3.7 \(\int \text{csch}^{\frac{5}{2}}(a+b x) \, dx\)

Optimal. Leaf size=80 \[ -\frac{2 \cosh (a+b x) \text{csch}^{\frac{3}{2}}(a+b x)}{3 b}+\frac{2 i \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right ),2\right )}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0365541, antiderivative size = 80, 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 = {3768, 3771, 2641} \[ -\frac{2 \cosh (a+b x) \text{csch}^{\frac{3}{2}}(a+b x)}{3 b}+\frac{2 i \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)} F\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]^(5/2),x]

[Out]

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

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 \text{csch}^{\frac{5}{2}}(a+b x) \, dx &=-\frac{2 \cosh (a+b x) \text{csch}^{\frac{3}{2}}(a+b x)}{3 b}-\frac{1}{3} \int \sqrt{\text{csch}(a+b x)} \, dx\\ &=-\frac{2 \cosh (a+b x) \text{csch}^{\frac{3}{2}}(a+b x)}{3 b}-\frac{1}{3} \left (\sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}\right ) \int \frac{1}{\sqrt{i \sinh (a+b x)}} \, dx\\ &=-\frac{2 \cosh (a+b x) \text{csch}^{\frac{3}{2}}(a+b x)}{3 b}+\frac{2 i \sqrt{\text{csch}(a+b x)} F\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right ) \sqrt{i \sinh (a+b x)}}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.118607, size = 61, normalized size = 0.76 \[ -\frac{2 \sqrt{\text{csch}(a+b x)} \left (\coth (a+b x)+i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 i a-2 i b x+\pi ),2\right )\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x]^(5/2),x]

[Out]

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

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Maple [A]  time = 0.199, size = 101, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,\cosh \left ( bx+a \right ) b} \left ( i\sqrt{1-i\sinh \left ( bx+a \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( bx+a \right ) }\sqrt{i\sinh \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( bx+a \right ) },{\frac{\sqrt{2}}{2}} \right ) \sinh \left ( bx+a \right ) +2\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)^(5/2),x)

[Out]

-1/3/sinh(b*x+a)^(3/2)*(I*(1-I*sinh(b*x+a))^(1/2)*2^(1/2)*(1+I*sinh(b*x+a))^(1/2)*(I*sinh(b*x+a))^(1/2)*Ellipt
icF((1-I*sinh(b*x+a))^(1/2),1/2*2^(1/2))*sinh(b*x+a)+2*cosh(b*x+a)^2)/cosh(b*x+a)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}\left (b x + a\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(csch(b*x + a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{csch}\left (b x + a\right )^{\frac{5}{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

integral(csch(b*x + a)^(5/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}^{\frac{5}{2}}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)**(5/2),x)

[Out]

Integral(csch(a + b*x)**(5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}\left (b x + a\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate(csch(b*x + a)^(5/2), x)