3.2.35 \(\int \cos (a+b x) \text {CosIntegral}(c+d x) \, dx\) [135]

Optimal. Leaf size=153 \[ -\frac {\text {CosIntegral}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b}-\frac {\text {CosIntegral}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b}+\frac {\text {CosIntegral}(c+d x) \sin (a+b x)}{b}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6647, 4515, 3384, 3380, 3383} \begin {gather*} -\frac {\sin \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {c (b-d)}{d}+x (b-d)\right )}{2 b}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {c (b+d)}{d}+x (b+d)\right )}{2 b}+\frac {\sin (a+b x) \text {CosIntegral}(c+d x)}{b}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(CosIntegral[(c*(b - d))/d + (b - d)*x]*Sin[a - (b*c)/d])/b - (CosIntegral[(c*(b + d))/d + (b + d)*x]*Sin
[a - (b*c)/d])/(2*b) + (CosIntegral[c + d*x]*Sin[a + b*x])/b - (Cos[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (
b - d)*x])/(2*b) - (Cos[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b)

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 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 6647

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.91, size = 153, normalized size = 1.00 \begin {gather*} \frac {i e^{-\frac {i (b c+a d)}{d}} \left (-e^{\frac {2 i b c}{d}} \text {Ei}\left (-\frac {i (b-d) (c+d x)}{d}\right )+e^{2 i a} \text {Ei}\left (\frac {i (b-d) (c+d x)}{d}\right )-e^{\frac {2 i b c}{d}} \text {Ei}\left (-\frac {i (b+d) (c+d x)}{d}\right )+e^{2 i a} \text {Ei}\left (\frac {i (b+d) (c+d x)}{d}\right )\right )+4 \text {CosIntegral}(c+d x) \sin (a+b x)}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.91, size = 290, normalized size = 1.90

method	result	size
default	\(\frac {\frac {\cosineIntegral \left (d x +c \right ) d \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{b}-\frac {d \left (\frac {d \left (-\frac {\sinIntegral \left (-\frac {\left (b -d \right ) \left (d x +c \right )}{d}-\frac {a d -c b}{d}-\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}-\frac {\cosineIntegral \left (\frac {\left (b -d \right ) \left (d x +c \right )}{d}+\frac {a d -c b}{d}+\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}\right )}{2}+\frac {d \left (-\frac {\sinIntegral \left (-\frac {\left (b +d \right ) \left (d x +c \right )}{d}-\frac {a d -c b}{d}-\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}-\frac {\cosineIntegral \left (\frac {\left (b +d \right ) \left (d x +c \right )}{d}+\frac {a d -c b}{d}+\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}\right )}{2}\right )}{b}}{d}\)	\(290\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Ci(d*x+c)/b*d*sin(1/d*b*(d*x+c)+(a*d-b*c)/d)-1/b*d*(1/2*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*(-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)))/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(fresnel_cos(d*x+c)*cos(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.39, size = 239, normalized size = 1.56 \begin {gather*} \frac {2 \, d \operatorname {C}\left (d x + c\right ) \sin \left (b x + a\right ) - \sqrt {d^{2}} \cos \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right ) \operatorname {S}\left (\frac {{\left (\pi d^{2} x + \pi c d + b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) + \sqrt {d^{2}} \cos \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right ) \operatorname {S}\left (\frac {{\left (\pi d^{2} x + \pi c d - b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) - \sqrt {d^{2}} \operatorname {C}\left (\frac {{\left (\pi d^{2} x + \pi c d - b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) \sin \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right ) - \sqrt {d^{2}} \operatorname {C}\left (\frac {{\left (\pi d^{2} x + \pi c d + b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) \sin \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right )}{2 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*d*fresnel_cos(d*x + c)*sin(b*x + a) - sqrt(d^2)*cos(a - b*c/d - 1/2*b^2/(pi*d^2))*fresnel_sin((pi*d^2*x
+ pi*c*d + b)*sqrt(d^2)/(pi*d^2)) + sqrt(d^2)*cos(a - b*c/d + 1/2*b^2/(pi*d^2))*fresnel_sin((pi*d^2*x + pi*c*
d - b)*sqrt(d^2)/(pi*d^2)) - sqrt(d^2)*fresnel_cos((pi*d^2*x + pi*c*d - b)*sqrt(d^2)/(pi*d^2))*sin(a - b*c/d +
1/2*b^2/(pi*d^2)) - sqrt(d^2)*fresnel_cos((pi*d^2*x + pi*c*d + b)*sqrt(d^2)/(pi*d^2))*sin(a - b*c/d - 1/2*b^2
/(pi*d^2)))/(b*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(a + b*x)*Ci(c + d*x), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {cosint}\left (c+d\,x\right )\,\cos \left (a+b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosint(c + d*x)*cos(a + b*x),x)

[Out]

int(cosint(c + d*x)*cos(a + b*x), x)

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