Optimal. Leaf size=25 \[ -\frac {1}{2} \text {Ci}\left (\frac {b}{x^2}\right ) \sin (a)-\frac {1}{2} \cos (a) \text {Si}\left (\frac {b}{x^2}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3458, 3457,
3456} \begin {gather*} -\frac {1}{2} \sin (a) \text {CosIntegral}\left (\frac {b}{x^2}\right )-\frac {1}{2} \cos (a) \text {Si}\left (\frac {b}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 3456
Rule 3457
Rule 3458
Rubi steps
\begin {align*} \int \frac {\sin \left (a+\frac {b}{x^2}\right )}{x} \, dx &=\cos (a) \int \frac {\sin \left (\frac {b}{x^2}\right )}{x} \, dx+\sin (a) \int \frac {\cos \left (\frac {b}{x^2}\right )}{x} \, dx\\ &=-\frac {1}{2} \text {Ci}\left (\frac {b}{x^2}\right ) \sin (a)-\frac {1}{2} \cos (a) \text {Si}\left (\frac {b}{x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 25, normalized size = 1.00 \begin {gather*} \frac {1}{2} \left (-\text {Ci}\left (\frac {b}{x^2}\right ) \sin (a)-\cos (a) \text {Si}\left (\frac {b}{x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 22, normalized size = 0.88
method | result | size |
derivativedivides | \(-\frac {\cos \left (a \right ) \sinIntegral \left (\frac {b}{x^{2}}\right )}{2}-\frac {\cosineIntegral \left (\frac {b}{x^{2}}\right ) \sin \left (a \right )}{2}\) | \(22\) |
default | \(-\frac {\cos \left (a \right ) \sinIntegral \left (\frac {b}{x^{2}}\right )}{2}-\frac {\cosineIntegral \left (\frac {b}{x^{2}}\right ) \sin \left (a \right )}{2}\) | \(22\) |
risch | \(-\frac {i {\mathrm e}^{i a} \expIntegral \left (1, -\frac {i b}{x^{2}}\right )}{4}+\frac {{\mathrm e}^{-i a} \pi \,\mathrm {csgn}\left (\frac {b}{x^{2}}\right )}{4}-\frac {{\mathrm e}^{-i a} \sinIntegral \left (\frac {b}{x^{2}}\right )}{2}+\frac {i \expIntegral \left (1, -\frac {i b}{x^{2}}\right ) {\mathrm e}^{-i a}}{4}\) | \(63\) |
meijerg | \(-\frac {\cos \left (a \right ) \sinIntegral \left (\frac {b}{x^{2}}\right )}{2}-\frac {\sqrt {\pi }\, \sin \left (a \right ) \left (\frac {2 \gamma -4 \ln \left (x \right )+\ln \left (b^{2}\right )}{\sqrt {\pi }}-\frac {2 \gamma }{\sqrt {\pi }}-\frac {2 \ln \left (2\right )}{\sqrt {\pi }}-\frac {2 \ln \left (\frac {b}{2 x^{2}}\right )}{\sqrt {\pi }}+\frac {2 \cosineIntegral \left (\frac {b}{x^{2}}\right )}{\sqrt {\pi }}\right )}{4}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.35, size = 43, normalized size = 1.72 \begin {gather*} \frac {1}{4} \, {\left (i \, {\rm Ei}\left (\frac {i \, b}{x^{2}}\right ) - i \, {\rm Ei}\left (-\frac {i \, b}{x^{2}}\right )\right )} \cos \left (a\right ) - \frac {1}{4} \, {\left ({\rm Ei}\left (\frac {i \, b}{x^{2}}\right ) + {\rm Ei}\left (-\frac {i \, b}{x^{2}}\right )\right )} \sin \left (a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 29, normalized size = 1.16 \begin {gather*} -\frac {1}{4} \, {\left (\operatorname {Ci}\left (\frac {b}{x^{2}}\right ) + \operatorname {Ci}\left (-\frac {b}{x^{2}}\right )\right )} \sin \left (a\right ) - \frac {1}{2} \, \cos \left (a\right ) \operatorname {Si}\left (\frac {b}{x^{2}}\right ) \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 \frac {\sin {\left (a + \frac {b}{x^{2}} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.04 \begin {gather*} -\frac {\sin \left (a\right )\,\mathrm {cosint}\left (\frac {b}{x^2}\right )}{2}-\frac {\cos \left (a\right )\,\mathrm {sinint}\left (\frac {b}{x^2}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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