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

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

[Out]

-ArcTan[Cot[a + b*x]]^2/(2*b)

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Rubi [A]  time = 0.0030273, 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} \[ -\frac{\tan ^{-1}(\cot (a+b x))^2}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[Cot[a + b*x]],x]

[Out]

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

Mathematica [A]  time = 0.007294, size = 18, normalized size = 1.12 \[ x \tan ^{-1}(\cot (a+b x))+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[Cot[a + b*x]],x]

[Out]

(b*x^2)/2 + x*ArcTan[Cot[a + b*x]]

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Maple [B]  time = 0.056, size = 51, normalized size = 3.2 \begin{align*}{\frac{\pi \,x}{2}}-{\frac{1}{b} \left ( - \left ({\frac{\pi }{2}}-{\rm arccot} \left (\cot \left ( bx+a \right ) \right ) \right ){\rm arccot} \left (\cot \left ( bx+a \right ) \right )-{\frac{1}{2} \left ({\frac{\pi }{2}}-{\rm arccot} \left (\cot \left ( bx+a \right ) \right ) \right ) ^{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.951275, size = 20, normalized size = 1.25 \begin{align*} -\frac{1}{2} \, b x^{2} + \frac{1}{2} \, \pi x - a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*x^2 + 1/2*pi*x - a*x

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Fricas [A]  time = 1.97813, 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-arccot(cot(b*x+a)),x, algorithm="fricas")

[Out]

-1/2*b*x^2 + 1/2*(pi - 2*a)*x

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Sympy [A]  time = 0.150933, size = 24, normalized size = 1.5 \begin{align*} \frac{\pi x}{2} - \begin{cases} \frac{\operatorname{acot}^{2}{\left (\cot{\left (a + b x \right )} \right )}}{2 b} & \text{for}\: b \neq 0 \\x \operatorname{acot}{\left (\cot{\left (a \right )} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

pi*x/2 - Piecewise((acot(cot(a + b*x))**2/(2*b), Ne(b, 0)), (x*acot(cot(a)), True))

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Giac [A]  time = 1.08506, size = 20, normalized size = 1.25 \begin{align*} -\frac{1}{2} \, b x^{2} + \frac{1}{2} \, \pi x - a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*x^2 + 1/2*pi*x - a*x