\(\int \frac {1}{\sqrt {a \cot ^4(x)}} \, dx\) [35]

Optimal result

Integrand size = 10, antiderivative size = 31 \[ \int \frac {1}{\sqrt {a \cot ^4(x)}} \, dx=\frac {\cot (x)}{\sqrt {a \cot ^4(x)}}-\frac {x \cot ^2(x)}{\sqrt {a \cot ^4(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3739, 3554, 8} \[ \int \frac {1}{\sqrt {a \cot ^4(x)}} \, dx=\frac {\cot (x)}{\sqrt {a \cot ^4(x)}}-\frac {x \cot ^2(x)}{\sqrt {a \cot ^4(x)}} \]

[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 3554

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 3739

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*} \text {integral}& = \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] (verified)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {a \cot ^4(x)}} \, dx=\frac {\cot (x)-\arctan (\tan (x)) \cot ^2(x)}{\sqrt {a \cot ^4(x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {\cot \left (x \right ) \left (\left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (x \right )\right )\right ) \cot \left (x \right )+1\right )}{\sqrt {a \cot \left (x \right )^{4}}}\)	\(26\)
default	\(\frac {\cot \left (x \right ) \left (\left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (x \right )\right )\right ) \cot \left (x \right )+1\right )}{\sqrt {a \cot \left (x \right )^{4}}}\)	\(26\)
risch	\(\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2} x}{\sqrt {\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{4}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}-1\right )^{2}}-\frac {2 i \left ({\mathrm e}^{2 i x}+1\right )}{\sqrt {\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{4}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}-1\right )^{2}}\)	\(85\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (27) = 54\).

Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.58 \[ \int \frac {1}{\sqrt {a \cot ^4(x)}} \, dx=\frac {{\left (x \cos \left (2 \, x\right )^{2} - {\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right ) - x\right )} \sqrt {\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a} \]

[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)

Sympy [F]

\[ \int \frac {1}{\sqrt {a \cot ^4(x)}} \, dx=\int \frac {1}{\sqrt {a \cot ^{4}{\left (x \right )}}}\, dx \]

[In]

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

[Out]

Integral(1/sqrt(a*cot(x)**4), x)

Maxima [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\sqrt {a \cot ^4(x)}} \, dx=-\frac {x}{\sqrt {a}} + \frac {\tan \left (x\right )}{\sqrt {a}} \]

[In]

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

[Out]

-x/sqrt(a) + tan(x)/sqrt(a)

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {a \cot ^4(x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a \cot ^4(x)}} \, dx=\int \frac {1}{\sqrt {a\,{\mathrm {cot}\left (x\right )}^4}} \,d x \]

[In]

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

[Out]

int(1/(a*cot(x)^4)^(1/2), x)