Optimal. Leaf size=15 \[ -\frac {\text {Ci}(2 x)}{2}+\frac {\log (x)}{2} \]
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Rubi [A]
time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3393, 3383}
\begin {gather*} \frac {\log (x)}{2}-\frac {1}{2} \text {CosIntegral}(2 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 3383
Rule 3393
Rubi steps
\begin {align*} \int \frac {\sin ^2(x)}{x} \, dx &=\int \left (\frac {1}{2 x}-\frac {\cos (2 x)}{2 x}\right ) \, dx\\ &=\frac {\log (x)}{2}-\frac {1}{2} \int \frac {\cos (2 x)}{x} \, dx\\ &=-\frac {\text {Ci}(2 x)}{2}+\frac {\log (x)}{2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -\frac {\text {Ci}(2 x)}{2}+\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 12, normalized size = 0.80
method | result | size |
default | \(-\frac {\cosineIntegral \left (2 x \right )}{2}+\frac {\ln \left (x \right )}{2}\) | \(12\) |
risch | \(-\frac {\cosineIntegral \left (2 x \right )}{2}+\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (x \right )}{4}-\frac {i \pi \,\mathrm {csgn}\left (i x \right )}{4}+\frac {\ln \left (x \right )}{2}\) | \(32\) |
meijerg | \(\frac {\sqrt {\pi }\, \left (\frac {2 \gamma }{\sqrt {\pi }}+\frac {2 \ln \left (2\right )}{\sqrt {\pi }}+\frac {2 \ln \left (x \right )}{\sqrt {\pi }}-\frac {2 \cosineIntegral \left (2 x \right )}{\sqrt {\pi }}\right )}{4}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 1.07, size = 17, normalized size = 1.13 \begin {gather*} -\frac {1}{4} \, {\rm Ei}\left (2 i \, x\right ) - \frac {1}{4} \, {\rm Ei}\left (-2 i \, x\right ) + \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 17, normalized size = 1.13 \begin {gather*} -\frac {1}{4} \, \operatorname {Ci}\left (2 \, x\right ) - \frac {1}{4} \, \operatorname {Ci}\left (-2 \, x\right ) + \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.58, size = 10, normalized size = 0.67 \begin {gather*} \frac {\log {\left (x \right )}}{2} - \frac {\operatorname {Ci}{\left (2 x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 11, normalized size = 0.73 \begin {gather*} -\frac {1}{2} \, \operatorname {Ci}\left (2 \, x\right ) + \frac {1}{2} \, \log \left (x\right ) \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 = -0.07 \begin {gather*} \frac {\ln \left (x\right )}{2}-\frac {\mathrm {cosint}\left (2\,x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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