3.122 \(\int \cos (5 x) \text{CosIntegral}(2 x) \, dx\)

Optimal. Leaf size=29 \[ \frac{1}{5} \text{CosIntegral}(2 x) \sin (5 x)-\frac{\text{Si}(3 x)}{10}-\frac{\text{Si}(7 x)}{10} \]

[Out]

(CosIntegral[2*x]*Sin[5*x])/5 - SinIntegral[3*x]/10 - SinIntegral[7*x]/10

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Rubi [A]  time = 0.0534955, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {6512, 12, 4430, 3299} \[ \frac{1}{5} \text{CosIntegral}(2 x) \sin (5 x)-\frac{\text{Si}(3 x)}{10}-\frac{\text{Si}(7 x)}{10} \]

Antiderivative was successfully verified.

[In]

Int[Cos[5*x]*CosIntegral[2*x],x]

[Out]

(CosIntegral[2*x]*Sin[5*x])/5 - SinIntegral[3*x]/10 - SinIntegral[7*x]/10

Rule 6512

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]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 \cos (5 x) \text{Ci}(2 x) \, dx &=\frac{1}{5} \text{Ci}(2 x) \sin (5 x)-\frac{2}{5} \int \frac{\cos (2 x) \sin (5 x)}{2 x} \, dx\\ &=\frac{1}{5} \text{Ci}(2 x) \sin (5 x)-\frac{1}{5} \int \frac{\cos (2 x) \sin (5 x)}{x} \, dx\\ &=\frac{1}{5} \text{Ci}(2 x) \sin (5 x)-\frac{1}{5} \int \left (\frac{\sin (3 x)}{2 x}+\frac{\sin (7 x)}{2 x}\right ) \, dx\\ &=\frac{1}{5} \text{Ci}(2 x) \sin (5 x)-\frac{1}{10} \int \frac{\sin (3 x)}{x} \, dx-\frac{1}{10} \int \frac{\sin (7 x)}{x} \, dx\\ &=\frac{1}{5} \text{Ci}(2 x) \sin (5 x)-\frac{\text{Si}(3 x)}{10}-\frac{\text{Si}(7 x)}{10}\\ \end{align*}

Mathematica [A]  time = 0.024018, size = 27, normalized size = 0.93 \[ \frac{1}{10} (2 \text{CosIntegral}(2 x) \sin (5 x)-\text{Si}(3 x)-\text{Si}(7 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[5*x]*CosIntegral[2*x],x]

[Out]

(2*CosIntegral[2*x]*Sin[5*x] - SinIntegral[3*x] - SinIntegral[7*x])/10

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Maple [A]  time = 0.095, size = 24, normalized size = 0.8 \begin{align*} -{\frac{{\it Si} \left ( 3\,x \right ) }{10}}-{\frac{{\it Si} \left ( 7\,x \right ) }{10}}+{\frac{{\it Ci} \left ( 2\,x \right ) \sin \left ( 5\,x \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Ci(2*x)*cos(5*x),x)

[Out]

-1/10*Si(3*x)-1/10*Si(7*x)+1/5*Ci(2*x)*sin(5*x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Ci}\left (2 \, x\right ) \cos \left (5 \, x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(2*x)*cos(5*x),x, algorithm="maxima")

[Out]

integrate(Ci(2*x)*cos(5*x), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(2*x)*cos(5*x),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (5 x \right )} \operatorname{Ci}{\left (2 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(2*x)*cos(5*x),x)

[Out]

Integral(cos(5*x)*Ci(2*x), x)

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Giac [A]  time = 1.2446, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{5} \, \operatorname{Ci}\left (2 \, x\right ) \sin \left (5 \, x\right ) - \frac{1}{10} \, \operatorname{Si}\left (7 \, x\right ) - \frac{1}{10} \, \operatorname{Si}\left (3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(2*x)*cos(5*x),x, algorithm="giac")

[Out]

1/5*cos_integral(2*x)*sin(5*x) - 1/10*sin_integral(7*x) - 1/10*sin_integral(3*x)