Optimal. Leaf size=12 \[ -\frac {\tanh ^{-1}(\cos (a+b x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3770} \[ -\frac {\tanh ^{-1}(\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3770
Rubi steps
\begin {align*} \int \csc (a+b x) \, dx &=-\frac {\tanh ^{-1}(\cos (a+b x))}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.02, size = 38, normalized size = 3.17 \[ \frac {\log \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}-\frac {\log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.67, size = 30, normalized size = 2.50 \[ -\frac {\log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 15, normalized size = 1.25 \[ \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 20, normalized size = 1.67 \[ -\frac {\ln \left (\csc \left (b x +a \right )+\cot \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 19, normalized size = 1.58 \[ -\frac {\log \left (\cot \left (b x + a\right ) + \csc \left (b x + a\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.40, size = 12, normalized size = 1.00 \[ -\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.93, size = 37, normalized size = 3.08 \[ \begin {cases} - \frac {\log {\left (\cot {\left (a + b x \right )} + \csc {\left (a + b x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x \left (\cot {\relax (a )} \csc {\relax (a )} + \csc ^{2}{\relax (a )}\right )}{\cot {\relax (a )} + \csc {\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________