Optimal. Leaf size=32 \[ \frac{\sqrt{\cos ^2(x)+1} F\left (\left .x+\frac{\pi }{2}\right |-1\right )}{\sqrt{-\cos ^2(x)-1}} \]
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Rubi [A] time = 0.0183404, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3183, 3182} \[ \frac{\sqrt{\cos ^2(x)+1} F\left (\left .x+\frac{\pi }{2}\right |-1\right )}{\sqrt{-\cos ^2(x)-1}} \]
Antiderivative was successfully verified.
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Rule 3183
Rule 3182
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1-\cos ^2(x)}} \, dx &=\frac{\sqrt{1+\cos ^2(x)} \int \frac{1}{\sqrt{1+\cos ^2(x)}} \, dx}{\sqrt{-1-\cos ^2(x)}}\\ &=\frac{\sqrt{1+\cos ^2(x)} F\left (\left .\frac{\pi }{2}+x\right |-1\right )}{\sqrt{-1-\cos ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0403404, size = 33, normalized size = 1.03 \[ \frac{\sqrt{\cos (2 x)+3} F\left (x\left |\frac{1}{2}\right .\right )}{\sqrt{2} \sqrt{-\cos (2 x)-3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.943, size = 62, normalized size = 1.9 \begin{align*}{\frac{i{\it EllipticF} \left ( i\cos \left ( x \right ) ,i \right ) }{\sin \left ( x \right ) }\sqrt{- \left ( 1+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \left ( \sin \left ( x \right ) \right ) ^{2}}\sqrt{1+ \left ( \cos \left ( x \right ) \right ) ^{2}}\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{4}-1}}}{\frac{1}{\sqrt{-1- \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\cos \left (x\right )^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{2}{\sqrt{e^{\left (4 i \, x\right )} + 6 \, e^{\left (2 i \, x\right )} + 1}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \cos ^{2}{\left (x \right )} - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\cos \left (x\right )^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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