Optimal. Leaf size=14 \[ -\frac{\tanh ^{-1}(\cos (a+b x))}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0163863, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4287, 3770} \[ -\frac{\tanh ^{-1}(\cos (a+b x))}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4287
Rule 3770
Rubi steps
\begin{align*} \int \cos (a+b x) \csc (2 a+2 b x) \, dx &=\frac{1}{2} \int \csc (a+b x) \, dx\\ &=-\frac{\tanh ^{-1}(\cos (a+b x))}{2 b}\\ \end{align*}
Mathematica [B] time = 0.0124396, size = 42, normalized size = 3. \[ \frac{1}{2} \left (\frac{\log \left (\sin \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}-\frac{\log \left (\cos \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.019, size = 22, normalized size = 1.6 \begin{align*}{\frac{\ln \left ( \csc \left ( bx+a \right ) -\cot \left ( bx+a \right ) \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.15764, size = 113, normalized size = 8.07 \begin{align*} -\frac{\log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) - \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.491409, size = 93, normalized size = 6.64 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) - \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right )}{4 \, b} \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 [A] time = 1.19284, size = 22, normalized size = 1.57 \begin{align*} \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) \right |}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]