3.1.54 \(\int \cos (5 x) \text {Si}(2 x) \, dx\) [54]

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

[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 = 0.04, 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 = {6652, 12, 4513, 3383} \begin {gather*} -\frac {1}{10} \text {CosIntegral}(3 x)+\frac {1}{10} \text {CosIntegral}(7 x)+\frac {1}{5} \text {Si}(2 x) \sin (5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/10*CosIntegral[3*x] + 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 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 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 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]

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 = 0.02, size = 25, normalized size = 0.86 \begin {gather*} \frac {1}{10} (-\text {CosIntegral}(3 x)+\text {CosIntegral}(7 x)+2 \sin (5 x) \text {Si}(2 x)) \end {gather*}

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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Maple [A]
time = 0.93, size = 24, normalized size = 0.83

method	result	size
default	\(-\frac {\cosineIntegral \left (3 x \right )}{10}+\frac {\cosineIntegral \left (7 x \right )}{10}+\frac {\sinIntegral \left (2 x \right ) \sin \left (5 x \right )}{5}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(5*x)*Si(2*x),x,method=_RETURNVERBOSE)

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).
time = 0.36, size = 54, normalized size = 1.86 \begin {gather*} \frac {1}{5} \, {\left (16 \, \cos \left (x\right )^{4} \operatorname {Si}\left (2 \, x\right ) - 12 \, \cos \left (x\right )^{2} \operatorname {Si}\left (2 \, x\right ) + \operatorname {Si}\left (2 \, x\right )\right )} \sin \left (x\right ) + \frac {1}{20} \, \operatorname {Ci}\left (7 \, x\right ) - \frac {1}{20} \, \operatorname {Ci}\left (3 \, x\right ) - \frac {1}{20} \, \operatorname {Ci}\left (-3 \, x\right ) + \frac {1}{20} \, \operatorname {Ci}\left (-7 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(16*cos(x)^4*sin_integral(2*x) - 12*cos(x)^2*sin_integral(2*x) + sin_integral(2*x))*sin(x) + 1/20*cos_inte
gral(7*x) - 1/20*cos_integral(3*x) - 1/20*cos_integral(-3*x) + 1/20*cos_integral(-7*x)

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

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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Giac [A]
time = 0.41, size = 23, normalized size = 0.79 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5*x)*sin_integral(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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \mathrm {sinint}\left (2\,x\right )\,\cos \left (5\,x\right ) \,d x \end {gather*}

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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