Optimal. Leaf size=16 \[ \frac {\cos (x) \tanh ^{-1}(\sin (x))}{\sqrt {a \cos ^2(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 16, 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 = {3176, 3207, 3770} \[ \frac {\cos (x) \tanh ^{-1}(\sin (x))}{\sqrt {a \cos ^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3176
Rule 3207
Rule 3770
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a-a \sin ^2(x)}} \, dx &=\int \frac {1}{\sqrt {a \cos ^2(x)}} \, dx\\ &=\frac {\cos (x) \int \sec (x) \, dx}{\sqrt {a \cos ^2(x)}}\\ &=\frac {\tanh ^{-1}(\sin (x)) \cos (x)}{\sqrt {a \cos ^2(x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.02, size = 46, normalized size = 2.88 \[ \frac {\cos (x) \left (\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )\right )}{\sqrt {a \cos ^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.43, size = 65, normalized size = 4.06 \[ \left [-\frac {\sqrt {a \cos \relax (x)^{2}} \log \left (-\frac {\sin \relax (x) - 1}{\sin \relax (x) + 1}\right )}{2 \, a \cos \relax (x)}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {a \cos \relax (x)^{2}} \sqrt {-a} \sin \relax (x)}{a \cos \relax (x)}\right )}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-a \sin \relax (x)^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.12, size = 20, normalized size = 1.25 \[ \frac {\cos \relax (x ) \mathrm {am}^{-1}\left (x | 1\right )}{\sqrt {a \left (\cos ^{2}\relax (x )\right )}\, \mathrm {csgn}\left (\cos \relax (x )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.51, size = 38, normalized size = 2.38 \[ \frac {\log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) - \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right )}{2 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {a-a\,{\sin \relax (x)}^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- a \sin ^{2}{\relax (x )} + a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________