Optimal. Leaf size=35 \[ \frac {x \cos (b x)}{2 b}-\frac {\sin (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Si}(b x) \]
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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6638, 12, 3377,
2717} \begin {gather*} -\frac {\sin (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Si}(b x)+\frac {x \cos (b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2717
Rule 3377
Rule 6638
Rubi steps
\begin {align*} \int x \text {Si}(b x) \, dx &=\frac {1}{2} x^2 \text {Si}(b x)-\frac {1}{2} b \int \frac {x \sin (b x)}{b} \, dx\\ &=\frac {1}{2} x^2 \text {Si}(b x)-\frac {1}{2} \int x \sin (b x) \, dx\\ &=\frac {x \cos (b x)}{2 b}+\frac {1}{2} x^2 \text {Si}(b x)-\frac {\int \cos (b x) \, dx}{2 b}\\ &=\frac {x \cos (b x)}{2 b}-\frac {\sin (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Si}(b x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.00 \begin {gather*} \frac {x \cos (b x)}{2 b}-\frac {\sin (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Si}(b x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 32, normalized size = 0.91
method | result | size |
derivativedivides | \(\frac {\frac {b^{2} x^{2} \sinIntegral \left (b x \right )}{2}-\frac {\sin \left (b x \right )}{2}+\frac {b x \cos \left (b x \right )}{2}}{b^{2}}\) | \(32\) |
default | \(\frac {\frac {b^{2} x^{2} \sinIntegral \left (b x \right )}{2}-\frac {\sin \left (b x \right )}{2}+\frac {b x \cos \left (b x \right )}{2}}{b^{2}}\) | \(32\) |
meijerg | \(\frac {\sqrt {\pi }\, \left (\frac {b x \cos \left (b x \right )}{2 \sqrt {\pi }}-\frac {\sin \left (b x \right )}{2 \sqrt {\pi }}+\frac {b^{2} x^{2} \sinIntegral \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 29, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {Si}\left (b x\right ) + \frac {b x \cos \left (b x\right ) - \sin \left (b x\right )}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 30, normalized size = 0.86 \begin {gather*} \frac {b^{2} x^{2} \operatorname {Si}\left (b x\right ) + b x \cos \left (b x\right ) - \sin \left (b x\right )}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.42, size = 29, normalized size = 0.83 \begin {gather*} \frac {x^{2} \operatorname {Si}{\left (b x \right )}}{2} + \frac {x \cos {\left (b x \right )}}{2 b} - \frac {\sin {\left (b x \right )}}{2 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 29, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {Si}\left (b x\right ) + \frac {x \cos \left (b x\right )}{2 \, b} - \frac {\sin \left (b x\right )}{2 \, 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 = -0.03 \begin {gather*} \frac {x^2\,\mathrm {sinint}\left (b\,x\right )}{2}-\frac {\sin \left (b\,x\right )-b\,x\,\cos \left (b\,x\right )}{2\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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