3.42 \(\int \sqrt {\csc ^2(x)} \, dx\)

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 = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4122, 215} \[ -\sinh ^{-1}(\cot (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Csc[x]^2],x]

[Out]

-ArcSinh[Cot[x]]

Rule 215

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 4122

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 {\csc ^2(x)} \, dx &=-\operatorname {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]  time = 0.01, size = 28, normalized size = 5.60 \[ \sin (x) \sqrt {\csc ^2(x)} \left (\log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Csc[x]^2],x]

[Out]

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

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fricas [B]  time = 0.54, size = 19, normalized size = 3.80 \[ -\frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.30, size = 12, normalized size = 2.40 \[ \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{\mathrm {sgn}\left (\sin \relax (x)\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.61, size = 31, normalized size = 6.20 \[ \frac {\sin \relax (x ) \sqrt {-\frac {1}{-1+\cos ^{2}\relax (x )}}\, \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right ) \sqrt {4}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((csc(x)^2)^(1/2),x)

[Out]

1/2*sin(x)*(-1/(-1+cos(x)^2))^(1/2)*ln(-(-1+cos(x))/sin(x))*4^(1/2)

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maxima [B]  time = 0.58, size = 35, normalized size = 7.00 \[ -\frac {1}{2} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \frac {1}{2} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csc(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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mupad [F]  time = 0.00, size = -1, normalized size = -0.20 \[ \int \sqrt {\frac {1}{{\sin \relax (x)}^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sin(x)^2)^(1/2),x)

[Out]

int((1/sin(x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\csc ^{2}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csc(x)**2)**(1/2),x)

[Out]

Integral(sqrt(csc(x)**2), x)

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