3.54 \(\int \cos (5 x) \text {Si}(2 x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 = {6517, 12, 4428, 3302} \[ -\frac {1}{10} \text {CosIntegral}(3 x)+\frac {1}{10} \text {CosIntegral}(7 x)+\frac {1}{5} \text {Si}(2 x) \sin (5 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 4428

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 6517

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]

Rubi steps

\begin {align*} \int \cos (5 x) \text {Si}(2 x) \, dx &=\frac {1}{5} \sin (5 x) \text {Si}(2 x)-\frac {2}{5} \int \frac {\sin (2 x) \sin (5 x)}{2 x} \, dx\\ &=\frac {1}{5} \sin (5 x) \text {Si}(2 x)-\frac {1}{5} \int \frac {\sin (2 x) \sin (5 x)}{x} \, dx\\ &=\frac {1}{5} \sin (5 x) \text {Si}(2 x)-\frac {1}{5} \int \left (\frac {\cos (3 x)}{2 x}-\frac {\cos (7 x)}{2 x}\right ) \, dx\\ &=\frac {1}{5} \sin (5 x) \text {Si}(2 x)-\frac {1}{10} \int \frac {\cos (3 x)}{x} \, dx+\frac {1}{10} \int \frac {\cos (7 x)}{x} \, dx\\ &=-\frac {\text {Ci}(3 x)}{10}+\frac {\text {Ci}(7 x)}{10}+\frac {1}{5} \sin (5 x) \text {Si}(2 x)\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 25, normalized size = 0.86 \[ \frac {1}{10} (-\text {Ci}(3 x)+\text {Ci}(7 x)+2 \text {Si}(2 x) \sin (5 x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred when FriCAS evaluated '(operator(Si)((x)*(2)))*(cos((x)*(5)))
':   There are 1 exposed and 1 unexposed library operations named elt having      1 argument(s) but none was d
etermined to be applicable. Use HyperDoc      Browse, or issue                                )display op elt
to learn more about the available operations. Perhaps       package-calling the operation or using coerci
ons on the arguments       will allow you to apply the operation.    Cannot find application of object of type
BasicOperator to argument(s)       of type(s)                               Polynomial(Integer)

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giac [A]  time = 0.18, size = 23, normalized size = 0.79 \[ \frac {1}{5} \, \sin \left (5 \, x\right ) \operatorname {Si}\left (2 \, x\right ) + \frac {1}{10} \, \operatorname {Ci}\left (7 \, x\right ) - \frac {1}{10} \, \operatorname {Ci}\left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.16, size = 24, normalized size = 0.83 \[ -\frac {\Ci \left (3 x \right )}{10}+\frac {\Ci \left (7 x \right )}{10}+\frac {\Si \left (2 x \right ) \sin \left (5 x \right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Si}\left (2 \, x\right ) \cos \left (5 \, x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Si(2*x)*cos(5*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {sinint}\left (2\,x\right )\,\cos \left (5\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sinint(2*x)*cos(5*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\left (5 x \right )} \operatorname {Si}{\left (2 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(5*x)*Si(2*x), x)

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