3.11 \(\int \frac{1}{\text{csch}^{\frac{3}{2}}(a+b x)} \, dx\)

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

[Out]

(2*Cosh[a + b*x])/(3*b*Sqrt[Csch[a + b*x]]) + (((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 = 0.0324635, 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, 2641} \[ \frac{2 \cosh (a+b x)}{3 b \sqrt{\text{csch}(a+b x)}}+\frac{2 i \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)} F\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]^(-3/2),x]

[Out]

(2*Cosh[a + b*x])/(3*b*Sqrt[Csch[a + b*x]]) + (((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 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 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 \frac{1}{\text{csch}^{\frac{3}{2}}(a+b x)} \, dx &=\frac{2 \cosh (a+b x)}{3 b \sqrt{\text{csch}(a+b x)}}-\frac{1}{3} \int \sqrt{\text{csch}(a+b x)} \, dx\\ &=\frac{2 \cosh (a+b x)}{3 b \sqrt{\text{csch}(a+b x)}}-\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)}{3 b \sqrt{\text{csch}(a+b x)}}+\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 = 0.0649159, size = 63, normalized size = 0.79 \[ \frac{\sqrt{\text{csch}(a+b x)} \left (\sinh (2 (a+b x))-2 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 i a-2 i b x+\pi ),2\right )\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x]^(-3/2),x]

[Out]

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

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Maple [A]  time = 0.174, size = 100, normalized size = 1.3 \begin{align*}{\frac{1}{\cosh \left ( bx+a \right ) b} \left ( -{\frac{i}{3}}\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 ) +{\frac{2\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}} \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)^(3/2),x)

[Out]

(-1/3*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)*EllipticF((1-I*sinh(b*x+
a))^(1/2),1/2*2^(1/2))+2/3*sinh(b*x+a)*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{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csch(b*x + a)^(-3/2), 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{3}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(csch(b*x + a)^(-3/2), x)

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

[Out]

Integral(csch(a + b*x)**(-3/2), 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{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csch(b*x + a)^(-3/2), x)