\(\int \frac {\sin (b x) \text {Si}(b x)}{x} \, dx\) [41]

Optimal result

Integrand size = 12, antiderivative size = 10 \[ \int \frac {\sin (b x) \text {Si}(b x)}{x} \, dx=\frac {\text {Si}(b x)^2}{2} \]

[Out]

1/2*Si(b*x)^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6818} \[ \int \frac {\sin (b x) \text {Si}(b x)}{x} \, dx=\frac {\text {Si}(b x)^2}{2} \]

[In]

Int[(Sin[b*x]*SinIntegral[b*x])/x,x]

[Out]

SinIntegral[b*x]^2/2

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Si}(b x)^2}{2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (b x) \text {Si}(b x)}{x} \, dx=\frac {\text {Si}(b x)^2}{2} \]

[In]

Integrate[(Sin[b*x]*SinIntegral[b*x])/x,x]

[Out]

SinIntegral[b*x]^2/2

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\operatorname {Si}\left (b x \right )^{2}}{2}\)	\(9\)
default	\(\frac {\operatorname {Si}\left (b x \right )^{2}}{2}\)	\(9\)

[In]

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

[Out]

1/2*Si(b*x)^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\sin (b x) \text {Si}(b x)}{x} \, dx=\frac {1}{2} \, \operatorname {Si}\left (b x\right )^{2} \]

[In]

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

[Out]

1/2*sin_integral(b*x)^2

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {\sin (b x) \text {Si}(b x)}{x} \, dx=\frac {\operatorname {Si}^{2}{\left (b x \right )}}{2} \]

[In]

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

[Out]

Si(b*x)**2/2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\sin (b x) \text {Si}(b x)}{x} \, dx=\frac {1}{2} \, \operatorname {Si}\left (b x\right )^{2} \]

[In]

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

[Out]

1/2*sin_integral(b*x)^2

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin (b x) \text {Si}(b x)}{x} \, dx=\frac {{\mathrm {sinint}\left (b\,x\right )}^2}{2} \]

[In]

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

[Out]

sinint(b*x)^2/2