### 3.1030 $$\int \frac{\coth (\sqrt{x}) \text{csch}(\sqrt{x})}{\sqrt{x}} \, dx$$

Optimal. Leaf size=8 $-2 \text{csch}\left (\sqrt{x}\right )$

[Out]

-2*Csch[Sqrt[x]]

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Rubi [A]  time = 0.191162, 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} $-2 \text{csch}\left (\sqrt{x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(Coth[Sqrt[x]]*Csch[Sqrt[x]])/Sqrt[x],x]

[Out]

-2*Csch[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{\coth \left (\sqrt{x}\right ) \text{csch}\left (\sqrt{x}\right )}{\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \coth (x) \text{csch}(x) \, dx,x,\sqrt{x}\right )\\ &=-\left (2 i \operatorname{Subst}\left (\int 1 \, dx,x,-i \text{csch}\left (\sqrt{x}\right )\right )\right )\\ &=-2 \text{csch}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0168591, size = 8, normalized size = 1. $-2 \text{csch}\left (\sqrt{x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Coth[Sqrt[x]]*Csch[Sqrt[x]])/Sqrt[x],x]

[Out]

-2*Csch[Sqrt[x]]

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Maple [A]  time = 0.013, size = 7, normalized size = 0.9 \begin{align*} -2\,{\rm csch} \left (\sqrt{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x^(1/2))*csch(x^(1/2))/x^(1/2),x)

[Out]

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

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Maxima [B]  time = 1.07936, size = 23, normalized size = 2.88 \begin{align*} \frac{4}{e^{\left (-\sqrt{x}\right )} - e^{\sqrt{x}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x^(1/2))*csch(x^(1/2))/x^(1/2),x, algorithm="maxima")

[Out]

4/(e^(-sqrt(x)) - e^sqrt(x))

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Fricas [B]  time = 2.08235, size = 146, normalized size = 18.25 \begin{align*} -\frac{4 \,{\left (\cosh \left (\sqrt{x}\right ) + \sinh \left (\sqrt{x}\right )\right )}}{\cosh \left (\sqrt{x}\right )^{2} + 2 \, \cosh \left (\sqrt{x}\right ) \sinh \left (\sqrt{x}\right ) + \sinh \left (\sqrt{x}\right )^{2} - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x^(1/2))*csch(x^(1/2))/x^(1/2),x, algorithm="fricas")

[Out]

-4*(cosh(sqrt(x)) + sinh(sqrt(x)))/(cosh(sqrt(x))^2 + 2*cosh(sqrt(x))*sinh(sqrt(x)) + sinh(sqrt(x))^2 - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth{\left (\sqrt{x} \right )} \operatorname{csch}{\left (\sqrt{x} \right )}}{\sqrt{x}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x**(1/2))*csch(x**(1/2))/x**(1/2),x)

[Out]

Integral(coth(sqrt(x))*csch(sqrt(x))/sqrt(x), x)

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Giac [B]  time = 1.13097, size = 23, normalized size = 2.88 \begin{align*} \frac{4}{e^{\left (-\sqrt{x}\right )} - e^{\sqrt{x}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x^(1/2))*csch(x^(1/2))/x^(1/2),x, algorithm="giac")

[Out]

4/(e^(-sqrt(x)) - e^sqrt(x))