Optimal. Leaf size=46 \[ -\frac {b \cos (b x)}{4 x}-\frac {\sin (b x)}{4 x^2}-\frac {1}{4} b^2 \text {Si}(b x)-\frac {\text {Si}(b x)}{2 x^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6638, 12, 3378,
3380} \begin {gather*} -\frac {1}{4} b^2 \text {Si}(b x)-\frac {\text {Si}(b x)}{2 x^2}-\frac {\sin (b x)}{4 x^2}-\frac {b \cos (b x)}{4 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3378
Rule 3380
Rule 6638
Rubi steps
\begin {align*} \int \frac {\text {Si}(b x)}{x^3} \, dx &=-\frac {\text {Si}(b x)}{2 x^2}+\frac {1}{2} b \int \frac {\sin (b x)}{b x^3} \, dx\\ &=-\frac {\text {Si}(b x)}{2 x^2}+\frac {1}{2} \int \frac {\sin (b x)}{x^3} \, dx\\ &=-\frac {\sin (b x)}{4 x^2}-\frac {\text {Si}(b x)}{2 x^2}+\frac {1}{4} b \int \frac {\cos (b x)}{x^2} \, dx\\ &=-\frac {b \cos (b x)}{4 x}-\frac {\sin (b x)}{4 x^2}-\frac {\text {Si}(b x)}{2 x^2}-\frac {1}{4} b^2 \int \frac {\sin (b x)}{x} \, dx\\ &=-\frac {b \cos (b x)}{4 x}-\frac {\sin (b x)}{4 x^2}-\frac {1}{4} b^2 \text {Si}(b x)-\frac {\text {Si}(b x)}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 46, normalized size = 1.00 \begin {gather*} -\frac {b \cos (b x)}{4 x}-\frac {\sin (b x)}{4 x^2}-\frac {1}{4} b^2 \text {Si}(b x)-\frac {\text {Si}(b x)}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 48, normalized size = 1.04
method | result | size |
derivativedivides | \(b^{2} \left (-\frac {\sinIntegral \left (b x \right )}{2 b^{2} x^{2}}-\frac {\sin \left (b x \right )}{4 b^{2} x^{2}}-\frac {\cos \left (b x \right )}{4 b x}-\frac {\sinIntegral \left (b x \right )}{4}\right )\) | \(48\) |
default | \(b^{2} \left (-\frac {\sinIntegral \left (b x \right )}{2 b^{2} x^{2}}-\frac {\sin \left (b x \right )}{4 b^{2} x^{2}}-\frac {\cos \left (b x \right )}{4 b x}-\frac {\sinIntegral \left (b x \right )}{4}\right )\) | \(48\) |
meijerg | \(\frac {\sqrt {\pi }\, b^{2} \left (-\frac {4 \cos \left (b x \right )}{b x \sqrt {\pi }}-\frac {4 \sin \left (b x \right )}{b^{2} x^{2} \sqrt {\pi }}-\frac {4 \left (b^{2} x^{2}+2\right ) \sinIntegral \left (b x \right )}{b^{2} x^{2} \sqrt {\pi }}\right )}{16}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.37, size = 32, normalized size = 0.70 \begin {gather*} -\frac {1}{4} \, b^{2} {\left (-i \, \Gamma \left (-2, i \, b x\right ) + i \, \Gamma \left (-2, -i \, b x\right )\right )} - \frac {\operatorname {Si}\left (b x\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 31, normalized size = 0.67 \begin {gather*} -\frac {b x \cos \left (b x\right ) + {\left (b^{2} x^{2} + 2\right )} \operatorname {Si}\left (b x\right ) + \sin \left (b x\right )}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.47, size = 41, normalized size = 0.89 \begin {gather*} - \frac {b^{2} \operatorname {Si}{\left (b x \right )}}{4} - \frac {b \cos {\left (b x \right )}}{4 x} - \frac {\sin {\left (b x \right )}}{4 x^{2}} - \frac {\operatorname {Si}{\left (b x \right )}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.43, size = 149, normalized size = 3.24 \begin {gather*} -\frac {b^{2} x^{2} \Im \left ( \operatorname {Ci}\left (b x\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} - b^{2} x^{2} \Im \left ( \operatorname {Ci}\left (-b x\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} + 2 \, b^{2} x^{2} \operatorname {Si}\left (b x\right ) \tan \left (\frac {1}{2} \, b x\right )^{2} + b^{2} x^{2} \Im \left ( \operatorname {Ci}\left (b x\right ) \right ) - b^{2} x^{2} \Im \left ( \operatorname {Ci}\left (-b x\right ) \right ) + 2 \, b^{2} x^{2} \operatorname {Si}\left (b x\right ) - 2 \, b x \tan \left (\frac {1}{2} \, b x\right )^{2} + 2 \, b x + 4 \, \tan \left (\frac {1}{2} \, b x\right )}{8 \, {\left (x^{2} \tan \left (\frac {1}{2} \, b x\right )^{2} + x^{2}\right )}} - \frac {\operatorname {Si}\left (b x\right )}{2 \, x^{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 = -0.02 \begin {gather*} -\frac {\frac {\sin \left (b\,x\right )}{2}+\frac {b\,x\,\cos \left (b\,x\right )}{2}}{2\,x^2}-\frac {b^2\,\mathrm {sinint}\left (b\,x\right )}{4}-\frac {\mathrm {sinint}\left (b\,x\right )}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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