3.45 \(\int \sqrt{a \text{csch}^4(x)} \, dx\)

Optimal. Leaf size=16 \[ \sinh (x) (-\cosh (x)) \sqrt{a \text{csch}^4(x)} \]

[Out]

-(Cosh[x]*Sqrt[a*Csch[x]^4]*Sinh[x])

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Rubi [A]  time = 0.0167914, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4123, 3767, 8} \[ \sinh (x) (-\cosh (x)) \sqrt{a \text{csch}^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Csch[x]^4],x]

[Out]

-(Cosh[x]*Sqrt[a*Csch[x]^4]*Sinh[x])

Rule 4123

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^
FracPart[p])/(c*Sec[e + f*x])^(n*FracPart[p]), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

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

Rubi steps

\begin{align*} \int \sqrt{a \text{csch}^4(x)} \, dx &=\left (\sqrt{a \text{csch}^4(x)} \sinh ^2(x)\right ) \int \text{csch}^2(x) \, dx\\ &=-\left (\left (i \sqrt{a \text{csch}^4(x)} \sinh ^2(x)\right ) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (x))\right )\\ &=-\cosh (x) \sqrt{a \text{csch}^4(x)} \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.005449, size = 16, normalized size = 1. \[ \sinh (x) (-\cosh (x)) \sqrt{a \text{csch}^4(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Csch[x]^4],x]

[Out]

-(Cosh[x]*Sqrt[a*Csch[x]^4]*Sinh[x])

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Maple [A]  time = 0.068, size = 29, normalized size = 1.8 \begin{align*} -2\,\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{4}}}}{{\rm e}^{-2\,x}} \left ({{\rm e}^{2\,x}}-1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.7271, size = 18, normalized size = 1.12 \begin{align*} \frac{2 \, \sqrt{a}}{e^{\left (-2 \, x\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a)/(e^(-2*x) - 1)

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Fricas [B]  time = 1.38496, size = 230, normalized size = 14.38 \begin{align*} -\frac{2 \, \sqrt{\frac{a}{e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1}}{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (2 \, x\right )}}{2 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} \sinh \left (x\right ) + e^{\left (2 \, x\right )} \sinh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csch(x)^4)^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(a/(e^(8*x) - 4*e^(6*x) + 6*e^(4*x) - 4*e^(2*x) + 1))*(e^(4*x) - 2*e^(2*x) + 1)*e^(2*x)/(2*cosh(x)*e^(2
*x)*sinh(x) + e^(2*x)*sinh(x)^2 + (cosh(x)^2 - 1)*e^(2*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \operatorname{csch}^{4}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*csch(x)**4), x)

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Giac [A]  time = 1.16743, size = 18, normalized size = 1.12 \begin{align*} -\frac{2 \, \sqrt{a}}{e^{\left (2 \, x\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(a)/(e^(2*x) - 1)