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

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

[Out]

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

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Rubi [A]
time = 0.01, 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 = {2188, 30} \begin {gather*} \frac {\cot ^{-1}(\cot (a+b x))^2}{2 b} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \cot ^{-1}(\cot (a+b x)) \, dx &=\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*}

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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.12 \begin {gather*} -\frac {b x^2}{2}+x \cot ^{-1}(\cot (a+b x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[Cot[a + b*x]],x]

[Out]

-1/2*(b*x^2) + x*ArcCot[Cot[a + b*x]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(44\) vs. \(2(14)=28\).
time = 0.18, size = 45, normalized size = 2.81

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 = 0.25, size = 10, normalized size = 0.62 \begin {gather*} \frac {1}{2} \, b x^{2} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + a*x

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Fricas [A]
time = 3.88, size = 10, normalized size = 0.62 \begin {gather*} \frac {1}{2} x^{2} b + x a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*b + x*a

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Sympy [A]
time = 0.08, size = 19, normalized size = 1.19 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]
time = 0.41, size = 10, normalized size = 0.62 \begin {gather*} \frac {1}{2} \, b x^{2} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + a*x

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Mupad [B]
time = 0.64, size = 16, normalized size = 1.00 \begin {gather*} x\,\mathrm {acot}\left (\mathrm {cot}\left (a+b\,x\right )\right )-\frac {b\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(cot(a + b*x)),x)

[Out]

x*acot(cot(a + b*x)) - (b*x^2)/2

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