Optimal. Leaf size=30 \[ -\frac {1}{2} \cos (x) \sqrt {2-\cos ^2(x)}-\sin ^{-1}\left (\frac {\cos (x)}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3186, 195, 216} \[ -\frac {1}{2} \cos (x) \sqrt {2-\cos ^2(x)}-\sin ^{-1}\left (\frac {\cos (x)}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 3186
Rubi steps
\begin {align*} \int \sin (x) \sqrt {1+\sin ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \sqrt {2-x^2} \, dx,x,\cos (x)\right )\\ &=-\frac {1}{2} \cos (x) \sqrt {2-\cos ^2(x)}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2}} \, dx,x,\cos (x)\right )\\ &=-\sin ^{-1}\left (\frac {\cos (x)}{\sqrt {2}}\right )-\frac {1}{2} \cos (x) \sqrt {2-\cos ^2(x)}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 53, normalized size = 1.77 \[ -\frac {\cos (x) \sqrt {3-\cos (2 x)}}{2 \sqrt {2}}+i \log \left (\sqrt {3-\cos (2 x)}+i \sqrt {2} \cos (x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 71, normalized size = 2.37 \[ -\frac {1}{2} \, \sqrt {-\cos \relax (x)^{2} + 2} \cos \relax (x) + \frac {1}{2} \, \arctan \left (-\frac {\cos \relax (x) \sin \relax (x) - {\left (\cos \relax (x)^{3} - \cos \relax (x)\right )} \sqrt {-\cos \relax (x)^{2} + 2}}{\cos \relax (x)^{4} - 3 \, \cos \relax (x)^{2} + 1}\right ) - \frac {1}{2} \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 25, normalized size = 0.83 \[ -\frac {1}{2} \, \sqrt {-\cos \relax (x)^{2} + 2} \cos \relax (x) - \arcsin \left (\frac {1}{2} \, \sqrt {2} \cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.11, size = 51, normalized size = 1.70 \[ -\frac {\sqrt {\left (1+\sin ^{2}\relax (x )\right ) \left (\cos ^{2}\relax (x )\right )}\, \left (\sqrt {-\left (\cos ^{4}\relax (x )\right )+2 \left (\cos ^{2}\relax (x )\right )}+\arcsin \left (\cos ^{2}\relax (x )-1\right )\right )}{2 \cos \relax (x ) \sqrt {1+\sin ^{2}\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 25, normalized size = 0.83 \[ -\frac {1}{2} \, \sqrt {-\cos \relax (x)^{2} + 2} \cos \relax (x) - \arcsin \left (\frac {1}{2} \, \sqrt {2} \cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \sin \relax (x)\,\sqrt {{\sin \relax (x)}^2+1} \,d x \]
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 \[ \int \sqrt {\sin ^{2}{\relax (x )} + 1} \sin {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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