3.843 \(\int \frac{\cot (\sqrt{x}) \csc (\sqrt{x})}{\sqrt{x}} \, dx\)

Optimal. Leaf size=8 \[ -2 \csc \left (\sqrt{x}\right ) \]

[Out]

-2*Csc[Sqrt[x]]

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Rubi [A]  time = 0.196644, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6715, 2606, 8} \[ -2 \csc \left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Csc[Sqrt[x]]

Rule 6715

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]

Rule 2606

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 8

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

Rubi steps

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

Mathematica [A]  time = 0.0179433, size = 8, normalized size = 1. \[ -2 \csc \left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Csc[Sqrt[x]]

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Maple [A]  time = 0.016, size = 7, normalized size = 0.9 \begin{align*} -2\,\csc \left ( \sqrt{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.952835, size = 11, normalized size = 1.38 \begin{align*} -\frac{2}{\sin \left (\sqrt{x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/sin(sqrt(x))

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Fricas [A]  time = 2.0916, size = 23, normalized size = 2.88 \begin{align*} -\frac{2}{\sin \left (\sqrt{x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/sin(sqrt(x))

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Sympy [A]  time = 0.327789, size = 8, normalized size = 1. \begin{align*} - 2 \csc{\left (\sqrt{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*csc(sqrt(x))

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Giac [A]  time = 1.06237, size = 11, normalized size = 1.38 \begin{align*} -\frac{2}{\sin \left (\sqrt{x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/sin(sqrt(x))