3.1.63 \(\int x \sin (a+b x) \text {Si}(c+d x) \, dx\) [63]

Optimal. Leaf size=371 \[ \frac {\cos (a-c+(b-d) x)}{2 b (b-d)}-\frac {\cos (a+c+(b+d) x)}{2 b (b+d)}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2}+\frac {c \text {CosIntegral}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b d}-\frac {c \text {CosIntegral}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b d}+\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}+\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}-\frac {x \cos (a+b x) \text {Si}(c+d x)}{b}+\frac {\sin (a+b x) \text {Si}(c+d x)}{b^2}-\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2} \]

[Out]

-1/2*Ci(c*(b-d)/d+(b-d)*x)*cos(a-b*c/d)/b^2+1/2*Ci(c*(b+d)/d+(b+d)*x)*cos(a-b*c/d)/b^2+1/2*cos(a-c+(b-d)*x)/b/
(b-d)-1/2*cos(a+c+(b+d)*x)/b/(b+d)+1/2*c*cos(a-b*c/d)*Si(c*(b-d)/d+(b-d)*x)/b/d-x*cos(b*x+a)*Si(d*x+c)/b-1/2*c
*cos(a-b*c/d)*Si(c*(b+d)/d+(b+d)*x)/b/d+1/2*c*Ci(c*(b-d)/d+(b-d)*x)*sin(a-b*c/d)/b/d-1/2*c*Ci(c*(b+d)/d+(b+d)*
x)*sin(a-b*c/d)/b/d+1/2*Si(c*(b-d)/d+(b-d)*x)*sin(a-b*c/d)/b^2-1/2*Si(c*(b+d)/d+(b+d)*x)*sin(a-b*c/d)/b^2+Si(d
*x+c)*sin(b*x+a)/b^2

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Rubi [A]
time = 0.64, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6648, 6874, 4670, 2718, 4515, 3384, 3380, 3383, 6652, 4513} \begin {gather*} -\frac {\cos \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {c (b-d)}{d}+x (b-d)\right )}{2 b^2}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {c (b+d)}{d}+x (b+d)\right )}{2 b^2}+\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}+\frac {\sin (a+b x) \text {Si}(c+d x)}{b^2}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}+\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {c (b-d)}{d}+x (b-d)\right )}{2 b d}-\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {c (b+d)}{d}+x (b+d)\right )}{2 b d}+\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}-\frac {x \cos (a+b x) \text {Si}(c+d x)}{b}-\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}+\frac {\cos (a+x (b-d)-c)}{2 b (b-d)}-\frac {\cos (a+x (b+d)+c)}{2 b (b+d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sin[a + b*x]*SinIntegral[c + d*x],x]

[Out]

Cos[a - c + (b - d)*x]/(2*b*(b - d)) - Cos[a + c + (b + d)*x]/(2*b*(b + d)) - (Cos[a - (b*c)/d]*CosIntegral[(c
*(b - d))/d + (b - d)*x])/(2*b^2) + (Cos[a - (b*c)/d]*CosIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2) + (c*Cos
Integral[(c*(b - d))/d + (b - d)*x]*Sin[a - (b*c)/d])/(2*b*d) - (c*CosIntegral[(c*(b + d))/d + (b + d)*x]*Sin[
a - (b*c)/d])/(2*b*d) + (c*Cos[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) + (Sin[a - (b*c)/d
]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) - (x*Cos[a + b*x]*SinIntegral[c + d*x])/b + (Sin[a + b*x]*Si
nIntegral[c + d*x])/b^2 - (c*Cos[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d) - (Sin[a - (b*c)
/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2)

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 3384

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

Rule 4513

Int[((e_.) + (f_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(p_.)*Sin[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int[E
xpandTrigReduce[(e + f*x)^m, Sin[a + b*x]^p*Sin[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p,
0] && IGtQ[q, 0] && IntegerQ[m]

Rule 4515

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

Rule 4670

Int[Cos[w_]^(q_.)*Sin[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sin[v]^p*Cos[w]^q, x], x] /; IGtQ[p, 0] &&
IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w],
x]))

