Optimal. Leaf size=21 \[ \sin (x) \sqrt {\csc (x)+1}+\tanh ^{-1}\left (\sqrt {\csc (x)+1}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3873, 47, 63, 207} \[ \sin (x) \sqrt {\csc (x)+1}+\tanh ^{-1}\left (\sqrt {\csc (x)+1}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 207
Rule 3873
Rubi steps
\begin {align*} \int \cos (x) \sqrt {1+\csc (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x^2} \, dx,x,\csc (x)\right )\\ &=\sqrt {1+\csc (x)} \sin (x)-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\csc (x)\right )\\ &=\sqrt {1+\csc (x)} \sin (x)-\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\csc (x)}\right )\\ &=\tanh ^{-1}\left (\sqrt {1+\csc (x)}\right )+\sqrt {1+\csc (x)} \sin (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ \sin (x) \sqrt {\csc (x)+1}+\tanh ^{-1}\left (\sqrt {\csc (x)+1}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.99, size = 79, normalized size = 3.76 \[ \sqrt {\frac {\sin \relax (x) + 1}{\sin \relax (x)}} \sin \relax (x) + \frac {1}{2} \, \log \left (\frac {2 \, {\left (\sqrt {\frac {\sin \relax (x) + 1}{\sin \relax (x)}} \sin \relax (x) + \sin \relax (x) + 1\right )}}{\cos \relax (x) + \sin \relax (x) + 1}\right ) - \frac {1}{2} \, \log \left (-\frac {2 \, {\left (\sqrt {\frac {\sin \relax (x) + 1}{\sin \relax (x)}} \sin \relax (x) - \sin \relax (x) - 1\right )}}{\cos \relax (x) + \sin \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 38, normalized size = 1.81 \[ \frac {1}{2} \, {\left (2 \, \sqrt {\sin \relax (x)^{2} + \sin \relax (x)} - \log \left ({\left | 2 \, \sqrt {\sin \relax (x)^{2} + \sin \relax (x)} - 2 \, \sin \relax (x) - 1 \right |}\right )\right )} \mathrm {sgn}\left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 48, normalized size = 2.29 \[ \frac {1}{2 \sqrt {1+\csc \relax (x )}-2}-\frac {\ln \left (\sqrt {1+\csc \relax (x )}-1\right )}{2}+\frac {1}{2 \sqrt {1+\csc \relax (x )}+2}+\frac {\ln \left (\sqrt {1+\csc \relax (x )}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 38, normalized size = 1.81 \[ \sqrt {\frac {1}{\sin \relax (x)} + 1} \sin \relax (x) + \frac {1}{2} \, \log \left (\sqrt {\frac {1}{\sin \relax (x)} + 1} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {1}{\sin \relax (x)} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.10, size = 47, normalized size = 2.24 \[ \sin \relax (x)\,\sqrt {\frac {1}{\sin \relax (x)}+1}+\frac {\ln \left (\sin \relax (x)+\sqrt {{\sin \relax (x)}^2+\sin \relax (x)}+\frac {1}{2}\right )\,\sin \relax (x)\,\sqrt {\frac {1}{\sin \relax (x)}+1}}{2\,\sqrt {{\sin \relax (x)}^2+\sin \relax (x)}} \]
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 \[ \int \sqrt {\csc {\relax (x )} + 1} \cos {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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