Optimal. Leaf size=17 \[ -\frac {\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt {a \sin ^2(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3207, 3770} \[ -\frac {\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt {a \sin ^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3207
Rule 3770
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \sin ^2(x)}} \, dx &=\frac {\sin (x) \int \csc (x) \, dx}{\sqrt {a \sin ^2(x)}}\\ &=-\frac {\tanh ^{-1}(\cos (x)) \sin (x)}{\sqrt {a \sin ^2(x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 30, normalized size = 1.76 \[ \frac {\sin (x) \left (\log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )\right )\right )}{\sqrt {a \sin ^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 70, normalized size = 4.12 \[ \left [\frac {\sqrt {-a \cos \relax (x)^{2} + a} \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right )}{2 \, a \sin \relax (x)}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-a \cos \relax (x)^{2} + a} \sqrt {-a} \cos \relax (x)}{a \sin \relax (x)}\right )}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 15, normalized size = 0.88 \[ \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\sin \relax (x)\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.90, size = 49, normalized size = 2.88 \[ -\frac {\sin \relax (x ) \sqrt {a \left (\cos ^{2}\relax (x )\right )}\, \ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cos ^{2}\relax (x )\right )}+2 a}{\sin \relax (x )}\right )}{\sqrt {a}\, \cos \relax (x ) \sqrt {a \left (\sin ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.63, size = 26, normalized size = 1.53 \[ \frac {\sqrt {-a} {\left (\arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) - \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {a\,{\sin \relax (x)}^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \sin ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________