Optimal. Leaf size=18 \[ -\tanh ^{-1}\left (\frac {\cot (x)}{\sqrt {-\cot ^2(x)-2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4128, 377, 206} \[ -\tanh ^{-1}\left (\frac {\cot (x)}{\sqrt {-\cot ^2(x)-2}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 377
Rule 4128
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1-\csc ^2(x)}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\sqrt {-2-x^2} \left (1+x^2\right )} \, dx,x,\cot (x)\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\cot (x)}{\sqrt {-2-\cot ^2(x)}}\right )\\ &=-\tanh ^{-1}\left (\frac {\cot (x)}{\sqrt {-2-\cot ^2(x)}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.03, size = 51, normalized size = 2.83 \[ -\frac {\sqrt {\cos (2 x)-3} \csc (x) \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)-3}\right )}{\sqrt {2} \sqrt {-\csc ^2(x)-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.41, size = 65, normalized size = 3.61 \[ \frac {1}{2} \, \log \left (-2 \, \sqrt {e^{\left (4 i \, x\right )} - 6 \, e^{\left (2 i \, x\right )} + 1} {\left (e^{\left (2 i \, x\right )} - 1\right )} + 2 \, e^{\left (4 i \, x\right )} - 8 \, e^{\left (2 i \, x\right )} - 2\right ) - \frac {1}{2} \, \log \left (\sqrt {e^{\left (4 i \, x\right )} - 6 \, e^{\left (2 i \, x\right )} + 1} - e^{\left (2 i \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\csc \relax (x)^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.76, size = 75, normalized size = 4.17 \[ -\frac {\sin \relax (x ) \sqrt {\frac {\cos ^{2}\relax (x )-2}{\left (\cos \relax (x )+1\right )^{2}}}\, \arctanh \left (\frac {\cos \relax (x ) \sqrt {4}\, \left (-1+\cos \relax (x )\right )}{2 \sin \relax (x )^{2} \sqrt {\frac {\cos ^{2}\relax (x )-2}{\left (\cos \relax (x )+1\right )^{2}}}}\right )}{\sqrt {-\frac {\cos ^{2}\relax (x )-2}{-1+\cos ^{2}\relax (x )}}\, \left (-1+\cos \relax (x )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\csc \relax (x)^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {-\frac {1}{{\sin \relax (x)}^2}-1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \csc ^{2}{\relax (x )} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________