\(\int \frac {\cot (\sqrt {x}) \csc (\sqrt {x})}{\sqrt {x}} \, dx\) [843]

Optimal result

Integrand size = 18, antiderivative size = 8 \[ \int \frac {\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-2 \csc \left (\sqrt {x}\right ) \]

[Out]

-2*csc(x^(1/2))

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6847, 2686, 8} \[ \int \frac {\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-2 \csc \left (\sqrt {x}\right ) \]

[In]

Int[(Cot[Sqrt[x]]*Csc[Sqrt[x]])/Sqrt[x],x]

[Out]

-2*Csc[Sqrt[x]]

Rule 8

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

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \cot (x) \csc (x) \, dx,x,\sqrt {x}\right ) \\ & = -\left (2 \text {Subst}\left (\int 1 \, dx,x,\csc \left (\sqrt {x}\right )\right )\right ) \\ & = -2 \csc \left (\sqrt {x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-2 \csc \left (\sqrt {x}\right ) \]

[In]

Integrate[(Cot[Sqrt[x]]*Csc[Sqrt[x]])/Sqrt[x],x]

[Out]

-2*Csc[Sqrt[x]]

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(-2 \csc \left (\sqrt {x}\right )\)	\(7\)
default	\(-2 \csc \left (\sqrt {x}\right )\)	\(7\)

[In]

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

[Out]

-2*csc(x^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-\frac {2}{\sin \left (\sqrt {x}\right )} \]

[In]

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

[Out]

-2/sin(sqrt(x))

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=- 2 \csc {\left (\sqrt {x} \right )} \]

[In]

integrate(cot(x**(1/2))*csc(x**(1/2))/x**(1/2),x)

[Out]

-2*csc(sqrt(x))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-\frac {2}{\sin \left (\sqrt {x}\right )} \]

[In]

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

[Out]

-2/sin(sqrt(x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-\frac {2}{\sin \left (\sqrt {x}\right )} \]

[In]

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

[Out]

-2/sin(sqrt(x))

Mupad [B] (verification not implemented)

Time = 26.83 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-\frac {2}{\sin \left (\sqrt {x}\right )} \]

[In]

int(cot(x^(1/2))/(x^(1/2)*sin(x^(1/2))),x)

[Out]

-2/sin(x^(1/2))