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

Optimal. Leaf size=16 \[ -\frac{\cot ^{-1}(\tan (a+b x))^2}{2 b} \]

[Out]

-ArcCot[Tan[a + b*x]]^2/(2*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0082204, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2157, 30} \[ -\frac{\cot ^{-1}(\tan (a+b x))^2}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[Tan[a + b*x]],x]

[Out]

-ArcCot[Tan[a + b*x]]^2/(2*b)

Rule 2157

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \cot ^{-1}(\tan (a+b x)) \, dx &=-\frac{\operatorname{Subst}\left (\int x \, dx,x,\cot ^{-1}(\tan (a+b x))\right )}{b}\\ &=-\frac{\cot ^{-1}(\tan (a+b x))^2}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0053343, size = 18, normalized size = 1.12 \[ x \cot ^{-1}(\tan (a+b x))+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[Tan[a + b*x]],x]

[Out]

(b*x^2)/2 + x*ArcCot[Tan[a + b*x]]

________________________________________________________________________________________

Maple [A]  time = 0.042, size = 20, normalized size = 1.3 \begin{align*}{\frac{\pi \,x}{2}}-{\frac{ \left ( \arctan \left ( \tan \left ( bx+a \right ) \right ) \right ) ^{2}}{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*Pi-arctan(tan(b*x+a)),x)

[Out]

1/2*Pi*x-1/2/b*arctan(tan(b*x+a))^2

________________________________________________________________________________________

Maxima [A]  time = 0.975565, size = 23, normalized size = 1.44 \begin{align*} \frac{1}{2} \, \pi x - \frac{{\left (b x + a\right )}^{2}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*pi*x - 1/2*(b*x + a)^2/b

________________________________________________________________________________________

Fricas [A]  time = 2.35328, size = 42, normalized size = 2.62 \begin{align*} -\frac{1}{2} \, b x^{2} + \frac{1}{2} \,{\left (\pi - 2 \, a\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*x^2 + 1/2*(pi - 2*a)*x

________________________________________________________________________________________

Sympy [B]  time = 0.167083, size = 48, normalized size = 3. \begin{align*} \frac{\pi x}{2} - \begin{cases} \frac{\left (\operatorname{atan}{\left (\tan{\left (a + b x \right )} \right )} + \pi \left \lfloor{\frac{a + b x - \frac{\pi }{2}}{\pi }}\right \rfloor \right )^{2}}{2 b} & \text{for}\: b \neq 0 \\x \left (\operatorname{atan}{\left (\tan{\left (a \right )} \right )} + \pi \left \lfloor{\frac{a - \frac{\pi }{2}}{\pi }}\right \rfloor \right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*pi-atan(tan(b*x+a)),x)

[Out]

pi*x/2 - Piecewise(((atan(tan(a + b*x)) + pi*floor((a + b*x - pi/2)/pi))**2/(2*b), Ne(b, 0)), (x*(atan(tan(a))
+ pi*floor((a - pi/2)/pi)), True))

________________________________________________________________________________________

Giac [A]  time = 1.12254, size = 41, normalized size = 2.56 \begin{align*} -\frac{1}{2} \, b x^{2} + \pi x \left \lfloor \frac{b x + a}{\pi } + \frac{1}{2} \right \rfloor + \frac{1}{2} \, \pi x - a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*x^2 + pi*x*floor((b*x + a)/pi + 1/2) + 1/2*pi*x - a*x