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

Optimal. Leaf size=28 \[ -\tan ^{-1}\left (\frac{\cot (x)}{\sqrt{\cot ^2(x)+2}}\right )-\sinh ^{-1}\left (\frac{\cot (x)}{\sqrt{2}}\right ) \]

[Out]

-ArcSinh[Cot[x]/Sqrt[2]] - ArcTan[Cot[x]/Sqrt[2 + Cot[x]^2]]

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Rubi [A]  time = 0.0211046, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4128, 402, 215, 377, 203} \[ -\tan ^{-1}\left (\frac{\cot (x)}{\sqrt{\cot ^2(x)+2}}\right )-\sinh ^{-1}\left (\frac{\cot (x)}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Csc[x]^2],x]

[Out]

-ArcSinh[Cot[x]/Sqrt[2]] - 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 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

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

Mathematica [B]  time = 0.0557428, size = 68, normalized size = 2.43 \[ \frac{\sqrt{2} \sin (x) \sqrt{\csc ^2(x)+1} \left (\log \left (\sqrt{2} \cos (x)+\sqrt{\cos (2 x)-3}\right )+\tan ^{-1}\left (\frac{\sqrt{2} \cos (x)}{\sqrt{\cos (2 x)-3}}\right )\right )}{\sqrt{\cos (2 x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Csc[x]^2],x]

[Out]

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

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Maple [B]  time = 0.184, size = 166, normalized size = 5.9 \begin{align*} -{\frac{\sqrt{4} \left ( -1+\cos \left ( x \right ) \right ) }{4\,\sin \left ( x \right ) }\sqrt{{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-2}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}}} \left ( \ln \left ( -2\,{\frac{1}{ \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt{-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-2}{ \left ( \cos \left ( x \right ) +1 \right ) ^{2}}}}+ \left ( \cos \left ( x \right ) \right ) ^{2}+\cos \left ( x \right ) -\sqrt{-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-2}{ \left ( \cos \left ( x \right ) +1 \right ) ^{2}}}}-2 \right ) } \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 ) -{\it Artanh} \left ({\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-3\,\cos \left ( x \right ) +2}{ \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 ) \right ){\frac{1}{\sqrt{-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}-2}{ \left ( \cos \left ( x \right ) +1 \right ) ^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*4^(1/2)*((cos(x)^2-2)/(cos(x)^2-1))^(1/2)*(-1+cos(x))*(ln(-2*(cos(x)^2*(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)
+cos(x)^2+cos(x)-(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)-2)/sin(x)^2)+2*arctan(cos(x)*(-1+cos(x))/(-(cos(x)^2-2)/(c
os(x)+1)^2)^(1/2)/sin(x)^2)-arctanh((cos(x)^2-3*cos(x)+2)/(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)/sin(x)^2))/sin(x)
/(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(csc(x)^2 + 1), x)

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Fricas [B]  time = 0.51618, size = 500, normalized size = 17.86 \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 ) - \frac{1}{2} \, \log \left (-\cos \left (x\right )^{2} + \cos \left (x\right ) \sin \left (x\right ) -{\left (\cos \left (x\right )^{2} - \cos \left (x\right ) \sin \left (x\right ) - 1\right )} \sqrt{\frac{\cos \left (x\right )^{2} - 2}{\cos \left (x\right )^{2} - 1}} + 2\right ) + \frac{1}{2} \, \log \left (-\cos \left (x\right )^{2} - \cos \left (x\right ) \sin \left (x\right ) -{\left (\cos \left (x\right )^{2} + \cos \left (x\right ) \sin \left (x\right ) - 1\right )} \sqrt{\frac{\cos \left (x\right )^{2} - 2}{\cos \left (x\right )^{2} - 1}} + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(csc(x)**2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(csc(x)^2 + 1), x)