3.39 \(\int \frac{1}{\sqrt{a \text{csch}^3(x)}} \, dx\)

Optimal. Leaf size=62 \[ \frac{2 \coth (x)}{3 \sqrt{a \text{csch}^3(x)}}-\frac{2 i \sqrt{i \sinh (x)} \text{csch}^2(x) \text{EllipticF}\left (\frac{\pi }{4}-\frac{i x}{2},2\right )}{3 \sqrt{a \text{csch}^3(x)}} \]

[Out]

(2*Coth[x])/(3*Sqrt[a*Csch[x]^3]) - (((2*I)/3)*Csch[x]^2*EllipticF[Pi/4 - (I/2)*x, 2]*Sqrt[I*Sinh[x]])/Sqrt[a*
Csch[x]^3]

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Rubi [A]  time = 0.0341976, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3769, 3771, 2641} \[ \frac{2 \coth (x)}{3 \sqrt{a \text{csch}^3(x)}}-\frac{2 i \sqrt{i \sinh (x)} \text{csch}^2(x) F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right )}{3 \sqrt{a \text{csch}^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Csch[x]^3],x]

[Out]

(2*Coth[x])/(3*Sqrt[a*Csch[x]^3]) - (((2*I)/3)*Csch[x]^2*EllipticF[Pi/4 - (I/2)*x, 2]*Sqrt[I*Sinh[x]])/Sqrt[a*
Csch[x]^3]

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 3769

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

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a \text{csch}^3(x)}} \, dx &=\frac{(i \text{csch}(x))^{3/2} \int \frac{1}{(i \text{csch}(x))^{3/2}} \, dx}{\sqrt{a \text{csch}^3(x)}}\\ &=\frac{2 \coth (x)}{3 \sqrt{a \text{csch}^3(x)}}+\frac{(i \text{csch}(x))^{3/2} \int \sqrt{i \text{csch}(x)} \, dx}{3 \sqrt{a \text{csch}^3(x)}}\\ &=\frac{2 \coth (x)}{3 \sqrt{a \text{csch}^3(x)}}-\frac{\left (\text{csch}^2(x) \sqrt{i \sinh (x)}\right ) \int \frac{1}{\sqrt{i \sinh (x)}} \, dx}{3 \sqrt{a \text{csch}^3(x)}}\\ &=\frac{2 \coth (x)}{3 \sqrt{a \text{csch}^3(x)}}-\frac{2 i \text{csch}^2(x) F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{i \sinh (x)}}{3 \sqrt{a \text{csch}^3(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0731304, size = 43, normalized size = 0.69 \[ \frac{2 \left (\coth (x)+\frac{\text{csch}(x) \text{EllipticF}\left (\frac{1}{4} (\pi -2 i x),2\right )}{\sqrt{i \sinh (x)}}\right )}{3 \sqrt{a \text{csch}^3(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Csch[x]^3],x]

[Out]

(2*(Coth[x] + (Csch[x]*EllipticF[(Pi - (2*I)*x)/4, 2])/Sqrt[I*Sinh[x]]))/(3*Sqrt[a*Csch[x]^3])

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{a \left ({\rm csch} \left (x\right ) \right ) ^{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*csch(x)^3)^(1/2),x)

[Out]

int(1/(a*csch(x)^3)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \operatorname{csch}\left (x\right )^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(a*csch(x)^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \operatorname{csch}\left (x\right )^{3}}}{a \operatorname{csch}\left (x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)^3)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*csch(x)^3)/(a*csch(x)^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \operatorname{csch}^{3}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)**3)**(1/2),x)

[Out]

Integral(1/sqrt(a*csch(x)**3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \operatorname{csch}\left (x\right )^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)^3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(a*csch(x)^3), x)