3.35 \(\int \frac{1}{\sqrt{a \cot ^4(x)}} \, dx\)

Optimal. Leaf size=31 \[ \frac{\cot (x)}{\sqrt{a \cot ^4(x)}}-\frac{x \cot ^2(x)}{\sqrt{a \cot ^4(x)}} \]

[Out]

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

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Rubi [A]  time = 0.0161043, antiderivative size = 31, 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} \[ \frac{\cot (x)}{\sqrt{a \cot ^4(x)}}-\frac{x \cot ^2(x)}{\sqrt{a \cot ^4(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Cot[x]^4],x]

[Out]

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

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

Mathematica [A]  time = 0.0224911, size = 21, normalized size = 0.68 \[ \frac{\cot (x)-x \cot ^2(x)}{\sqrt{a \cot ^4(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Cot[x]^4],x]

[Out]

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

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Maple [A]  time = 0.084, size = 26, normalized size = 0.8 \begin{align*}{\cot \left ( x \right ) \left ( \left ({\frac{\pi }{2}}-{\rm arccot} \left (\cot \left ( x \right ) \right ) \right ) \cot \left ( x \right ) +1 \right ){\frac{1}{\sqrt{a \left ( \cot \left ( x \right ) \right ) ^{4}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.52214, size = 18, normalized size = 0.58 \begin{align*} -\frac{x}{\sqrt{a}} + \frac{\tan \left (x\right )}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/sqrt(a) + tan(x)/sqrt(a)

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Fricas [B]  time = 2.10897, size = 203, normalized size = 6.55 \begin{align*} \frac{{\left (x \cos \left (2 \, x\right )^{2} -{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right ) - x\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}}}{a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*cos(2*x)^2 - (cos(2*x) - 1)*sin(2*x) - x)*sqrt((a*cos(2*x)^2 + 2*a*cos(2*x) + a)/(cos(2*x)^2 - 2*cos(2*x) +
1))/(a*cos(2*x)^2 + 2*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 ^{4}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*cot(x)**4), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError