\(\int \sin (b x) \text {Si}(b x) \, dx\) [42]

Optimal result

Integrand size = 9, antiderivative size = 26 \[ \int \sin (b x) \text {Si}(b x) \, dx=-\frac {\cos (b x) \text {Si}(b x)}{b}+\frac {\text {Si}(2 b x)}{2 b} \]

[Out]

-cos(b*x)*Si(b*x)/b+1/2*Si(2*b*x)/b

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6646, 12, 4491, 3380} \[ \int \sin (b x) \text {Si}(b x) \, dx=\frac {\text {Si}(2 b x)}{2 b}-\frac {\text {Si}(b x) \cos (b x)}{b} \]

[In]

Int[Sin[b*x]*SinIntegral[b*x],x]

[Out]

-((Cos[b*x]*SinIntegral[b*x])/b) + SinIntegral[2*b*x]/(2*b)

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 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 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*} \text {integral}& = -\frac {\cos (b x) \text {Si}(b x)}{b}+\int \frac {\cos (b x) \sin (b x)}{b x} \, dx \\ & = -\frac {\cos (b x) \text {Si}(b x)}{b}+\frac {\int \frac {\cos (b x) \sin (b x)}{x} \, dx}{b} \\ & = -\frac {\cos (b x) \text {Si}(b x)}{b}+\frac {\int \frac {\sin (2 b x)}{2 x} \, dx}{b} \\ & = -\frac {\cos (b x) \text {Si}(b x)}{b}+\frac {\int \frac {\sin (2 b x)}{x} \, dx}{2 b} \\ & = -\frac {\cos (b x) \text {Si}(b x)}{b}+\frac {\text {Si}(2 b x)}{2 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \sin (b x) \text {Si}(b x) \, dx=-\frac {\cos (b x) \text {Si}(b x)}{b}+\frac {\text {Si}(2 b x)}{2 b} \]

[In]

Integrate[Sin[b*x]*SinIntegral[b*x],x]

[Out]

-((Cos[b*x]*SinIntegral[b*x])/b) + SinIntegral[2*b*x]/(2*b)

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {-\cos \left (b x \right ) \operatorname {Si}\left (b x \right )+\frac {\operatorname {Si}\left (2 b x \right )}{2}}{b}\)	\(23\)
default	\(\frac {-\cos \left (b x \right ) \operatorname {Si}\left (b x \right )+\frac {\operatorname {Si}\left (2 b x \right )}{2}}{b}\)	\(23\)

[In]

int(Si(b*x)*sin(b*x),x,method=_RETURNVERBOSE)

[Out]

1/b*(-cos(b*x)*Si(b*x)+1/2*Si(2*b*x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \sin (b x) \text {Si}(b x) \, dx=-\frac {2 \, \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) - \operatorname {Si}\left (2 \, b x\right )}{2 \, b} \]

[In]

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

[Out]

-1/2*(2*cos(b*x)*sin_integral(b*x) - sin_integral(2*b*x))/b

Sympy [F]

\[ \int \sin (b x) \text {Si}(b x) \, dx=\int \sin {\left (b x \right )} \operatorname {Si}{\left (b x \right )}\, dx \]

[In]

integrate(Si(b*x)*sin(b*x),x)

[Out]

Integral(sin(b*x)*Si(b*x), x)

Maxima [F]

\[ \int \sin (b x) \text {Si}(b x) \, dx=\int { \sin \left (b x\right ) \operatorname {Si}\left (b x\right ) \,d x } \]

[In]

integrate(sin_integral(b*x)*sin(b*x),x, algorithm="maxima")

[Out]

integrate(sin(b*x)*sin_integral(b*x), x)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \sin (b x) \text {Si}(b x) \, dx=-\frac {\cos \left (b x\right ) \operatorname {Si}\left (b x\right )}{b} + \frac {\Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) - \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) + 2 \, \operatorname {Si}\left (2 \, b x\right )}{4 \, b} \]

[In]

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

[Out]

-cos(b*x)*sin_integral(b*x)/b + 1/4*(imag_part(cos_integral(2*b*x)) - imag_part(cos_integral(-2*b*x)) + 2*sin_
integral(2*b*x))/b

Mupad [F(-1)]

Timed out. \[ \int \sin (b x) \text {Si}(b x) \, dx=\int \mathrm {sinint}\left (b\,x\right )\,\sin \left (b\,x\right ) \,d x \]

[In]

int(sinint(b*x)*sin(b*x),x)

[Out]

int(sinint(b*x)*sin(b*x), x)