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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 8

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

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 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
]])

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 = 0.02, 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]

________________________________________________________________________________________

fricas [B]  time = 1.35, size = 80, normalized size = 2.58 \[ \frac {{\left (x \cos \left (2 \, x\right )^{2} - {\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right ) - x\right )} \sqrt {\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.31, size = 13, normalized size = 0.42 \[ -\frac {x}{\sqrt {a}} + \frac {\tan \relax (x)}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/sqrt(a) + tan(x)/sqrt(a)

________________________________________________________________________________________

maple [A]  time = 0.22, size = 26, normalized size = 0.84 \[ \frac {\cot \relax (x ) \left (\left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \relax (x )\right )\right ) \cot \relax (x )+1\right )}{\sqrt {a \left (\cot ^{4}\relax (x )\right )}} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 13, normalized size = 0.42 \[ -\frac {x}{\sqrt {a}} + \frac {\tan \relax (x)}{\sqrt {a}} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {a\,{\mathrm {cot}\relax (x)}^4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a*cot(x)^4)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \cot ^{4}{\relax (x )}}}\, dx \]

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)

________________________________________________________________________________________