3.11.30 \(\int \frac {\coth (\sqrt {x}) \text {csch}(\sqrt {x})}{\sqrt {x}} \, dx\) [1030]

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

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 = {6847, 2686, 8} \begin {gather*} -2 \text {csch}\left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.01, size = 8, normalized size = 1.00 \begin {gather*} -2 \text {csch}\left (\sqrt {x}\right ) \end {gather*}

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 = 1.09, size = 7, normalized size = 0.88

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (6) = 12\).
time = 0.27, size = 17, normalized size = 2.12 \begin {gather*} \frac {4}{e^{\left (-\sqrt {x}\right )} - e^{\sqrt {x}}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (6) = 12\).
time = 0.41, size = 37, normalized size = 4.62 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth {\left (\sqrt {x} \right )} \operatorname {csch}{\left (\sqrt {x} \right )}}{\sqrt {x}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (6) = 12\).
time = 0.40, size = 17, normalized size = 2.12 \begin {gather*} \frac {4}{e^{\left (-\sqrt {x}\right )} - e^{\sqrt {x}}} \end {gather*}

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))

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Mupad [B]
time = 1.75, size = 8, normalized size = 1.00 \begin {gather*} -\frac {2}{\mathrm {sinh}\left (\sqrt {x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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