3.37 \(\int (a \text{csch}^3(x))^{3/2} \, dx\)

Optimal. Leaf size=81 \[ \frac{10}{21} i a \sqrt{i \sinh (x)} \sinh (x) \text{EllipticF}\left (\frac{\pi }{4}-\frac{i x}{2},2\right ) \sqrt{a \text{csch}^3(x)}+\frac{10}{21} a \cosh (x) \sqrt{a \text{csch}^3(x)}-\frac{2}{7} a \coth (x) \text{csch}(x) \sqrt{a \text{csch}^3(x)} \]

[Out]

(10*a*Cosh[x]*Sqrt[a*Csch[x]^3])/21 - (2*a*Coth[x]*Csch[x]*Sqrt[a*Csch[x]^3])/7 + ((10*I)/21)*a*Sqrt[a*Csch[x]
^3]*EllipticF[Pi/4 - (I/2)*x, 2]*Sqrt[I*Sinh[x]]*Sinh[x]

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Rubi [A]  time = 0.0452946, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3768, 3771, 2641} \[ \frac{10}{21} a \cosh (x) \sqrt{a \text{csch}^3(x)}-\frac{2}{7} a \coth (x) \text{csch}(x) \sqrt{a \text{csch}^3(x)}+\frac{10}{21} i a \sqrt{i \sinh (x)} \sinh (x) F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{a \text{csch}^3(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csch[x]^3)^(3/2),x]

[Out]

(10*a*Cosh[x]*Sqrt[a*Csch[x]^3])/21 - (2*a*Coth[x]*Csch[x]*Sqrt[a*Csch[x]^3])/7 + ((10*I)/21)*a*Sqrt[a*Csch[x]
^3]*EllipticF[Pi/4 - (I/2)*x, 2]*Sqrt[I*Sinh[x]]*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 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 \left (a \text{csch}^3(x)\right )^{3/2} \, dx &=\frac{\left (i a \sqrt{a \text{csch}^3(x)}\right ) \int (i \text{csch}(x))^{9/2} \, dx}{(i \text{csch}(x))^{3/2}}\\ &=-\frac{2}{7} a \coth (x) \text{csch}(x) \sqrt{a \text{csch}^3(x)}+\frac{\left (5 i a \sqrt{a \text{csch}^3(x)}\right ) \int (i \text{csch}(x))^{5/2} \, dx}{7 (i \text{csch}(x))^{3/2}}\\ &=\frac{10}{21} a \cosh (x) \sqrt{a \text{csch}^3(x)}-\frac{2}{7} a \coth (x) \text{csch}(x) \sqrt{a \text{csch}^3(x)}+\frac{\left (5 i a \sqrt{a \text{csch}^3(x)}\right ) \int \sqrt{i \text{csch}(x)} \, dx}{21 (i \text{csch}(x))^{3/2}}\\ &=\frac{10}{21} a \cosh (x) \sqrt{a \text{csch}^3(x)}-\frac{2}{7} a \coth (x) \text{csch}(x) \sqrt{a \text{csch}^3(x)}+\frac{1}{21} \left (5 a \sqrt{a \text{csch}^3(x)} \sqrt{i \sinh (x)} \sinh (x)\right ) \int \frac{1}{\sqrt{i \sinh (x)}} \, dx\\ &=\frac{10}{21} a \cosh (x) \sqrt{a \text{csch}^3(x)}-\frac{2}{7} a \coth (x) \text{csch}(x) \sqrt{a \text{csch}^3(x)}+\frac{10}{21} i a \sqrt{a \text{csch}^3(x)} F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{i \sinh (x)} \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.113659, size = 56, normalized size = 0.69 \[ -\frac{2}{21} a \sinh (x) \sqrt{a \text{csch}^3(x)} \left (\coth (x) \left (3 \text{csch}^2(x)-5\right )-5 i \sqrt{i \sinh (x)} \text{EllipticF}\left (\frac{1}{4} (\pi -2 i x),2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csch[x]^3)^(3/2),x]

[Out]

(-2*a*Sqrt[a*Csch[x]^3]*(Coth[x]*(-5 + 3*Csch[x]^2) - (5*I)*EllipticF[(Pi - (2*I)*x)/4, 2]*Sqrt[I*Sinh[x]])*Si
nh[x])/21

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ({\rm csch} \left (x\right ) \right ) ^{3} \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \operatorname{csch}\left (x\right )^{3}\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*csch(x)^3)^(3/2), 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}} 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)^(3/2),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 \left (a \operatorname{csch}^{3}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*csch(x)**3)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \operatorname{csch}\left (x\right )^{3}\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*csch(x)^3)^(3/2), x)