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 6874

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 3.28, size = 345, normalized size = 0.93 \begin {gather*} \frac {e^{-i (a+c)} \left (b d \left (-\frac {e^{-i (b+d) x}}{b+d}+\frac {e^{i (2 a+b x-d x)}}{b-d}\right )-i (b c-i d) e^{\frac {i (-b c+(2 a+c) d)}{d}} \text {Ei}\left (\frac {i (b-d) (c+d x)}{d}\right )+(-i b c+d) e^{\frac {i c (b+d)}{d}} \text {Ei}\left (-\frac {i (b+d) (c+d x)}{d}\right )\right )}{4 b^2 d}+\frac {e^{-i (a-c)} \left (b d \left (\frac {e^{-i (b-d) x}}{b-d}-\frac {e^{i (2 a+(b+d) x)}}{b+d}\right )+i (b c+i d) e^{\frac {i c (b-d)}{d}} \text {Ei}\left (-\frac {i (b-d) (c+d x)}{d}\right )+(i b c+d) e^{-\frac {i (b c-2 a d+c d)}{d}} \text {Ei}\left (\frac {i (b+d) (c+d x)}{d}\right )\right )}{4 b^2 d}-\frac {(b x \cos (a+b x)-\sin (a+b x)) \text {Si}(c+d x)}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Sin[a + b*x]*SinIntegral[c + d*x],x]

[Out]

(b*d*(-(1/((b + d)*E^(I*(b + d)*x))) + E^(I*(2*a + b*x - d*x))/(b - d)) - I*(b*c - I*d)*E^((I*(-(b*c) + (2*a +
c)*d))/d)*ExpIntegralEi[(I*(b - d)*(c + d*x))/d] + ((-I)*b*c + d)*E^((I*c*(b + d))/d)*ExpIntegralEi[((-I)*(b
+ d)*(c + d*x))/d])/(4*b^2*d*E^(I*(a + c))) + (b*d*(1/((b - d)*E^(I*(b - d)*x)) - E^(I*(2*a + (b + d)*x))/(b +
d)) + I*(b*c + I*d)*E^((I*c*(b - d))/d)*ExpIntegralEi[((-I)*(b - d)*(c + d*x))/d] + ((I*b*c + d)*ExpIntegralE
i[(I*(b + d)*(c + d*x))/d])/E^((I*(b*c - 2*a*d + c*d))/d))/(4*b^2*d*E^(I*(a - c))) - ((b*x*Cos[a + b*x] - Sin[
a + b*x])*SinIntegral[c + d*x])/b^2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1245\) vs. \(2(351)=702\).
time = 2.03, size = 1246, normalized size = 3.36

method	result	size
default	\(\text {Expression too large to display}\)	\(1246\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*Si(d*x+c)*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

