\(\int \frac {1}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx\) [11]

Optimal result

Integrand size = 10, antiderivative size = 80 \[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\frac {2 \cosh (a+b x)}{3 b \sqrt {\text {csch}(a+b x)}}+\frac {2 i \sqrt {\text {csch}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right ),2\right ) \sqrt {i \sinh (a+b x)}}{3 b} \]

[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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3854, 3856, 2720} \[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\frac {2 \cosh (a+b x)}{3 b \sqrt {\text {csch}(a+b x)}}+\frac {2 i \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right ),2\right )}{3 b} \]

[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 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3854

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 3856

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*} \text {integral}& = \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)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right ),2\right ) \sqrt {i \sinh (a+b x)}}{3 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\frac {\sqrt {\text {csch}(a+b x)} \left (-2 i \operatorname {EllipticF}\left (\frac {1}{4} (-2 i a+\pi -2 i b x),2\right ) \sqrt {i \sinh (a+b x)}+\sinh (2 (a+b x))\right )}{3 b} \]

[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)

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.25

method	result	size
default	\(\frac {-\frac {i \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (b x +a \right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )}{3}+\frac {2 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{3}}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b}\)	\(100\)

[In]

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

[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*cosh(b*x+a)^2*sinh(b*x+a))/cosh(b*x+a)/sinh(b*x+a)^(1/2)/b

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.79 \[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\frac {\sqrt {2} {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} - 1\right )} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}} - 4 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sqrt {2} \sinh \left (b x + a\right )^{2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{6 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]

[In]

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

[Out]

1/6*(sqrt(2)*(cosh(b*x + a)^4 + 4*cosh(b*x + a)^3*sinh(b*x + a) + 6*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*cosh(b
*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 - 1)*sqrt((cosh(b*x + a) + sinh(b*x + a))/(cosh(b*x + a)^2 + 2*cosh(
b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)) - 4*(sqrt(2)*cosh(b*x + a)^2 + 2*sqrt(2)*cosh(b*x + a)*sinh(b*x
+ a) + sqrt(2)*sinh(b*x + a)^2)*weierstrassPInverse(4, 0, cosh(b*x + a) + sinh(b*x + a)))/(b*cosh(b*x + a)^2
+ 2*b*cosh(b*x + a)*sinh(b*x + a) + b*sinh(b*x + a)^2)

Sympy [F]

\[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {1}{\operatorname {csch}^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]

[In]

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

[Out]

Integral(csch(a + b*x)**(-3/2), x)

Maxima [F]

\[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {1}{\operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate(csch(b*x + a)^(-3/2), x)

Giac [F]

\[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {1}{\operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate(csch(b*x + a)^(-3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {1}{{\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int(1/(1/sinh(a + b*x))^(3/2),x)

[Out]

int(1/(1/sinh(a + b*x))^(3/2), x)