)) - 4*b*x*tan(b*x + a) - 8*log(abs(b*x + a))*tan(b*x + a)^2 +
4*real_part(cos_integral(2*b*x + 2*a))*tan(b*x + a)^2 + 4*real_part(cos_integral(-2*b*x - 2*a))*tan(b*x + a)^2
- 4*a*imag_part(cos_integral(2*b*x + 2*a)) + 4*a*imag_part(cos_integral(-2*b*x - 2*a)) - 8*a*sin_integral(2*b
*x + 2*a) + 4*a*tan(b*x + a) + 5*tan(b*x + a)^2 - 8*log(abs(b*x + a)) + 4*real_part(cos_integral(2*b*x + 2*a))
+ 4*real_part(cos_integral(-2*b*x - 2*a)) - 5)/(b^3*tan(b*x + a)^2 + b^3)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\mathrm {sinint}\left (a+b\,x\right )\,\cos \left (a+b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sinint(a + b*x)*cos(a + b*x),x)

[Out]

int(x^2*sinint(a + b*x)*cos(a + b*x), x)

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