(-Si(d*x+c)/b*(-d/b*a*cos(1/d*b*(d*x+c)+(a*d-b*c)/d)-1/b*d*(sin(1/d*b*(d*x+c)+(a*d-b*c)/d)-(1/d*b*(d*x+c)+(a*d
-b*c)/d)*cos(1/d*b*(d*x+c)+(a*d-b*c)/d)))+1/b*(1/2/(b-d)*d^2*a*(-Si(-(b-d)/d*(d*x+c)-(a*d-b*c)/d-(-a*d+b*c)/d)
*cos((-a*d+b*c)/d)/d-Ci((b-d)/d*(d*x+c)+(a*d-b*c)/d+(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d)-1/2*d^2*c/(b-d)*(-Si(-(
b-d)/d*(d*x+c)-(a*d-b*c)/d-(-a*d+b*c)/d)*cos((-a*d+b*c)/d)/d-Ci((b-d)/d*(d*x+c)+(a*d-b*c)/d+(-a*d+b*c)/d)*sin(
(-a*d+b*c)/d)/d)-1/2*(a*d-b*c)/(b-d)*d*(-Si(-(b-d)/d*(d*x+c)-(a*d-b*c)/d-(-a*d+b*c)/d)*cos((-a*d+b*c)/d)/d-Ci(
(b-d)/d*(d*x+c)+(a*d-b*c)/d+(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d)+1/2/(b-d)*d*cos((b-d)/d*(d*x+c)+(a*d-b*c)/d)-1/
2*d^2*a/(b+d)*(-Si(-(b+d)/d*(d*x+c)-(a*d-b*c)/d-(-a*d+b*c)/d)*cos((-a*d+b*c)/d)/d-Ci((b+d)/d*(d*x+c)+(a*d-b*c)
/d+(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d)-1/2*d^2*c/(b+d)*(-Si(-(b+d)/d*(d*x+c)-(a*d-b*c)/d-(-a*d+b*c)/d)*cos((-a*
d+b*c)/d)/d-Ci((b+d)/d*(d*x+c)+(a*d-b*c)/d+(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d)+1/2*(a*d-b*c)*d/(b+d)*(-Si(-(b+d
)/d*(d*x+c)-(a*d-b*c)/d-(-a*d+b*c)/d)*cos((-a*d+b*c)/d)/d-Ci((b+d)/d*(d*x+c)+(a*d-b*c)/d+(-a*d+b*c)/d)*sin((-a
*d+b*c)/d)/d)-1/2/(b+d)*d*cos((b+d)/d*(d*x+c)+(a*d-b*c)/d)-1/2/b*d^2*(-Si(-(b-d)/d*(d*x+c)-(a*d-b*c)/d-(-a*d+b
*c)/d)*sin((-a*d+b*c)/d)/d+Ci((b-d)/d*(d*x+c)+(a*d-b*c)/d+(-a*d+b*c)/d)*cos((-a*d+b*c)/d)/d)+1/2/b*d^2*(-Si(-(
b+d)/d*(d*x+c)-(a*d-b*c)/d-(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d+Ci((b+d)/d*(d*x+c)+(a*d-b*c)/d+(-a*d+b*c)/d)*cos(
(-a*d+b*c)/d)/d)))/d

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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*sin_integral(d*x+c)*sin(b*x+a),x, algorithm="maxima")

[Out]

integrate(x*sin(b*x + a)*sin_integral(d*x + c), x)

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Fricas [A]
time = 0.39, size = 580, normalized size = 1.56 \begin {gather*} \frac {4 \, b d^{2} \cos \left (b x + a\right ) \cos \left (d x + c\right ) + 4 \, b^{2} d \sin \left (b x + a\right ) \sin \left (d x + c\right ) - 4 \, {\left (b^{3} d - b d^{3}\right )} x \cos \left (b x + a\right ) \operatorname {Si}\left (d x + c\right ) + 4 \, {\left (b^{2} d - d^{3}\right )} \sin \left (b x + a\right ) \operatorname {Si}\left (d x + c\right ) + {\left ({\left (b^{2} d - d^{3}\right )} \operatorname {Ci}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) + {\left (b^{2} d - d^{3}\right )} \operatorname {Ci}\left (-\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) - {\left (b^{2} d - d^{3}\right )} \operatorname {Ci}\left (\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right ) - {\left (b^{2} d - d^{3}\right )} \operatorname {Ci}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right ) - 2 \, {\left (b^{3} c - b c d^{2}\right )} \operatorname {Si}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) - 2 \, {\left (b^{3} c - b c d^{2}\right )} \operatorname {Si}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) - {\left ({\left (b^{3} c - b c d^{2}\right )} \operatorname {Ci}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) + {\left (b^{3} c - b c d^{2}\right )} \operatorname {Ci}\left (-\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) - {\left (b^{3} c - b c d^{2}\right )} \operatorname {Ci}\left (\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right ) - {\left (b^{3} c - b c d^{2}\right )} \operatorname {Ci}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right ) + 2 \, {\left (b^{2} d - d^{3}\right )} \operatorname {Si}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) + 2 \, {\left (b^{2} d - d^{3}\right )} \operatorname {Si}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )}{4 \, {\left (b^{4} d - b^{2} d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin_integral(d*x+c)*sin(b*x+a),x, algorithm="fricas")

[Out]

