3.121 \(\int \text {Ci}(2 x) \sin (5 x) \, dx\)

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

[Out]

1/10*Ci(3*x)+1/10*Ci(7*x)-1/5*Ci(2*x)*cos(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 = {6518, 12, 4429, 3302} \[ \frac {1}{10} \text {CosIntegral}(3 x)+\frac {1}{10} \text {CosIntegral}(7 x)-\frac {1}{5} \text {CosIntegral}(2 x) \cos (5 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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]

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]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 23, normalized size = 0.79 \[ \frac {1}{10} (\text {Ci}(3 x)+\text {Ci}(7 x)-2 \text {Ci}(2 x) \cos (5 x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cos[5*x]*CosIntegral[2*x] + CosIntegral[3*x] + CosIntegral[7*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(Ci(2*x)*sin(5*x),x, algorithm="fricas")

[Out]

Exception raised: TypeError >> An error occurred when FriCAS evaluated '(operator(Ci)((x)*(2)))*(sin((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 = 0.20, size = 23, normalized size = 0.79 \[ -\frac {1}{5} \, \cos \left (5 \, x\right ) \operatorname {Ci}\left (2 \, x\right ) + \frac {1}{10} \, \operatorname {Ci}\left (7 \, x\right ) + \frac {1}{10} \, \operatorname {Ci}\left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 24, normalized size = 0.83 \[ \frac {\Ci \left (3 x \right )}{10}+\frac {\Ci \left (7 x \right )}{10}-\frac {\Ci \left (2 x \right ) \cos \left (5 x \right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Ci(2*x)*sin(5*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \frac {\mathrm {cosint}\left (3\,x\right )}{10}+\frac {\mathrm {cosint}\left (7\,x\right )}{10}-\frac {\mathrm {cosint}\left (2\,x\right )\,\cos \left (5\,x\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosint(2*x)*sin(5*x),x)

[Out]

cosint(3*x)/10 + cosint(7*x)/10 - (cosint(2*x)*cos(5*x))/5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(5*x)*Ci(2*x), x)

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