3.46 \(\int \frac{\tan ^{-1}(\cot (a+b x))}{x} \, dx\)

Optimal. Leaf size=19 \[ \log (x) \left (\tan ^{-1}(\cot (a+b x))+b x\right )-b x \]

[Out]

-(b*x) + (b*x + ArcTan[Cot[a + b*x]])*Log[x]

________________________________________________________________________________________

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]

Int[ArcTan[Cot[a + b*x]]/x,x]

[Out]

-(b*x) + (b*x + ArcTan[Cot[a + b*x]])*Log[x]

Rule 2158

Int[(v_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(b*x)/a, x] - Dist[(b*u
- a*v)/a, Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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]

Integrate[ArcTan[Cot[a + b*x]]/x,x]

[Out]

-(b*x) + (b*x + ArcTan[Cot[a + b*x]])*Log[x]

________________________________________________________________________________________

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]

int((1/2*Pi-arccot(cot(b*x+a)))/x,x)

[Out]

1/2*Pi*ln(x)-b*x-ln(x)*a-ln(x)*(arccot(cot(b*x+a))-b*x-a)

________________________________________________________________________________________

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]

integrate((1/2*pi-arccot(cot(b*x+a)))/x,x, algorithm="maxima")

[Out]

-b*x + 1/2*(pi - 2*a)*log(x)

________________________________________________________________________________________

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]

integrate((1/2*pi-arccot(cot(b*x+a)))/x,x, algorithm="fricas")

[Out]

-b*x + 1/2*(pi - 2*a)*log(x)

________________________________________________________________________________________

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]

integrate((1/2*pi-acot(cot(b*x+a)))/x,x)

[Out]

-(Integral(-pi/x, x) + Integral(2*acot(cot(a + b*x))/x, x))/2

________________________________________________________________________________________

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]

integrate((1/2*pi-arccot(cot(b*x+a)))/x,x, algorithm="giac")

[Out]

-b*x + 1/2*(pi - 2*a)*log(abs(x))