1/4*(4*b*d^2*cos(b*x + a)*cos(d*x + c) + 4*b^2*d*sin(b*x + a)*sin(d*x + c) - 4*(b^3*d - b*d^3)*x*cos(b*x + a)*
sin_integral(d*x + c) + 4*(b^2*d - d^3)*sin(b*x + a)*sin_integral(d*x + c) + ((b^2*d - d^3)*cos_integral((b*c
+ c*d + (b*d + d^2)*x)/d) + (b^2*d - d^3)*cos_integral(-(b*c + c*d + (b*d + d^2)*x)/d) - (b^2*d - d^3)*cos_int
egral((b*c - c*d + (b*d - d^2)*x)/d) - (b^2*d - d^3)*cos_integral(-(b*c - c*d + (b*d - d^2)*x)/d) - 2*(b^3*c -
b*c*d^2)*sin_integral((b*c + c*d + (b*d + d^2)*x)/d) - 2*(b^3*c - b*c*d^2)*sin_integral(-(b*c - c*d + (b*d -
d^2)*x)/d))*cos(-(b*c - a*d)/d) - ((b^3*c - b*c*d^2)*cos_integral((b*c + c*d + (b*d + d^2)*x)/d) + (b^3*c - b*
c*d^2)*cos_integral(-(b*c + c*d + (b*d + d^2)*x)/d) - (b^3*c - b*c*d^2)*cos_integral((b*c - c*d + (b*d - d^2)*
x)/d) - (b^3*c - b*c*d^2)*cos_integral(-(b*c - c*d + (b*d - d^2)*x)/d) + 2*(b^2*d - d^3)*sin_integral((b*c + c
*d + (b*d + d^2)*x)/d) + 2*(b^2*d - d^3)*sin_integral(-(b*c - c*d + (b*d - d^2)*x)/d))*sin(-(b*c - a*d)/d))/(b
^4*d - b^2*d^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*Si(d*x+c)*sin(b*x+a),x)

[Out]

Integral(x*sin(a + b*x)*Si(c + d*x), x)

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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.72, size = 200182, normalized size = 539.57 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin_integral(d*x+c)*sin(b*x+a),x, algorithm="giac")

[Out]

-(x*cos(b*x + a)/b - sin(b*x + a)/b^2)*sin_integral(d*x + c) - 1/4*(b^3*c*imag_part(cos_integral(b*x + d*x + c
+ b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2
*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - b*c*d^2*imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*b*
x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan
(1/2*(b*c - c*d)/d)^2 - b^3*c*imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*
b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 +
b*c*d^2*imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(
1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + b^3*c*imag_part(cos_
integral(-b*x + d*x + c - b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1
/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - b*c*d^2*imag_part(cos_integral(-b*x + d*x
+ c - b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(
1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - b^3*c*imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*
b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*t
an(1/2*(b*c - c*d)/d)^2 + b*c*d^2*imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan
(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d
)^2 + 2*b^3*c*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*ta
n(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - 2*b*c*d^2*sin_inte
gral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan
(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - 2*b^3*c*sin_integral((b*d*x - d^2*x + b*
c - c*d)/d)*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/
2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + 2*b*c*d^2*sin_integral((b*d*x - d^2*x + b*c - c*d)/d)*tan(1/2*b*
x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan
(1/2*(b*c - c*d)/d)^2 - 2*b^3*c*real_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/
2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d) +
2*b*c*d^2*real_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*ta
n(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d) - 2*b^3*c*real_part(co
s_integral(-b*x + d*x + c - b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan
(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d) + 2*b*c*d^2*real_part(cos_integral(-b*x + d*
x + c - b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*ta
n(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d) + 2*b^3*c*real_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2
*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)*ta
n(1/2*(b*c - c*d)/d)^2 - 2*b*c*d^2*real_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan
(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^
2 + 2*b^3*c*real_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*
tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 - 2*b*c*d^2*real_par
t(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2
*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 + 2*b^3*c*real_part(cos_integral(b*x - d
*x - c + b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)*tan
(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - 2*b*c*d^2*real_part(cos_integral(b*x - d*x - c + b*c/d))*tan(
1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2
*tan(1/2*(b*c - c*d)/d)^2 + 2*b^3*c*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*t
an(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d
)^2 - 2*b*c*d^2*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x
)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - 2*b^3*c*real_p
art(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)*
tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*t...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinint(c + d*x)*sin(a + b*x),x)

[Out]

int(x*sinint(c + d*x)*sin(a + b*x), x)

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