Optimal. Leaf size=16 \[ -\frac {\tanh ^{-1}\left (\frac {\cos (x)}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3186, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\cos (x)}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3186
Rubi steps
\begin {align*} \int \frac {\sin (x)}{1+\sin ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\cos (x)\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\cos (x)}{\sqrt {2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 46, normalized size = 2.88 \[ -\frac {i \left (\tan ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )-i}{\sqrt {2}}\right )-\tan ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )+i}{\sqrt {2}}\right )\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 29, normalized size = 1.81 \[ \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\cos \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) + 2}{\cos \relax (x)^{2} - 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.00, size = 27, normalized size = 1.69 \[ -\frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} + \cos \relax (x)\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} - \cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 14, normalized size = 0.88 \[ -\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {2}\, \cos \relax (x )}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 24, normalized size = 1.50 \[ \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \cos \relax (x)}{\sqrt {2} + \cos \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 13, normalized size = 0.81 \[ -\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\cos \relax (x)}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 22.87, size = 46, normalized size = 2.88 \[ \frac {\sqrt {2} \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} - 2 \sqrt {2} + 3 \right )}}{4} - \frac {\sqrt {2} \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 2 \sqrt {2} + 3 \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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