Optimal. Leaf size=23 \[ \frac {\cos (a+b x)}{b}-\frac {\tanh ^{-1}(\cos (a+b x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2592, 321, 206} \[ \frac {\cos (a+b x)}{b}-\frac {\tanh ^{-1}(\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 321
Rule 2592
Rubi steps
\begin {align*} \int \cos (a+b x) \cot (a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (a+b x)\right )}{b}\\ &=\frac {\cos (a+b x)}{b}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\tanh ^{-1}(\cos (a+b x))}{b}+\frac {\cos (a+b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 42, normalized size = 1.83 \[ \frac {\cos (a+b x)}{b}+\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{b}-\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 38, normalized size = 1.65 \[ \frac {2 \, \cos \left (b x + a\right ) - \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.24, size = 57, normalized size = 2.48 \[ -\frac {\frac {4}{\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1} - \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 32, normalized size = 1.39 \[ \frac {\cos \left (b x +a \right )}{b}+\frac {\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.36, size = 34, normalized size = 1.48 \[ \frac {2 \, \cos \left (b x + a\right ) - \log \left (\cos \left (b x + a\right ) + 1\right ) + \log \left (\cos \left (b x + a\right ) - 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.46, size = 35, normalized size = 1.52 \[ \frac {2}{b\,\left ({\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2+1\right )}+\frac {\ln \left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.31, size = 92, normalized size = 4.00 \[ \begin {cases} \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )} \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} + \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )}}{b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} + \frac {2}{b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} & \text {for}\: b \neq 0 \\\frac {x \cos ^{2}{\relax (a )}}{\sin {\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________