Optimal. Leaf size=26 \[ \frac {\text {Si}(2 b x)}{2 b}-\frac {\text {Si}(b x) \cos (b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6511, 12, 4406, 3299} \[ \frac {\text {Si}(2 b x)}{2 b}-\frac {\text {Si}(b x) \cos (b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3299
Rule 4406
Rule 6511
Rubi steps
\begin {align*} \int \sin (b x) \text {Si}(b x) \, dx &=-\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] time = 0.01, size = 26, normalized size = 1.00 \[ \frac {\text {Si}(2 b x)}{2 b}-\frac {\text {Si}(b x) \cos (b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sin \left (b x\right ) \operatorname {Si}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.23, size = 41, normalized size = 1.58 \[ -\frac {\cos \left (b x\right ) \operatorname {Si}\left (b x\right )}{b} + \frac {\Im \left (\operatorname {Ci}\left (2 \, b x\right ) \right ) - \Im \left (\operatorname {Ci}\left (-2 \, b x\right ) \right ) + 2 \, \operatorname {Si}\left (2 \, b x\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 23, normalized size = 0.88 \[ \frac {-\cos \left (b x \right ) \Si \left (b x \right )+\frac {\Si \left (2 b x \right )}{2}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Si}\left (b x\right ) \sin \left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \mathrm {sinint}\left (b\,x\right )\,\sin \left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin {\left (b x \right )} \operatorname {Si}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________