3.1.9 \(\int \sqrt {1+\cot ^2(x)} \, dx\) [9]

Optimal. Leaf size=5 \[ -\sinh ^{-1}(\cot (x)) \]

[Out]

-arcsinh(cot(x))

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Rubi [A]
time = 0.01, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3738, 4207, 221} \begin {gather*} -\sinh ^{-1}(\cot (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Cot[x]^2],x]

[Out]

-ArcSinh[Cot[x]]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 3738

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 4207

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[b*(ff/
f), Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rubi steps

\begin {align*} \int \sqrt {1+\cot ^2(x)} \, dx &=\int \sqrt {\csc ^2(x)} \, dx\\ &=-\text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\cot (x)\right )\\ &=-\sinh ^{-1}(\cot (x))\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(28\) vs. \(2(5)=10\).
time = 0.02, size = 28, normalized size = 5.60 \begin {gather*} \sqrt {\csc ^2(x)} \left (-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sin (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Cot[x]^2],x]

[Out]

Sqrt[Csc[x]^2]*(-Log[Cos[x/2]] + Log[Sin[x/2]])*Sin[x]

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Maple [A]
time = 0.20, size = 6, normalized size = 1.20

method	result	size
derivativedivides	\(-\arcsinh \left (\cot \left (x \right )\right )\)	\(6\)
default	\(-\arcsinh \left (\cot \left (x \right )\right )\)	\(6\)
risch	\(2 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )-2 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+cot(x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-arcsinh(cot(x))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (5) = 10\).
time = 0.60, size = 35, normalized size = 7.00 \begin {gather*} -\frac {1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac {1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cot(x)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/2*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 1/2*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (5) = 10\).
time = 2.26, size = 53, normalized size = 10.60 \begin {gather*} -\frac {1}{2} \, \log \left (\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) + \frac {1}{2} \, \log \left (-\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cot(x)^2)^(1/2),x, algorithm="fricas")

[Out]

-1/2*log(1/2*sqrt(2)*sqrt(-1/(cos(2*x) - 1))*sin(2*x) + 1) + 1/2*log(-1/2*sqrt(2)*sqrt(-1/(cos(2*x) - 1))*sin(
2*x) + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\cot ^{2}{\left (x \right )} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cot(x)**2)**(1/2),x)

[Out]

Integral(sqrt(cot(x)**2 + 1), x)

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Giac [A]
time = 0.45, size = 10, normalized size = 2.00 \begin {gather*} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cot(x)^2)^(1/2),x, algorithm="giac")

[Out]

log(abs(tan(1/2*x)))*sgn(sin(x))

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Mupad [B]
time = 0.32, size = 5, normalized size = 1.00 \begin {gather*} -\mathrm {asinh}\left (\mathrm {cot}\left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(x)^2 + 1)^(1/2),x)

[Out]

-asinh(cot(x))

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