Optimal. Leaf size=74 \[ -\frac {\text {Ci}(2 b x)}{2 b^2}-\frac {\text {Si}(b x) \sin (b x)}{b^2}+\frac {\log (x)}{2 b^2}-\frac {\sin ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {x \text {Si}(b x) \cos (b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6507, 6513, 12, 2564, 30, 6517, 3312, 3302} \[ -\frac {\text {CosIntegral}(2 b x)}{2 b^2}-\frac {\text {Si}(b x) \sin (b x)}{b^2}+\frac {\log (x)}{2 b^2}-\frac {\sin ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {x \text {Si}(b x) \cos (b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 2564
Rule 3302
Rule 3312
Rule 6507
Rule 6513
Rule 6517
Rubi steps
\begin {align*} \int x \text {Si}(b x)^2 \, dx &=\frac {1}{2} x^2 \text {Si}(b x)^2-\int x \sin (b x) \text {Si}(b x) \, dx\\ &=\frac {x \cos (b x) \text {Si}(b x)}{b}+\frac {1}{2} x^2 \text {Si}(b x)^2-\frac {\int \cos (b x) \text {Si}(b x) \, dx}{b}-\int \frac {\cos (b x) \sin (b x)}{b} \, dx\\ &=\frac {x \cos (b x) \text {Si}(b x)}{b}-\frac {\sin (b x) \text {Si}(b x)}{b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2-\frac {\int \cos (b x) \sin (b x) \, dx}{b}+\frac {\int \frac {\sin ^2(b x)}{b x} \, dx}{b}\\ &=\frac {x \cos (b x) \text {Si}(b x)}{b}-\frac {\sin (b x) \text {Si}(b x)}{b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {\int \frac {\sin ^2(b x)}{x} \, dx}{b^2}-\frac {\operatorname {Subst}(\int x \, dx,x,\sin (b x))}{b^2}\\ &=-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \cos (b x) \text {Si}(b x)}{b}-\frac {\sin (b x) \text {Si}(b x)}{b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {\int \left (\frac {1}{2 x}-\frac {\cos (2 b x)}{2 x}\right ) \, dx}{b^2}\\ &=\frac {\log (x)}{2 b^2}-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \cos (b x) \text {Si}(b x)}{b}-\frac {\sin (b x) \text {Si}(b x)}{b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2-\frac {\int \frac {\cos (2 b x)}{x} \, dx}{2 b^2}\\ &=-\frac {\text {Ci}(2 b x)}{2 b^2}+\frac {\log (x)}{2 b^2}-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \cos (b x) \text {Si}(b x)}{b}-\frac {\sin (b x) \text {Si}(b x)}{b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 58, normalized size = 0.78 \[ \frac {2 b^2 x^2 \text {Si}(b x)^2-2 \text {Ci}(2 b x)+4 \text {Si}(b x) (b x \cos (b x)-\sin (b x))+\cos (2 b x)+2 \log (x)}{4 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {Si}\left (b x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.60, size = 65, normalized size = 0.88 \[ \frac {1}{2} \, x^{2} \operatorname {Si}\left (b x\right )^{2} + {\left (\frac {x \cos \left (b x\right )}{b} - \frac {\sin \left (b x\right )}{b^{2}}\right )} \operatorname {Si}\left (b x\right ) + \frac {\cos \left (2 \, b x\right ) - \operatorname {Ci}\left (2 \, b x\right ) - \operatorname {Ci}\left (-2 \, b x\right ) + 2 \, \log \relax (x)}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 69, normalized size = 0.93 \[ \frac {x^{2} \Si \left (b x \right )^{2}}{2}-\frac {\Si \left (b x \right ) \sin \left (b x \right )}{b^{2}}+\frac {x \cos \left (b x \right ) \Si \left (b x \right )}{b}+\frac {\cos ^{2}\left (b x \right )}{2 b^{2}}+\frac {\ln \left (b x \right )}{2 b^{2}}-\frac {\Ci \left (2 b x \right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Si}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {sinint}\left (b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {Si}^{2}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________