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

Optimal. Leaf size=76 \[ -\frac{2 \cosh (a+b x) \sqrt{\text{csch}(a+b x)}}{b}-\frac{2 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}} \]

[Out]

(-2*Cosh[a + b*x]*Sqrt[Csch[a + b*x]])/b - ((2*I)*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 = 0.0301108, antiderivative size = 76, 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, 2639} \[ -\frac{2 \cosh (a+b x) \sqrt{\text{csch}(a+b x)}}{b}-\frac{2 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]^(3/2),x]

[Out]

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

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

Mathematica [A]  time = 0.204587, size = 57, normalized size = 0.75 \[ -\frac{2 \sqrt{\text{csch}(a+b x)} \left (\cosh (a+b x)-\sqrt{i \sinh (a+b x)} E\left (\left .\frac{1}{4} (-2 i a-2 i b x+\pi )\right |2\right )\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x]^(3/2),x]

[Out]

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

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Maple [A]  time = 0.151, size = 154, normalized size = 2. \begin{align*}{\frac{1}{\cosh \left ( bx+a \right ) b} \left ( 2\,\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 EllipticE} \left ( \sqrt{1-i\sinh \left ( bx+a \right ) },1/2\,\sqrt{2} \right ) -\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 ) -2\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{\sinh \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)^(3/2),x)

[Out]

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

[Out]

integrate(csch(b*x + a)^(3/2), x)