Optimal. Leaf size=74 \[ -\frac {\text {CosIntegral}(2 b x)}{2 b^2}+\frac {\log (x)}{2 b^2}-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \cos (b x) \text {Si}(b x)}{b}-\frac {\sin (b x) \text {Si}(b x)}{b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2 \]
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Rubi [A]
time = 0.06, 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 = {6642, 6648, 12,
2644, 30, 6652, 3393, 3383} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2644
Rule 3383
Rule 3393
Rule 6642
Rule 6648
Rule 6652
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 {\text {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*}
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Mathematica [A]
time = 0.04, size = 58, normalized size = 0.78 \begin {gather*} \frac {\cos (2 b x)-2 \text {CosIntegral}(2 b x)+2 \log (x)+4 (b x \cos (b x)-\sin (b x)) \text {Si}(b x)+2 b^2 x^2 \text {Si}(b x)^2}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 62, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {b^{2} x^{2} \sinIntegral \left (b x \right )^{2}}{2}-2 \sinIntegral \left (b x \right ) \left (\frac {\sin \left (b x \right )}{2}-\frac {b x \cos \left (b x \right )}{2}\right )+\frac {\left (\cos ^{2}\left (b x \right )\right )}{2}+\frac {\ln \left (b x \right )}{2}-\frac {\cosineIntegral \left (2 b x \right )}{2}}{b^{2}}\) | \(62\) |
default | \(\frac {\frac {b^{2} x^{2} \sinIntegral \left (b x \right )^{2}}{2}-2 \sinIntegral \left (b x \right ) \left (\frac {\sin \left (b x \right )}{2}-\frac {b x \cos \left (b x \right )}{2}\right )+\frac {\left (\cos ^{2}\left (b x \right )\right )}{2}+\frac {\ln \left (b x \right )}{2}-\frac {\cosineIntegral \left (2 b x \right )}{2}}{b^{2}}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
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.
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Fricas [A]
time = 0.37, size = 68, normalized size = 0.92 \begin {gather*} \frac {2 \, b^{2} x^{2} \operatorname {Si}\left (b x\right )^{2} + 4 \, b x \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) + 2 \, \cos \left (b x\right )^{2} - 4 \, \sin \left (b x\right ) \operatorname {Si}\left (b x\right ) - \operatorname {Ci}\left (2 \, b x\right ) - \operatorname {Ci}\left (-2 \, b x\right ) + 2 \, \log \left (x\right )}{4 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \operatorname {Si}^{2}{\left (b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 65, normalized size = 0.88 \begin {gather*} \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 \left (x\right )}{4 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\mathrm {sinint}\left (b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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