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 = 0.01, 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 = 0.46, 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 [B] time = 1.16, size = 51, normalized size = 4.25 \[ \frac {\log \left ({\left | -\frac {\cos \left (b x + a\right )}{b} + \frac {1}{{\left | b \right |}} \right |}\right )}{2 \, {\left | b \right |}} - \frac {\log \left ({\left | -\frac {\cos \left (b x + a\right )}{b} - \frac {1}{{\left | b \right |}} \right |}\right )}{2 \, {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 21, normalized size = 1.75 \[ \frac {\ln \left (-\cot \left (b x +a \right )+\csc \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.42, size = 26, normalized size = 2.17 \[ -\frac {\log \left (\cos \left (b x + a\right ) + 1\right ) - \log \left (\cos \left (b x + a\right ) - 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.12, 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 = 0.53, size = 17, normalized size = 1.42 \[ \begin {cases} \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\sin {\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________