3.46 \(\int \frac{1}{\sqrt{a \text{csch}^4(x)}} \, dx\)

Optimal. Leaf size=36 \[ \frac{\coth (x)}{2 \sqrt{a \text{csch}^4(x)}}-\frac{x \text{csch}^2(x)}{2 \sqrt{a \text{csch}^4(x)}} \]

[Out]

Coth[x]/(2*Sqrt[a*Csch[x]^4]) - (x*Csch[x]^2)/(2*Sqrt[a*Csch[x]^4])

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Rubi [A]  time = 0.0161932, antiderivative size = 36, 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, 2635, 8} \[ \frac{\coth (x)}{2 \sqrt{a \text{csch}^4(x)}}-\frac{x \text{csch}^2(x)}{2 \sqrt{a \text{csch}^4(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Csch[x]^4],x]

[Out]

Coth[x]/(2*Sqrt[a*Csch[x]^4]) - (x*Csch[x]^2)/(2*Sqrt[a*Csch[x]^4])

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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a \text{csch}^4(x)}} \, dx &=\frac{\text{csch}^2(x) \int \sinh ^2(x) \, dx}{\sqrt{a \text{csch}^4(x)}}\\ &=\frac{\coth (x)}{2 \sqrt{a \text{csch}^4(x)}}-\frac{\text{csch}^2(x) \int 1 \, dx}{2 \sqrt{a \text{csch}^4(x)}}\\ &=\frac{\coth (x)}{2 \sqrt{a \text{csch}^4(x)}}-\frac{x \text{csch}^2(x)}{2 \sqrt{a \text{csch}^4(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0242194, size = 24, normalized size = 0.67 \[ \frac{\coth (x)-x \text{csch}^2(x)}{2 \sqrt{a \text{csch}^4(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Csch[x]^4],x]

[Out]

(Coth[x] - x*Csch[x]^2)/(2*Sqrt[a*Csch[x]^4])

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Maple [B]  time = 0.069, size = 89, normalized size = 2.5 \begin{align*} -{\frac{{{\rm e}^{2\,x}}x}{2\, \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{4}}}}}}}+{\frac{{{\rm e}^{4\,x}}}{8\, \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{4}}}}}}}-{\frac{1}{8\, \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{4}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)/(exp(2*x)-1)^2*exp(2*x)*x+1/8/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)/(exp(2*
x)-1)^2*exp(4*x)-1/8/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)/(exp(2*x)-1)^2

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Maxima [A]  time = 1.58208, size = 30, normalized size = 0.83 \begin{align*} -\frac{{\left (e^{\left (-4 \, x\right )} - 1\right )} e^{\left (2 \, x\right )}}{8 \, \sqrt{a}} - \frac{x}{2 \, \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(e^(-4*x) - 1)*e^(2*x)/sqrt(a) - 1/2*x/sqrt(a)

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Fricas [B]  time = 1.63992, size = 771, normalized size = 21.42 \begin{align*} \frac{{\left ({\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{4} + \cosh \left (x\right )^{4} + 4 \,{\left (\cosh \left (x\right ) e^{\left (4 \, x\right )} - 2 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 4 \, x \cosh \left (x\right )^{2} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} +{\left (3 \, \cosh \left (x\right )^{2} - 2 \, x\right )} e^{\left (4 \, x\right )} - 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 2 \, x\right )} e^{\left (2 \, x\right )} - 2 \, x\right )} \sinh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{4} - 4 \, x \cosh \left (x\right )^{2} - 1\right )} e^{\left (4 \, x\right )} - 2 \,{\left (\cosh \left (x\right )^{4} - 4 \, x \cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )} + 4 \,{\left (\cosh \left (x\right )^{3} - 2 \, x \cosh \left (x\right ) +{\left (\cosh \left (x\right )^{3} - 2 \, x \cosh \left (x\right )\right )} e^{\left (4 \, x\right )} - 2 \,{\left (\cosh \left (x\right )^{3} - 2 \, x \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right ) - 1\right )} \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}} e^{\left (2 \, x\right )}}{8 \,{\left (a \cosh \left (x\right )^{2} e^{\left (2 \, x\right )} + 2 \, a \cosh \left (x\right ) e^{\left (2 \, x\right )} \sinh \left (x\right ) + a e^{\left (2 \, x\right )} \sinh \left (x\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((e^(4*x) - 2*e^(2*x) + 1)*sinh(x)^4 + cosh(x)^4 + 4*(cosh(x)*e^(4*x) - 2*cosh(x)*e^(2*x) + cosh(x))*sinh(
x)^3 - 4*x*cosh(x)^2 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 - 2*x)*e^(4*x) - 2*(3*cosh(x)^2 - 2*x)*e^(2*x) - 2*x)*sin
h(x)^2 + (cosh(x)^4 - 4*x*cosh(x)^2 - 1)*e^(4*x) - 2*(cosh(x)^4 - 4*x*cosh(x)^2 - 1)*e^(2*x) + 4*(cosh(x)^3 -
2*x*cosh(x) + (cosh(x)^3 - 2*x*cosh(x))*e^(4*x) - 2*(cosh(x)^3 - 2*x*cosh(x))*e^(2*x))*sinh(x) - 1)*sqrt(a/(e^
(8*x) - 4*e^(6*x) + 6*e^(4*x) - 4*e^(2*x) + 1))*e^(2*x)/(a*cosh(x)^2*e^(2*x) + 2*a*cosh(x)*e^(2*x)*sinh(x) + a
*e^(2*x)*sinh(x)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \operatorname{csch}^{4}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*csch(x)**4), x)

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Giac [A]  time = 1.20209, size = 35, normalized size = 0.97 \begin{align*} \frac{{\left (2 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )} - 4 \, x + e^{\left (2 \, x\right )}}{8 \, \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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