3.21 \(\int \frac{1}{\sqrt{1-\csc ^2(x)}} \, dx\)

Optimal. Leaf size=17 \[ -\frac{\cot (x) \log (\cos (x))}{\sqrt{-\cot ^2(x)}} \]

[Out]

-((Cot[x]*Log[Cos[x]])/Sqrt[-Cot[x]^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0243337, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4121, 3658, 3475} \[ -\frac{\cot (x) \log (\cos (x))}{\sqrt{-\cot ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 - Csc[x]^2],x]

[Out]

-((Cot[x]*Log[Cos[x]])/Sqrt[-Cot[x]^2])

Rule 4121

Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(b*tan[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1-\csc ^2(x)}} \, dx &=\int \frac{1}{\sqrt{-\cot ^2(x)}} \, dx\\ &=\frac{\cot (x) \int \tan (x) \, dx}{\sqrt{-\cot ^2(x)}}\\ &=-\frac{\cot (x) \log (\cos (x))}{\sqrt{-\cot ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.008956, size = 17, normalized size = 1. \[ -\frac{\cot (x) \log (\cos (x))}{\sqrt{-\cot ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - Csc[x]^2],x]

[Out]

-((Cot[x]*Log[Cos[x]])/Sqrt[-Cot[x]^2])

________________________________________________________________________________________

Maple [B]  time = 0.162, size = 67, normalized size = 3.9 \begin{align*} -{\frac{\cos \left ( x \right ) \sqrt{4}}{2\,\sin \left ( x \right ) } \left ( -\ln \left ( 2\, \left ( \cos \left ( x \right ) +1 \right ) ^{-1} \right ) +\ln \left ( -{\frac{-1+\cos \left ( x \right ) -\sin \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +\ln \left ( -{\frac{-1+\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }} \right ) \right ){\frac{1}{\sqrt{{\frac{ \left ( \cos \left ( x \right ) \right ) ^{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]

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

________________________________________________________________________________________

Maxima [C]  time = 1.49634, size = 12, normalized size = 0.71 \begin{align*} -\frac{1}{2} i \, \log \left (\tan \left (x\right )^{2} + 1\right ) \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*I*log(tan(x)^2 + 1)

________________________________________________________________________________________

Fricas [A]  time = 0.487555, size = 36, normalized size = 2.12 \begin{align*} x - \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]

x - arctan(sin(x)/cos(x))

________________________________________________________________________________________

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

________________________________________________________________________________________

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)