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

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

[Out]

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

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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, 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 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]

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 {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*}

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Mathematica [A]  time = 0.07, size = 63, normalized size = 0.79 \[ \frac {\sqrt {\text {csch}(a+b x)} \left (\sinh (2 (a+b x))-2 i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{4} (-2 i a-2 i b x+\pi )\right |2\right )\right )}{3 b} \]

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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fricas [F]  time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]

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)

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maple [A]  time = 0.41, size = 100, normalized size = 1.25 \[ \frac {-\frac {i \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {i \sinh \left (b x +a \right )+1}\, \sqrt {i \sinh \left (b x +a \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )}{3}+\frac {2 \left (\cosh ^{2}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{3}}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/3*I*(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))+2/3*cosh(b*x+a)^2*sinh(b*x+a))/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 \[ \int \frac {1}{\operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/sinh(a + b*x))^(3/2),x)

[Out]

int(1/(1/sinh(a + b*x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {csch}^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]

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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