\(\int (b \text {csch}(c+d x))^{3/2} \, dx\) [15]

Optimal result

Integrand size = 12, antiderivative size = 84 \[ \int (b \text {csch}(c+d x))^{3/2} \, dx=-\frac {2 b \cosh (c+d x) \sqrt {b \text {csch}(c+d x)}}{d}-\frac {2 i b^2 E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{d \sqrt {b \text {csch}(c+d x)} \sqrt {i \sinh (c+d x)}} \]

[Out]

-2*b*cosh(d*x+c)*(b*csch(d*x+c))^(1/2)/d+2*I*b^2*(sin(1/2*I*c+1/4*Pi+1/2*I*d*x)^2)^(1/2)/sin(1/2*I*c+1/4*Pi+1/
2*I*d*x)*EllipticE(cos(1/2*I*c+1/4*Pi+1/2*I*d*x),2^(1/2))/d/(b*csch(d*x+c))^(1/2)/(I*sinh(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3853, 3856, 2719} \[ \int (b \text {csch}(c+d x))^{3/2} \, dx=-\frac {2 b \cosh (c+d x) \sqrt {b \text {csch}(c+d x)}}{d}-\frac {2 i b^2 E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{d \sqrt {i \sinh (c+d x)} \sqrt {b \text {csch}(c+d x)}} \]

[In]

Int[(b*Csch[c + d*x])^(3/2),x]

[Out]

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

Rule 2719

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

Rule 3853

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int (b \text {csch}(c+d x))^{3/2} \, dx=-\frac {2 b \sqrt {b \text {csch}(c+d x)} \left (\cosh (c+d x)-E\left (\left .\frac {1}{4} (-2 i c+\pi -2 i d x)\right |2\right ) \sqrt {i \sinh (c+d x)}\right )}{d} \]

[In]

Integrate[(b*Csch[c + d*x])^(3/2),x]

[Out]

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

Maple [F]

\[\int \left (b \,\operatorname {csch}\left (d x +c \right )\right )^{\frac {3}{2}}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.27 \[ \int (b \text {csch}(c+d x))^{3/2} \, dx=-\frac {2 \, {\left (\sqrt {2} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )\right ) + \sqrt {2} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1}}\right )}}{d} \]

[In]

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

[Out]

-2*(sqrt(2)*b^(3/2)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cosh(d*x + c) + sinh(d*x + c))) + sqrt(2)*
(b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((b*cosh(d*x + c) + b*sinh(d*x + c))/(cosh(d*x + c)^2 + 2*cosh(d*x + c
)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)))/d

Sympy [F]

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

[In]

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

[Out]

Integral((b*csch(c + d*x))**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((b*csch(d*x + c))^(3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((b*csch(d*x + c))^(3/2), x)

Mupad [F(-1)]

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

[In]

int((b/sinh(c + d*x))^(3/2),x)

[Out]

int((b/sinh(c + d*x))^(3/2), x)