3.28 \(\int \frac{1}{(a \cot ^2(x))^{3/2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{\tan (x)}{2 a \sqrt{a \cot ^2(x)}}+\frac{\cot (x) \log (\cos (x))}{a \sqrt{a \cot ^2(x)}} \]

[Out]

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

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Rubi [A]  time = 0.0181165, antiderivative size = 39, 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 = {3658, 3473, 3475} \[ \frac{\tan (x)}{2 a \sqrt{a \cot ^2(x)}}+\frac{\cot (x) \log (\cos (x))}{a \sqrt{a \cot ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cot[x]^2)^(-3/2),x]

[Out]

(Cot[x]*Log[Cos[x]])/(a*Sqrt[a*Cot[x]^2]) + Tan[x]/(2*a*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 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

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}{\left (a \cot ^2(x)\right )^{3/2}} \, dx &=\frac{\cot (x) \int \tan ^3(x) \, dx}{a \sqrt{a \cot ^2(x)}}\\ &=\frac{\tan (x)}{2 a \sqrt{a \cot ^2(x)}}-\frac{\cot (x) \int \tan (x) \, dx}{a \sqrt{a \cot ^2(x)}}\\ &=\frac{\cot (x) \log (\cos (x))}{a \sqrt{a \cot ^2(x)}}+\frac{\tan (x)}{2 a \sqrt{a \cot ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.027751, size = 30, normalized size = 0.77 \[ \frac{\csc (x) \sec (x)+2 \cot (x) \log (\cos (x))}{2 a \sqrt{a \cot ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cot[x]^2)^(-3/2),x]

[Out]

(2*Cot[x]*Log[Cos[x]] + Csc[x]*Sec[x])/(2*a*Sqrt[a*Cot[x]^2])

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Maple [A]  time = 0.046, size = 36, normalized size = 0.9 \begin{align*} -{\frac{\cot \left ( x \right ) \left ( \ln \left ( \left ( \cot \left ( x \right ) \right ) ^{2}+1 \right ) \left ( \cot \left ( x \right ) \right ) ^{2}-2\,\ln \left ( \cot \left ( x \right ) \right ) \left ( \cot \left ( x \right ) \right ) ^{2}-1 \right ) }{2} \left ( a \left ( \cot \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cot(x)^2)^(3/2),x)

[Out]

-1/2*cot(x)*(ln(cot(x)^2+1)*cot(x)^2-2*ln(cot(x))*cot(x)^2-1)/(a*cot(x)^2)^(3/2)

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Maxima [A]  time = 1.5459, size = 30, normalized size = 0.77 \begin{align*} \frac{\tan \left (x\right )^{2}}{2 \, a^{\frac{3}{2}}} - \frac{\log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)^2)^(3/2),x, algorithm="maxima")

[Out]

1/2*tan(x)^2/a^(3/2) - 1/2*log(tan(x)^2 + 1)/a^(3/2)

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Fricas [B]  time = 1.68903, size = 198, normalized size = 5.08 \begin{align*} \frac{{\left ({\left (\cos \left (2 \, x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (2 \, x\right ) + \frac{1}{2}\right ) \sin \left (2 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} \sqrt{-\frac{a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}}}{2 \,{\left (a^{2} \cos \left (2 \, x\right )^{2} + 2 \, a^{2} \cos \left (2 \, x\right ) + a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \cot ^{2}{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)**2)**(3/2),x)

[Out]

Integral((a*cot(x)**2)**(-3/2), x)

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Giac [B]  time = 1.53959, size = 188, normalized size = 4.82 \begin{align*} -\frac{\frac{\log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, x\right )^{2}} + 2\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )}{\mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )} - \frac{\log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, x\right )^{2}} - 2\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )}{\mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )} + \frac{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, x\right )^{2}}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right ) - 6 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )}{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, x\right )^{2}} - 2\right )} \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )} - \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )}{2 \, a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)^2)^(3/2),x, algorithm="giac")

[Out]

-1/2*(log(tan(1/2*x)^2 + 1/tan(1/2*x)^2 + 2)*sgn(tan(1/2*x))/sgn(-tan(1/2*x)^2 + 1) - log(tan(1/2*x)^2 + 1/tan
(1/2*x)^2 - 2)*sgn(tan(1/2*x))/sgn(-tan(1/2*x)^2 + 1) + ((tan(1/2*x)^2 + 1/tan(1/2*x)^2)*sgn(tan(1/2*x)) - 6*s
gn(tan(1/2*x)))/((tan(1/2*x)^2 + 1/tan(1/2*x)^2 - 2)*sgn(-tan(1/2*x)^2 + 1)) - sgn(tan(1/2*x)))/a^(3/2)