Optimal. Leaf size=16 \[ -\frac{1}{2} \tanh ^{-1}(\cos (x))-\frac{1}{2} \cot (x) \csc (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.006974, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3768, 3770} \[ -\frac{1}{2} \tanh ^{-1}(\cos (x))-\frac{1}{2} \cot (x) \csc (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \csc ^3(x) \, dx &=-\frac{1}{2} \cot (x) \csc (x)+\frac{1}{2} \int \csc (x) \, dx\\ &=-\frac{1}{2} \tanh ^{-1}(\cos (x))-\frac{1}{2} \cot (x) \csc (x)\\ \end{align*}
Mathematica [B] time = 0.0045829, size = 47, normalized size = 2.94 \[ -\frac{1}{8} \csc ^2\left (\frac{x}{2}\right )+\frac{1}{8} \sec ^2\left (\frac{x}{2}\right )+\frac{1}{2} \log \left (\sin \left (\frac{x}{2}\right )\right )-\frac{1}{2} \log \left (\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.031, size = 18, normalized size = 1.1 \begin{align*} -{\frac{\cot \left ( x \right ) \csc \left ( x \right ) }{2}}+{\frac{\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.932211, size = 36, normalized size = 2.25 \begin{align*} \frac{\cos \left (x\right )}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} - \frac{1}{4} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{4} \, \log \left (\cos \left (x\right ) - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.04442, size = 150, normalized size = 9.38 \begin{align*} -\frac{{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 2 \, \cos \left (x\right )}{4 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.112021, size = 27, normalized size = 1.69 \begin{align*} \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{4} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{4} + \frac{\cos{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.06475, size = 73, normalized size = 4.56 \begin{align*} -\frac{{\left (\frac{2 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 1\right )}{\left (\cos \left (x\right ) + 1\right )}}{8 \,{\left (\cos \left (x\right ) - 1\right )}} - \frac{\cos \left (x\right ) - 1}{8 \,{\left (\cos \left (x\right ) + 1\right )}} + \frac{1}{4} \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]