3.134 \(\int x \cos (a+b x) \text{CosIntegral}(c+d x) \, dx\)

Optimal. Leaf size=370 \[ -\frac{\cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b-d)}{d}+x (b-d)\right )}{2 b^2}+\frac{\cos (a+b x) \text{CosIntegral}(c+d x)}{b^2}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b+d)}{d}+x (b+d)\right )}{2 b^2}+\frac{\sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b^2}+\frac{\sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b^2}+\frac{c \sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b-d)}{d}+x (b-d)\right )}{2 b d}+\frac{c \sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b+d)}{d}+x (b+d)\right )}{2 b d}+\frac{x \sin (a+b x) \text{CosIntegral}(c+d x)}{b}+\frac{c \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b d}+\frac{c \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b d}+\frac{\cos (a+x (b-d)-c)}{2 b (b-d)}+\frac{\cos (a+x (b+d)+c)}{2 b (b+d)} \]

[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*x]*CosIntegral[c + d*x])/b^2 - (Cos[a - (b*c)/d]*CosIntegral[(c
*(b + d))/d + (b + d)*x])/(2*b^2) + (c*CosIntegral[(c*(b - d))/d + (b - d)*x]*Sin[a - (b*c)/d])/(2*b*d) + (c*C
osIntegral[(c*(b + d))/d + (b + d)*x]*Sin[a - (b*c)/d])/(2*b*d) + (x*CosIntegral[c + d*x]*Sin[a + b*x])/b + (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) + (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)

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Rubi [A]  time = 0.801469, antiderivative size = 370, normalized size of antiderivative = 1., 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 = {6514, 6742, 4574, 2638, 4430, 3303, 3299, 3302, 6518, 4429} \[ -\frac{\cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b-d)}{d}+x (b-d)\right )}{2 b^2}+\frac{\cos (a+b x) \text{CosIntegral}(c+d x)}{b^2}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b+d)}{d}+x (b+d)\right )}{2 b^2}+\frac{\sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b^2}+\frac{\sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b^2}+\frac{c \sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b-d)}{d}+x (b-d)\right )}{2 b d}+\frac{c \sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b+d)}{d}+x (b+d)\right )}{2 b d}+\frac{x \sin (a+b x) \text{CosIntegral}(c+d x)}{b}+\frac{c \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b d}+\frac{c \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b d}+\frac{\cos (a+x (b-d)-c)}{2 b (b-d)}+\frac{\cos (a+x (b+d)+c)}{2 b (b+d)} \]

Antiderivative was successfully verified.

[In]

Int[x*Cos[a + b*x]*CosIntegral[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*x]*CosIntegral[c + d*x])/b^2 - (Cos[a - (b*c)/d]*CosIntegral[(c
*(b + d))/d + (b + d)*x])/(2*b^2) + (c*CosIntegral[(c*(b - d))/d + (b - d)*x]*Sin[a - (b*c)/d])/(2*b*d) + (c*C
osIntegral[(c*(b + d))/d + (b + d)*x]*Sin[a - (b*c)/d])/(2*b*d) + (x*CosIntegral[c + d*x]*Sin[a + b*x])/b + (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) + (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 6514

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

Rule 6742

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

Rule 4574

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 2638

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

Rule 4430

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 3303

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 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]

Rule 3302

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 6518

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

Rule 4429

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

Rubi steps

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

Mathematica [C]  time = 4.40565, size = 433, normalized size = 1.17 \[ \frac{-i \exp \left (-\frac{i (d (a+c+d x)+b (c+d x))}{d}\right ) \left (\left (b^2-d^2\right ) (b c-i d) e^{i (2 a+x (b+d)+c)} \text{ExpIntegralEi}\left (\frac{i (b-d) (c+d x)}{d}\right )+e^{\frac{i b c}{d}} \left (i b d \left (d \left (-1+e^{2 i (a+b x)}\right )+b \left (1+e^{2 i (a+b x)}\right )\right )-\left (b^2-d^2\right ) (b c+i d) e^{\frac{i (b+d) (c+d x)}{d}} \text{ExpIntegralEi}\left (-\frac{i (b+d) (c+d x)}{d}\right )\right )\right )-i e^{-\frac{i (d (a-d x)+b (c+d x))}{d}} \left (\left (b^2-d^2\right ) (b c-i d) e^{i (2 a+x (b-d))} \text{ExpIntegralEi}\left (\frac{i (b+d) (c+d x)}{d}\right )+i b d e^{\frac{i c (b+d)}{d}} \left (-d e^{2 i (a+b x)}+b e^{2 i (a+b x)}+b+d\right )-\left (b^2-d^2\right ) (b c+i d) e^{i b \left (\frac{2 c}{d}+x\right )-i d x} \text{ExpIntegralEi}\left (-\frac{i (b-d) (c+d x)}{d}\right )\right )+4 d (b-d) (b+d) \text{CosIntegral}(c+d x) (b x \sin (a+b x)+\cos (a+b x))}{4 b^2 d (b-d) (b+d)} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cos[a + b*x]*CosIntegral[c + d*x],x]

[Out]

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

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Maple [B]  time = 0.105, size = 1212, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*Ci(d*x+c)*cos(b*x+a),x)

[Out]

(Ci(d*x+c)/b*(1/b*d*(sin(b/d*(d*x+c)+(a*d-b*c)/d)*(b/d*(d*x+c)+(a*d-b*c)/d)+cos(b/d*(d*x+c)+(a*d-b*c)/d))-a*d/
b*sin(b/d*(d*x+c)+(a*d-b*c)/d))-1/b*(-1/2/(b-d)*d*cos((b-d)/d*(d*x+c)+(a*d-b*c)/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^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/(b-d)*c*d^2*(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*(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^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/(b+d)*c*d^2*(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^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., size = 0, normalized size = 0. \begin{align*} \int x{\rm Ci}\left (d x + c\right ) \cos \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*Ci(d*x+c)*cos(b*x+a),x, algorithm="maxima")

[Out]

integrate(x*Ci(d*x + c)*cos(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*Ci(d*x+c)*cos(b*x+a),x, algorithm="fricas")

[Out]

integral(x*cos(b*x + a)*cos_integral(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*Ci(d*x+c)*cos(b*x+a),x)

[Out]

Integral(x*cos(a + b*x)*Ci(c + d*x), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*Ci(d*x+c)*cos(b*x+a),x, algorithm="giac")

[Out]

Timed out