3.18 \(\int \frac{1}{\sqrt{1+\csc ^2(x)}} \, dx\)

Optimal. Leaf size=16 \[ -\tan ^{-1}\left (\frac{\cot (x)}{\sqrt{\cot ^2(x)+2}}\right ) \]

[Out]

-ArcTan[Cot[x]/Sqrt[2 + Cot[x]^2]]

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Rubi [A]  time = 0.0169859, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4128, 377, 203} \[ -\tan ^{-1}\left (\frac{\cot (x)}{\sqrt{\cot ^2(x)+2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + Csc[x]^2],x]

[Out]

-ArcTan[Cot[x]/Sqrt[2 + Cot[x]^2]]

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 203

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

Rubi steps

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

Mathematica [B]  time = 0.0508819, size = 49, normalized size = 3.06 \[ -\frac{\sqrt{\cos (2 x)-3} \csc (x) \log \left (\sqrt{2} \cos (x)+\sqrt{\cos (2 x)-3}\right )}{\sqrt{2} \sqrt{\csc ^2(x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + Csc[x]^2],x]

[Out]

-((Sqrt[-3 + Cos[2*x]]*Csc[x]*Log[Sqrt[2]*Cos[x] + Sqrt[-3 + Cos[2*x]]])/(Sqrt[2]*Sqrt[1 + Csc[x]^2]))

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Maple [B]  time = 0.143, size = 72, normalized size = 4.5 \begin{align*} -{\frac{\sin \left ( x \right ) }{-1+\cos \left ( x \right ) }\sqrt{-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-2}{ \left ( \cos \left ( x \right ) +1 \right ) ^{2}}}}\arctan \left ({\frac{\cos \left ( x \right ) \left ( -1+\cos \left ( x \right ) \right ) }{ \left ( \sin \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-2}{ \left ( \cos \left ( x \right ) +1 \right ) ^{2}}}}}}} \right ){\frac{1}{\sqrt{{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-2}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.87107, size = 524, normalized size = 32.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan2(2*(-2*(6*cos(2*x) - 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 - 12*sin(4*x)*sin(2*x) +
36*sin(2*x)^2 - 12*cos(2*x) + 1)^(1/4)*sin(1/2*arctan2(sin(4*x) - 6*sin(2*x), cos(4*x) - 6*cos(2*x) + 1)), 2*
(-2*(6*cos(2*x) - 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 - 12*sin(4*x)*sin(2*x) + 36*sin(2*x)^2
- 12*cos(2*x) + 1)^(1/4)*cos(1/2*arctan2(sin(4*x) - 6*sin(2*x), cos(4*x) - 6*cos(2*x) + 1)) - 4) + 1/2*arctan
2(2*(-2*(6*cos(2*x) - 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 - 12*sin(4*x)*sin(2*x) + 36*sin(2*
x)^2 - 12*cos(2*x) + 1)^(1/4)*sin(1/2*arctan2(sin(4*x) - 6*sin(2*x), cos(4*x) - 6*cos(2*x) + 1)) + 2*sin(2*x),
2*(-2*(6*cos(2*x) - 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 - 12*sin(4*x)*sin(2*x) + 36*sin(2*x
)^2 - 12*cos(2*x) + 1)^(1/4)*cos(1/2*arctan2(sin(4*x) - 6*sin(2*x), cos(4*x) - 6*cos(2*x) + 1)) + 2*cos(2*x) -
6)

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Fricas [B]  time = 0.493589, size = 203, normalized size = 12.69 \begin{align*} \frac{1}{2} \, \arctan \left (\frac{{\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sqrt{\frac{\cos \left (x\right )^{2} - 2}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right ) - \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2} + 1}\right ) - \frac{1}{2} \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(((cos(x)^3 - cos(x))*sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1))*sin(x) - cos(x)*sin(x))/(cos(x)^4 - 3*cos(
x)^2 + 1)) - 1/2*arctan(sin(x)/cos(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(csc(x)**2 + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\csc \left (x\right )^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(csc(x)^2 + 1), x)