Optimal. Leaf size=11 \[ -\sin ^{-1}\left (\frac{\cos (x)}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0263391, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3186, 216} \[ -\sin ^{-1}\left (\frac{\cos (x)}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 3186
Rule 216
Rubi steps
\begin{align*} \int \frac{\sin (x)}{\sqrt{1+\sin ^2(x)}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2}} \, dx,x,\cos (x)\right )\\ &=-\sin ^{-1}\left (\frac{\cos (x)}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0637318, size = 29, normalized size = 2.64 \[ i \log \left (\sqrt{3-\cos (2 x)}+i \sqrt{2} \cos (x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.463, size = 33, normalized size = 3. \begin{align*}{\frac{\arcsin \left ( \left ( \sin \left ( x \right ) \right ) ^{2} \right ) }{2\,\cos \left ( x \right ) }\sqrt{ \left ( 1+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \left ( \cos \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{1+ \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41358, size = 14, normalized size = 1.27 \begin{align*} -\arcsin \left (\frac{1}{2} \, \sqrt{2} \cos \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.63381, size = 173, normalized size = 15.73 \begin{align*} \frac{1}{2} \, \arctan \left (-\frac{\cos \left (x\right ) \sin \left (x\right ) -{\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sqrt{-\cos \left (x\right )^{2} + 2}}{\cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2} + 1}\right ) - \frac{1}{2} \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1363, size = 14, normalized size = 1.27 \begin{align*} -\arcsin \left (\frac{1}{2} \, \sqrt{2} \cos \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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