3.18 \(\int \frac{1}{(b \text{csch}(c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=90 \[ \frac{2 \cosh (c+d x)}{3 b d \sqrt{b \text{csch}(c+d x)}}+\frac{2 i \sqrt{i \sinh (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right ),2\right ) \sqrt{b \text{csch}(c+d x)}}{3 b^2 d} \]

[Out]

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

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Rubi [A]  time = 0.0385371, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3769, 3771, 2641} \[ \frac{2 \cosh (c+d x)}{3 b d \sqrt{b \text{csch}(c+d x)}}+\frac{2 i \sqrt{i \sinh (c+d x)} F\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \text{csch}(c+d x)}}{3 b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(b*Csch[c + d*x])^(-3/2),x]

[Out]

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

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

Mathematica [A]  time = 0.088462, size = 73, normalized size = 0.81 \[ \frac{\text{csch}^2(c+d x) \left (\sinh (2 (c+d x))-2 i \sqrt{i \sinh (c+d x)} \text{EllipticF}\left (\frac{1}{4} (-2 i c-2 i d x+\pi ),2\right )\right )}{3 d (b \text{csch}(c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Csch[c + d*x])^(-3/2),x]

[Out]

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

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Maple [F]  time = 0.071, size = 0, normalized size = 0. \begin{align*} \int \left ( b{\rm csch} \left (dx+c\right ) \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*csch(d*x+c))^(3/2),x)

[Out]

int(1/(b*csch(d*x+c))^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \operatorname{csch}\left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*csch(d*x + c))^(-3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*csch(d*x + c))/(b^2*csch(d*x + c)^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \operatorname{csch}{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*csch(d*x+c))**(3/2),x)

[Out]

Integral((b*csch(c + d*x))**(-3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \operatorname{csch}\left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*csch(d*x + c))^(-3/2), x)