3.12 \(\int \frac{1}{\text{csch}^{\frac{5}{2}}(a+b x)} \, dx\)

Optimal. Leaf size=80 \[ \frac{2 \cosh (a+b x)}{5 b \text{csch}^{\frac{3}{2}}(a+b x)}+\frac{6 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{5 b \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}} \]

[Out]

(2*Cosh[a + b*x])/(5*b*Csch[a + b*x]^(3/2)) + (((6*I)/5)*EllipticE[(I*a - Pi/2 + I*b*x)/2, 2])/(b*Sqrt[Csch[a
+ b*x]]*Sqrt[I*Sinh[a + b*x]])

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Rubi [A]  time = 0.0324466, 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 = {3769, 3771, 2639} \[ \frac{2 \cosh (a+b x)}{5 b \text{csch}^{\frac{3}{2}}(a+b x)}+\frac{6 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{5 b \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]^(-5/2),x]

[Out]

(2*Cosh[a + b*x])/(5*b*Csch[a + b*x]^(3/2)) + (((6*I)/5)*EllipticE[(I*a - Pi/2 + I*b*x)/2, 2])/(b*Sqrt[Csch[a
+ b*x]]*Sqrt[I*Sinh[a + b*x]])

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 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 \frac{1}{\text{csch}^{\frac{5}{2}}(a+b x)} \, dx &=\frac{2 \cosh (a+b x)}{5 b \text{csch}^{\frac{3}{2}}(a+b x)}-\frac{3}{5} \int \frac{1}{\sqrt{\text{csch}(a+b x)}} \, dx\\ &=\frac{2 \cosh (a+b x)}{5 b \text{csch}^{\frac{3}{2}}(a+b x)}-\frac{3 \int \sqrt{i \sinh (a+b x)} \, dx}{5 \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ &=\frac{2 \cosh (a+b x)}{5 b \text{csch}^{\frac{3}{2}}(a+b x)}+\frac{6 i E\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right )}{5 b \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.125116, size = 67, normalized size = 0.84 \[ \frac{2 \left (\cosh (a+b x)-3 \sqrt{i \sinh (a+b x)} \text{csch}^2(a+b x) E\left (\left .\frac{1}{4} (-2 i a-2 i b x+\pi )\right |2\right )\right )}{5 b \text{csch}^{\frac{3}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x]^(-5/2),x]

[Out]

(2*(Cosh[a + b*x] - 3*Csch[a + b*x]^2*EllipticE[((-2*I)*a + Pi - (2*I)*b*x)/4, 2]*Sqrt[I*Sinh[a + b*x]]))/(5*b
*Csch[a + b*x]^(3/2))

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Maple [A]  time = 0.237, size = 164, normalized size = 2.1 \begin{align*}{\frac{1}{\cosh \left ( bx+a \right ) b} \left ( -{\frac{6\,\sqrt{2}}{5}\sqrt{1-i\sinh \left ( bx+a \right ) }\sqrt{1+i\sinh \left ( bx+a \right ) }\sqrt{i\sinh \left ( bx+a \right ) }{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( bx+a \right ) },{\frac{\sqrt{2}}{2}} \right ) }+{\frac{3\,\sqrt{2}}{5}\sqrt{1-i\sinh \left ( bx+a \right ) }\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 ) }+{\frac{2\, \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}{5}}-{\frac{2\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{5}} \right ){\frac{1}{\sqrt{\sinh \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/csch(b*x+a)^(5/2),x)

[Out]

(-6/5*(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)*EllipticE((1-I*sinh(b*x+a)
)^(1/2),1/2*2^(1/2))+3/5*(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)*Ellipti
cF((1-I*sinh(b*x+a))^(1/2),1/2*2^(1/2))+2/5*cosh(b*x+a)^4-2/5*cosh(b*x+a)^2)/cosh(b*x+a)/sinh(b*x+a)^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/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 (\frac{1}{\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(1/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 \frac{1}{\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(1/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 \frac{1}{\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(1/csch(b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate(csch(b*x + a)^(-5/2), x)