3.34 \(\int \sqrt{a \cot ^4(x)} \, dx\)

Optimal. Leaf size=32 \[ -x \tan ^2(x) \sqrt{a \cot ^4(x)}-\tan (x) \sqrt{a \cot ^4(x)} \]

[Out]

-(Sqrt[a*Cot[x]^4]*Tan[x]) - x*Sqrt[a*Cot[x]^4]*Tan[x]^2

________________________________________________________________________________________

Rubi [A]  time = 0.0154425, antiderivative size = 32, 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, 8} \[ -x \tan ^2(x) \sqrt{a \cot ^4(x)}-\tan (x) \sqrt{a \cot ^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Cot[x]^4],x]

[Out]

-(Sqrt[a*Cot[x]^4]*Tan[x]) - x*Sqrt[a*Cot[x]^4]*Tan[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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \sqrt{a \cot ^4(x)} \, dx &=\left (\sqrt{a \cot ^4(x)} \tan ^2(x)\right ) \int \cot ^2(x) \, dx\\ &=-\sqrt{a \cot ^4(x)} \tan (x)-\left (\sqrt{a \cot ^4(x)} \tan ^2(x)\right ) \int 1 \, dx\\ &=-\sqrt{a \cot ^4(x)} \tan (x)-x \sqrt{a \cot ^4(x)} \tan ^2(x)\\ \end{align*}

Mathematica [A]  time = 0.0153746, size = 20, normalized size = 0.62 \[ \tan ^2(x) (x+\cot (x)) \left (-\sqrt{a \cot ^4(x)}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Cot[x]^4],x]

[Out]

-(Sqrt[a*Cot[x]^4]*(x + Cot[x])*Tan[x]^2)

________________________________________________________________________________________

Maple [A]  time = 0.078, size = 27, normalized size = 0.8 \begin{align*}{\frac{1}{ \left ( \cot \left ( x \right ) \right ) ^{2}}\sqrt{a \left ( \cot \left ( x \right ) \right ) ^{4}} \left ( -\cot \left ( x \right ) +{\frac{\pi }{2}}-{\rm arccot} \left (\cot \left ( x \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cot(x)^4)^(1/2),x)

[Out]

(a*cot(x)^4)^(1/2)/cot(x)^2*(-cot(x)+1/2*Pi-arccot(cot(x)))

________________________________________________________________________________________

Maxima [A]  time = 1.64365, size = 22, normalized size = 0.69 \begin{align*} -\sqrt{a} x - \frac{\sqrt{a}}{\tan \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cot(x)^4)^(1/2),x, algorithm="maxima")

[Out]

-sqrt(a)*x - sqrt(a)/tan(x)

________________________________________________________________________________________

Fricas [B]  time = 2.15884, size = 154, normalized size = 4.81 \begin{align*} \frac{{\left (x \cos \left (2 \, x\right ) - x - \sin \left (2 \, x\right )\right )} \sqrt{\frac{a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{\cos \left (2 \, x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cot(x)^4)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cot ^{4}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cot(x)**4)**(1/2),x)

[Out]

Integral(sqrt(a*cot(x)**4), x)

________________________________________________________________________________________

Giac [A]  time = 1.29934, size = 28, normalized size = 0.88 \begin{align*} -\frac{1}{2} \, \sqrt{a}{\left (2 \, x + \frac{1}{\tan \left (\frac{1}{2} \, x\right )} - \tan \left (\frac{1}{2} \, x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cot(x)^4)^(1/2),x, algorithm="giac")

[Out]

-1/2*sqrt(a)*(2*x + 1/tan(1/2*x) - tan(1/2*x))