Optimal. Leaf size=19 \[ \log (x) \left (\tan ^{-1}(\cot (a+b x))+b x\right )-b x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0327329, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2158, 29} \[ \log (x) \left (\tan ^{-1}(\cot (a+b x))+b x\right )-b x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2158
Rule 29
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(\cot (a+b x))}{x} \, dx &=-b x-\left (-b x-\tan ^{-1}(\cot (a+b x))\right ) \int \frac{1}{x} \, dx\\ &=-b x+\left (b x+\tan ^{-1}(\cot (a+b x))\right ) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0154947, size = 19, normalized size = 1. \[ \log (x) \left (\tan ^{-1}(\cot (a+b x))+b x\right )-b x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.059, size = 35, normalized size = 1.8 \begin{align*}{\frac{\pi \,\ln \left ( x \right ) }{2}}-bx-\ln \left ( x \right ) a-\ln \left ( x \right ) \left ({\rm arccot} \left (\cot \left ( bx+a \right ) \right )-bx-a \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.949681, size = 19, normalized size = 1. \begin{align*} -b x + \frac{1}{2} \,{\left (\pi - 2 \, a\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.02123, size = 41, normalized size = 2.16 \begin{align*} -b x + \frac{1}{2} \,{\left (\pi - 2 \, a\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int - \frac{\pi }{x}\, dx + \int \frac{2 \operatorname{acot}{\left (\cot{\left (a + b x \right )} \right )}}{x}\, dx}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11118, size = 20, normalized size = 1.05 \begin{align*} -b x + \frac{1}{2} \,{\left (\pi - 2 \, a\right )} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]