3.38 \(\int \sqrt{a \text{csch}^3(x)} \, dx\)

Optimal. Leaf size=56 \[ -2 \sinh (x) \cosh (x) \sqrt{a \text{csch}^3(x)}-2 i (i \sinh (x))^{3/2} E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{a \text{csch}^3(x)} \]

[Out]

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

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Rubi [A]  time = 0.0338612, antiderivative size = 60, normalized size of antiderivative = 1.07, 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, 3768, 3771, 2639} \[ -2 \sinh (x) \cosh (x) \sqrt{a \text{csch}^3(x)}+\frac{2 i \sinh ^2(x) E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{a \text{csch}^3(x)}}{\sqrt{i \sinh (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Cosh[x]*Sqrt[a*Csch[x]^3]*Sinh[x] + ((2*I)*Sqrt[a*Csch[x]^3]*EllipticE[Pi/4 - (I/2)*x, 2]*Sinh[x]^2)/Sqrt[I
*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 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[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 2639

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

Rubi steps

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

Mathematica [A]  time = 0.028467, size = 42, normalized size = 0.75 \[ -2 \sinh (x) \sqrt{a \text{csch}^3(x)} \left (\cosh (x)-\sqrt{i \sinh (x)} E\left (\left .\frac{1}{4} (\pi -2 i x)\right |2\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int \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((a*csch(x)^3)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*csch(x)^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*csch(x)^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*csch(x)**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*csch(x)^3), x)