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]

2/5*cosh(b*x+a)/b/csch(b*x+a)^(3/2)-6/5*I*(sin(1/2*I*a+1/4*Pi+1/2*I*b*x)^2)^(1/2)/sin(1/2*I*a+1/4*Pi+1/2*I*b*x
)*EllipticE(cos(1/2*I*a+1/4*Pi+1/2*I*b*x),2^(1/2))/b/csch(b*x+a)^(1/2)/(I*sinh(b*x+a))^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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]

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]

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*}

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Mathematica [A]  time = 0.13, 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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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\operatorname {csch}\left (b x + a\right )^{\frac {5}{2}}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {csch}\left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]

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)

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maple [A]  time = 0.39, size = 164, normalized size = 2.05 \[ \frac {-\frac {6 \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {i \sinh \left (b x +a \right )+1}\, \sqrt {i \sinh \left (b x +a \right )}\, \EllipticE \left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {i \sinh \left (b x +a \right )+1}\, \sqrt {i \sinh \left (b x +a \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 \left (\cosh ^{4}\left (b x +a \right )\right )}{5}-\frac {2 \left (\cosh ^{2}\left (b x +a \right )\right )}{5}}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b} \]

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)*(I*sinh(b*x+a)+1)^(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)*(I*sinh(b*x+a)+1)^(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {csch}\left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/sinh(a + b*x))^(5/2),x)

[Out]

int(1/(1/sinh(a + b*x))^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {csch}^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]

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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