3.1.40 \(\int \frac {\sin (b x) \text {Si}(b x)}{x^2} \, dx\) [40]

Optimal. Leaf size=49 \[ -\frac {\sin ^2(b x)}{x}-\frac {\sin (b x) \text {Si}(b x)}{x}+b \text {Si}(2 b x)+b \text {Int}\left (\frac {\cos (b x) \text {Si}(b x)}{x},x\right ) \]

[Out]

b*CannotIntegrate(cos(b*x)*Si(b*x)/x,x)+b*Si(2*b*x)-Si(b*x)*sin(b*x)/x-sin(b*x)^2/x

________________________________________________________________________________________

Rubi [A]
time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sin (b x) \text {Si}(b x)}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Sin[b*x]*SinIntegral[b*x])/x^2,x]

[Out]

-(Sin[b*x]^2/x) - (Sin[b*x]*SinIntegral[b*x])/x + b*SinIntegral[2*b*x] + b*Defer[Int][(Cos[b*x]*SinIntegral[b*
x])/x, x]

Rubi steps

\begin {align*} \int \frac {\sin (b x) \text {Si}(b x)}{x^2} \, dx &=-\frac {\sin (b x) \text {Si}(b x)}{x}+b \int \frac {\sin ^2(b x)}{b x^2} \, dx+b \int \frac {\cos (b x) \text {Si}(b x)}{x} \, dx\\ &=-\frac {\sin (b x) \text {Si}(b x)}{x}+b \int \frac {\cos (b x) \text {Si}(b x)}{x} \, dx+\int \frac {\sin ^2(b x)}{x^2} \, dx\\ &=-\frac {\sin ^2(b x)}{x}-\frac {\sin (b x) \text {Si}(b x)}{x}+b \int \frac {\cos (b x) \text {Si}(b x)}{x} \, dx+(2 b) \int \frac {\sin (2 b x)}{2 x} \, dx\\ &=-\frac {\sin ^2(b x)}{x}-\frac {\sin (b x) \text {Si}(b x)}{x}+b \int \frac {\sin (2 b x)}{x} \, dx+b \int \frac {\cos (b x) \text {Si}(b x)}{x} \, dx\\ &=-\frac {\sin ^2(b x)}{x}-\frac {\sin (b x) \text {Si}(b x)}{x}+b \text {Si}(2 b x)+b \int \frac {\cos (b x) \text {Si}(b x)}{x} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin (b x) \text {Si}(b x)}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Sin[b*x]*SinIntegral[b*x])/x^2,x]

[Out]

Integrate[(Sin[b*x]*SinIntegral[b*x])/x^2, x]

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\sinIntegral \left (b x \right ) \sin \left (b x \right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Si(b*x)*sin(b*x)/x^2,x)

[Out]

int(Si(b*x)*sin(b*x)/x^2,x)

________________________________________________________________________________________

Maxima [A]
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(b*x)*sin(b*x)/x^2,x, algorithm="maxima")

[Out]

integrate(sin(b*x)*sin_integral(b*x)/x^2, x)

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin_integral(b*x)*sin(b*x)/x^2,x, algorithm="fricas")

[Out]

integral(sin(b*x)*sin_integral(b*x)/x^2, x)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (b x \right )} \operatorname {Si}{\left (b x \right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Si(b*x)*sin(b*x)/x**2,x)

[Out]

Integral(sin(b*x)*Si(b*x)/x**2, x)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin_integral(b*x)*sin(b*x)/x^2,x, algorithm="giac")

[Out]

integrate(sin(b*x)*sin_integral(b*x)/x^2, x)

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {sinint}\left (b\,x\right )\,\sin \left (b\,x\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sinint(b*x)*sin(b*x))/x^2,x)

[Out]

int((sinint(b*x)*sin(b*x))/x^2, x)

________________________________________________________________________________________