3.27 \(\int \frac{1}{\sqrt{a \cot ^2(x)}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0117983, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3658, 3475} \[ -\frac{\cot (x) \log (\cos (x))}{\sqrt{a \cot ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Cot[x]^2],x]

[Out]

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

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

Mathematica [A]  time = 0.007814, size = 17, normalized size = 1. \[ -\frac{\cot (x) \log (\cos (x))}{\sqrt{a \cot ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Cot[x]^2],x]

[Out]

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

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Maple [A]  time = 0.086, size = 28, normalized size = 1.7 \begin{align*} -{\frac{\cot \left ( x \right ) \left ( -\ln \left ( \left ( \cot \left ( x \right ) \right ) ^{2}+1 \right ) +2\,\ln \left ( \cot \left ( x \right ) \right ) \right ) }{2}{\frac{1}{\sqrt{a \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cot(x)^2)^(1/2),x)

[Out]

-1/2*cot(x)*(-ln(cot(x)^2+1)+2*ln(cot(x)))/(a*cot(x)^2)^(1/2)

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Maxima [A]  time = 1.55324, size = 16, normalized size = 0.94 \begin{align*} \frac{\log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/2*log(tan(x)^2 + 1)/sqrt(a)

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Fricas [B]  time = 1.67252, size = 128, normalized size = 7.53 \begin{align*} -\frac{\sqrt{-\frac{a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}} \log \left (\frac{1}{2} \, \cos \left (2 \, x\right ) + \frac{1}{2}\right ) \sin \left (2 \, x\right )}{2 \,{\left (a \cos \left (2 \, x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)^2)^(1/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(-(a*cos(2*x) + a)/(cos(2*x) - 1))*log(1/2*cos(2*x) + 1/2)*sin(2*x)/(a*cos(2*x) + a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)**2)**(1/2),x)

[Out]

Integral(1/sqrt(a*cot(x)**2), x)

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Giac [A]  time = 1.26688, size = 26, normalized size = 1.53 \begin{align*} -\frac{\log \left ({\left | \cos \left (x\right ) \right |}\right )}{\sqrt{a} \mathrm{sgn}\left (\cos \left (x\right )\right ) \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)^2)^(1/2),x, algorithm="giac")

[Out]

-log(abs(cos(x)))/(sqrt(a)*sgn(cos(x))*sgn(sin(x)))