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]

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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 30

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

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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.67, size = 15, normalized size = 0.94 \[ -\frac {1}{2} \, b x^{2} + \frac {1}{2} \, {\left (\pi - 2 \, a\right )} x \]

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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giac [A]  time = 0.11, size = 15, normalized size = 0.94 \[ -\frac {1}{2} \, b x^{2} + \frac {1}{2} \, \pi x - a x \]

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

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maple [B]  time = 0.04, size = 51, normalized size = 3.19 \[ \frac {\pi x}{2}-\frac {-\left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (b x +a \right )\right )\right ) \mathrm {arccot}\left (\cot \left (b x +a \right )\right )-\frac {\left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (b x +a \right )\right )\right )^{2}}{2}}{b} \]

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 = 0.32, size = 15, normalized size = 0.94 \[ -\frac {1}{2} \, b x^{2} + \frac {1}{2} \, \pi x - a x \]

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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mupad [B]  time = 0.07, size = 21, normalized size = 1.31 \[ \frac {\Pi \,x}{2}-x\,\mathrm {acot}\left (\mathrm {cot}\left (a+b\,x\right )\right )+\frac {b\,x^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Pi/2 - acot(cot(a + b*x)),x)

[Out]

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

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sympy [A]  time = 0.13, size = 24, normalized size = 1.50 \[ \frac {\pi x}{2} - \begin {cases} \frac {\operatorname {acot}^{2}{\left (\cot {\left (a + b x \right )} \right )}}{2 b} & \text {for}\: b \neq 0 \\x \operatorname {acot}{\left (\cot {\relax (a )} \right )} & \text {otherwise} \end {cases} \]

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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