3.25 \(\int (a \cot ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=36 \[ -\frac{1}{2} a \cot (x) \sqrt{a \cot ^2(x)}-a \tan (x) \sqrt{a \cot ^2(x)} \log (\sin (x)) \]

[Out]

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

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Rubi [A]  time = 0.0182863, antiderivative size = 36, 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} \[ -\frac{1}{2} a \cot (x) \sqrt{a \cot ^2(x)}-a \tan (x) \sqrt{a \cot ^2(x)} \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \left (a \cot ^2(x)\right )^{3/2} \, dx &=\left (a \sqrt{a \cot ^2(x)} \tan (x)\right ) \int \cot ^3(x) \, dx\\ &=-\frac{1}{2} a \cot (x) \sqrt{a \cot ^2(x)}-\left (a \sqrt{a \cot ^2(x)} \tan (x)\right ) \int \cot (x) \, dx\\ &=-\frac{1}{2} a \cot (x) \sqrt{a \cot ^2(x)}-a \sqrt{a \cot ^2(x)} \log (\sin (x)) \tan (x)\\ \end{align*}

Mathematica [A]  time = 0.0193772, size = 27, normalized size = 0.75 \[ -\frac{1}{2} a \tan (x) \sqrt{a \cot ^2(x)} \left (\csc ^2(x)+2 \log (\sin (x))\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Sqrt[a*Cot[x]^2]*(Csc[x]^2 + 2*Log[Sin[x]])*Tan[x])/2

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Maple [A]  time = 0.055, size = 29, normalized size = 0.8 \begin{align*}{\frac{- \left ( \cot \left ( x \right ) \right ) ^{2}+\ln \left ( \left ( \cot \left ( x \right ) \right ) ^{2}+1 \right ) }{2\, \left ( \cot \left ( x \right ) \right ) ^{3}} \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((a*cot(x)^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.54505, size = 41, normalized size = 1.14 \begin{align*} \frac{1}{2} \, a^{\frac{3}{2}} \log \left (\tan \left (x\right )^{2} + 1\right ) - a^{\frac{3}{2}} \log \left (\tan \left (x\right )\right ) - \frac{a^{\frac{3}{2}}}{2 \, \tan \left (x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.62114, size = 139, normalized size = 3.86 \begin{align*} \frac{{\left ({\left (a \cos \left (2 \, x\right ) - a\right )} \log \left (-\frac{1}{2} \, \cos \left (2 \, x\right ) + \frac{1}{2}\right ) - 2 \, a\right )} \sqrt{-\frac{a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}}}{2 \, \sin \left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((a*cot(x)**2)**(3/2),x)

[Out]

Integral((a*cot(x)**2)**(3/2), x)

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Giac [A]  time = 1.20721, size = 42, normalized size = 1.17 \begin{align*} \frac{1}{2} \, a^{\frac{3}{2}}{\left (\frac{1}{\cos \left (x\right )^{2} - 1} - \log \left (-\cos \left (x\right )^{2} + 1\right )\right )} \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((a*cot(x)^2)^(3/2),x, algorithm="giac")

[Out]

1/2*a^(3/2)*(1/(cos(x)^2 - 1) - log(-cos(x)^2 + 1))*sgn(cos(x))*sgn(sin(x))