3.1.53 \(\int \sin (5 x) \text {Si}(2 x) \, dx\) [53]

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

[Out]

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

________________________________________________________________________________________

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 = {6646, 12, 4515, 3380} \begin {gather*} -\frac {\text {Si}(3 x)}{10}+\frac {\text {Si}(7 x)}{10}-\frac {1}{5} \text {Si}(2 x) \cos (5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(Cos[5*x]*SinIntegral[2*x]) - SinIntegral[3*x]/10 + SinIntegral[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 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 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 6646

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 25, normalized size = 0.86 \begin {gather*} \frac {1}{10} (-2 \cos (5 x) \text {Si}(2 x)-\text {Si}(3 x)+\text {Si}(7 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.90, size = 24, normalized size = 0.83

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 41, normalized size = 1.41 \begin {gather*} -\frac {16}{5} \, \cos \left (x\right )^{5} \operatorname {Si}\left (2 \, x\right ) + 4 \, \cos \left (x\right )^{3} \operatorname {Si}\left (2 \, x\right ) - \cos \left (x\right ) \operatorname {Si}\left (2 \, x\right ) + \frac {1}{10} \, \operatorname {Si}\left (7 \, x\right ) - \frac {1}{10} \, \operatorname {Si}\left (3 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/5*cos(x)^5*sin_integral(2*x) + 4*cos(x)^3*sin_integral(2*x) - cos(x)*sin_integral(2*x) + 1/10*sin_integral
(7*x) - 1/10*sin_integral(3*x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sin {\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(Si(2*x)*sin(5*x),x)

[Out]

Integral(sin(5*x)*Si(2*x), x)

________________________________________________________________________________________

Giac [A]
time = 0.40, size = 23, normalized size = 0.79 \begin {gather*} -\frac {1}{5} \, \cos \left (5 \, x\right ) \operatorname {Si}\left (2 \, x\right ) + \frac {1}{10} \, \operatorname {Si}\left (7 \, x\right ) - \frac {1}{10} \, \operatorname {Si}\left (3 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sinint(2*x)*sin(5*x), x)

________________________________________________________________________________________