\(\int \cot ^{-1}(\cot (a+b x)) \, dx\) [170]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [A] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 7, antiderivative size = 16 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=\frac {\cot ^{-1}(\cot (a+b x))^2}{2 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2188, 30} \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=\frac {\cot ^{-1}(\cot (a+b x))^2}{2 b} \]

[In]

Int[ArcCot[Cot[a + b*x]],x]

[Out]

ArcCot[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 2188

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=-\frac {b x^2}{2}+x \cot ^{-1}(\cot (a+b x)) \]

[In]

Integrate[ArcCot[Cot[a + b*x]],x]

[Out]

-1/2*(b*x^2) + x*ArcCot[Cot[a + b*x]]

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method result size
parallelrisch \(-\frac {x^{2} b}{2}+x \,\operatorname {arccot}\left (\cot \left (b x +a \right )\right )\) \(17\)
parts \(x \,\operatorname {arccot}\left (\cot \left (b x +a \right )\right )+\frac {-\frac {\left (b x +a \right )^{2}}{2}+\left (b x +a \right ) a}{b}\) \(32\)
derivativedivides \(\frac {-\left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (b x +a \right )\right )\right ) \operatorname {arccot}\left (\cot \left (b x +a \right )\right )-\frac {\left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (b x +a \right )\right )\right )^{2}}{2}}{b}\) \(45\)
default \(\frac {-\left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (b x +a \right )\right )\right ) \operatorname {arccot}\left (\cot \left (b x +a \right )\right )-\frac {\left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (b x +a \right )\right )\right )^{2}}{2}}{b}\) \(45\)
risch \(-i x \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )-\frac {\pi x \operatorname {csgn}\left (i {\mathrm e}^{i \left (b x +a \right )}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 i \left (b x +a \right )}\right )}{4}+\frac {\pi x \,\operatorname {csgn}\left (i {\mathrm e}^{i \left (b x +a \right )}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 i \left (b x +a \right )}\right )^{2}}{2}-\frac {\pi x \operatorname {csgn}\left (i {\mathrm e}^{2 i \left (b x +a \right )}\right )^{3}}{4}-\frac {\pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i \left (b x +a \right )}\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i \left (b x +a \right )}}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right ) x}{4}+\frac {\pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i \left (b x +a \right )}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i \left (b x +a \right )}}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right )^{2} x}{4}+\frac {\pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right )^{2} x}{2}-\frac {\pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right )^{3} x}{2}+\frac {\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i \left (b x +a \right )}}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right )^{2} x}{4}-\frac {\pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i \left (b x +a \right )}}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right )^{3} x}{4}-\frac {x^{2} b}{2}\) \(337\)

[In]

int(arccot(cot(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=\frac {1}{2} x^{2} b + x a \]

[In]

integrate(arccot(cot(b*x+a)),x, algorithm="fricas")

[Out]

1/2*x^2*b + x*a

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=\begin {cases} \frac {\operatorname {acot}^{2}{\left (\cot {\left (a + b x \right )} \right )}}{2 b} & \text {for}\: b \neq 0 \\x \operatorname {acot}{\left (\cot {\left (a \right )} \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(acot(cot(b*x+a)),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=\frac {1}{2} \, b x^{2} + a x \]

[In]

integrate(arccot(cot(b*x+a)),x, algorithm="maxima")

[Out]

1/2*b*x^2 + a*x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=\frac {1}{2} \, b x^{2} + a x \]

[In]

integrate(arccot(cot(b*x+a)),x, algorithm="giac")

[Out]

1/2*b*x^2 + a*x

Mupad [B] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=x\,\mathrm {acot}\left (\mathrm {cot}\left (a+b\,x\right )\right )-\frac {b\,x^2}{2} \]

[In]

int(acot(cot(a + b*x)),x)

[Out]

x*acot(cot(a + b*x)) - (b*x^2)/2