Optimal. Leaf size=36 \[ \frac{\cot (x)}{2 \sqrt{-\sin ^2(x)}}+\frac{\sin (x) \tanh ^{-1}(\cos (x))}{2 \sqrt{-\sin ^2(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.025746, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3176, 3204, 3207, 3770} \[ \frac{\cot (x)}{2 \sqrt{-\sin ^2(x)}}+\frac{\sin (x) \tanh ^{-1}(\cos (x))}{2 \sqrt{-\sin ^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3176
Rule 3204
Rule 3207
Rule 3770
Rubi steps
\begin{align*} \int \frac{1}{\left (-1+\cos ^2(x)\right )^{3/2}} \, dx &=\int \frac{1}{\left (-\sin ^2(x)\right )^{3/2}} \, dx\\ &=\frac{\cot (x)}{2 \sqrt{-\sin ^2(x)}}-\frac{1}{2} \int \frac{1}{\sqrt{-\sin ^2(x)}} \, dx\\ &=\frac{\cot (x)}{2 \sqrt{-\sin ^2(x)}}-\frac{\sin (x) \int \csc (x) \, dx}{2 \sqrt{-\sin ^2(x)}}\\ &=\frac{\cot (x)}{2 \sqrt{-\sin ^2(x)}}+\frac{\tanh ^{-1}(\cos (x)) \sin (x)}{2 \sqrt{-\sin ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0300801, size = 53, normalized size = 1.47 \[ \frac{\sin (x) \left (\csc ^2\left (\frac{x}{2}\right )-\sec ^2\left (\frac{x}{2}\right )-4 \log \left (\sin \left (\frac{x}{2}\right )\right )+4 \log \left (\cos \left (\frac{x}{2}\right )\right )\right )}{8 \sqrt{-\sin ^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.499, size = 51, normalized size = 1.4 \begin{align*} -{\frac{1}{2\,\cos \left ( x \right ) \sin \left ( x \right ) }\sqrt{- \left ( \cos \left ( x \right ) \right ) ^{2}} \left ( -\arctan \left ({\frac{1}{\sqrt{- \left ( \cos \left ( x \right ) \right ) ^{2}}}} \right ) \left ( \sin \left ( x \right ) \right ) ^{2}+\sqrt{- \left ( \cos \left ( x \right ) \right ) ^{2}} \right ){\frac{1}{\sqrt{- \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.67064, size = 383, normalized size = 10.64 \begin{align*} \frac{{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) -{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \,{\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \,{\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + 4 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \cos \left (x\right ) \sin \left (2 \, x\right ) - 4 \, \cos \left (2 \, x\right ) \sin \left (x\right ) + 2 \, \sin \left (x\right )}{2 \,{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] time = 1.29558, size = 122, normalized size = 3.39 \begin{align*} -\frac{i \, \tan \left (\frac{1}{2} \, x\right )^{2}}{8 \, \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{3} - \tan \left (\frac{1}{2} \, x\right )\right )} - \frac{i \, \log \left (\tan \left (\frac{1}{2} \, x\right )^{2}\right )}{4 \, \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{3} - \tan \left (\frac{1}{2} \, x\right )\right )} + \frac{2 i \, \tan \left (\frac{1}{2} \, x\right )^{2} + i}{8 \, \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{3} - \tan \left (\frac{1}{2} \, x\right )\right ) \tan \left (\frac{1}{2} \, x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]