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

Optimal. Leaf size=15 \[ -\frac{\cot (x) \log (\cos (x))}{\sqrt{\cot ^2(x)}} \]

[Out]

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

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Rubi [A]  time = 0.0185607, antiderivative size = 15, 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 = {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 = 0.0087518, size = 15, 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])

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Maple [B]  time = 0.134, size = 68, normalized size = 4.5 \begin{align*} -{\frac{\sqrt{4}\cos \left ( x \right ) }{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*4^(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)/(-cos(x)
^2/(cos(x)^2-1))^(1/2)/sin(x)

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Maxima [A]  time = 1.68719, size = 12, normalized size = 0.8 \begin{align*} \frac{1}{2} \, \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*log(tan(x)^2 + 1)

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Fricas [A]  time = 0.485637, size = 19, normalized size = 1.27 \begin{align*} \log \left (-\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]

log(-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)