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(b*x+a)*csch(b*x+a)^(1/2)/b+2*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)*E
llipticE(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 = 76, 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 = {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 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 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]

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*}

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Mathematica [A]  time = 0.22, 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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fricas [F]  time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}\,{d x} \]

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)

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maple [A]  time = 0.38, size = 154, normalized size = 2.03 \[ \frac {2 \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {i \sinh \left (b x +a \right )+1}\, \sqrt {i \sinh \left (b x +a \right )}\, \EllipticE \left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {i \sinh \left (b x +a \right )+1}\, \sqrt {i \sinh \left (b x +a \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cosh ^{2}\left (b x +a \right )\right )}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(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))-(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)*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 = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sinh(a + b*x))^(3/2),x)

[Out]

int((1/sinh(a + b*x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}^{\frac {3}{2}}{\left (a + b x \right )}\, dx \]

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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