\(\int \frac {1}{(b \text {csch}(c+d x))^{7/2}} \, dx\) [20]

Optimal result

Integrand size = 12, antiderivative size = 118 \[ \int \frac {1}{(b \text {csch}(c+d x))^{7/2}} \, dx=\frac {2 \cosh (c+d x)}{7 b d (b \text {csch}(c+d x))^{5/2}}-\frac {10 \cosh (c+d x)}{21 b^3 d \sqrt {b \text {csch}(c+d x)}}-\frac {10 i \sqrt {b \text {csch}(c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right ),2\right ) \sqrt {i \sinh (c+d x)}}{21 b^4 d} \] Output:

2/7*cosh(d*x+c)/b/d/(b*csch(d*x+c))^(5/2)-10/21*cosh(d*x+c)/b^3/d/(b*csch( 
d*x+c))^(1/2)-10/21*I*(b*csch(d*x+c))^(1/2)*InverseJacobiAM(1/2*I*c-1/4*Pi 
+1/2*I*d*x,2^(1/2))*(I*sinh(d*x+c))^(1/2)/b^4/d
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(b \text {csch}(c+d x))^{7/2}} \, dx=\frac {\sqrt {b \text {csch}(c+d x)} \left (40 i \operatorname {EllipticF}\left (\frac {1}{4} (-2 i c+\pi -2 i d x),2\right ) \sqrt {i \sinh (c+d x)}-26 \sinh (2 (c+d x))+3 \sinh (4 (c+d x))\right )}{84 b^4 d} \] Input:

Integrate[(b*Csch[c + d*x])^(-7/2),x]
 

Output:

(Sqrt[b*Csch[c + d*x]]*((40*I)*EllipticF[((-2*I)*c + Pi - (2*I)*d*x)/4, 2] 
*Sqrt[I*Sinh[c + d*x]] - 26*Sinh[2*(c + d*x)] + 3*Sinh[4*(c + d*x)]))/(84* 
b^4*d)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4256, 3042, 4256, 3042, 4258, 3042, 3120}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{(b \text {csch}(c+d x))^{7/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(i b \csc (i c+i d x))^{7/2}}dx\)

\(\Big \downarrow \) 4256

\(\displaystyle \frac {2 \cosh (c+d x)}{7 b d (b \text {csch}(c+d x))^{5/2}}-\frac {5 \int \frac {1}{(b \text {csch}(c+d x))^{3/2}}dx}{7 b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \cosh (c+d x)}{7 b d (b \text {csch}(c+d x))^{5/2}}-\frac {5 \int \frac {1}{(i b \csc (i c+i d x))^{3/2}}dx}{7 b^2}\)

\(\Big \downarrow \) 4256

\(\displaystyle \frac {2 \cosh (c+d x)}{7 b d (b \text {csch}(c+d x))^{5/2}}-\frac {5 \left (\frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}-\frac {\int \sqrt {b \text {csch}(c+d x)}dx}{3 b^2}\right )}{7 b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \cosh (c+d x)}{7 b d (b \text {csch}(c+d x))^{5/2}}-\frac {5 \left (\frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}-\frac {\int \sqrt {i b \csc (i c+i d x)}dx}{3 b^2}\right )}{7 b^2}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {2 \cosh (c+d x)}{7 b d (b \text {csch}(c+d x))^{5/2}}-\frac {5 \left (\frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}-\frac {\sqrt {i \sinh (c+d x)} \sqrt {b \text {csch}(c+d x)} \int \frac {1}{\sqrt {i \sinh (c+d x)}}dx}{3 b^2}\right )}{7 b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \cosh (c+d x)}{7 b d (b \text {csch}(c+d x))^{5/2}}-\frac {5 \left (\frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}-\frac {\sqrt {i \sinh (c+d x)} \sqrt {b \text {csch}(c+d x)} \int \frac {1}{\sqrt {\sin (i c+i d x)}}dx}{3 b^2}\right )}{7 b^2}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {2 \cosh (c+d x)}{7 b d (b \text {csch}(c+d x))^{5/2}}-\frac {5 \left (\frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}+\frac {2 i \sqrt {i \sinh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right ),2\right ) \sqrt {b \text {csch}(c+d x)}}{3 b^2 d}\right )}{7 b^2}\)

Input:

Int[(b*Csch[c + d*x])^(-7/2),x]
 

Output:

(2*Cosh[c + d*x])/(7*b*d*(b*Csch[c + d*x])^(5/2)) - (5*((2*Cosh[c + d*x])/ 
(3*b*d*Sqrt[b*Csch[c + d*x]]) + (((2*I)/3)*Sqrt[b*Csch[c + d*x]]*EllipticF 
[(I*c - Pi/2 + I*d*x)/2, 2]*Sqrt[I*Sinh[c + d*x]])/(b^2*d)))/(7*b^2)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3120

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 4256

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] + Simp[(n + 1)/(b^2*n)   Int[(b*Csc[c 
+ d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* 
n]
 

rule 4258

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(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]
 

Maple [F]

\[\int \frac {1}{\left (\operatorname {csch}\left (d x +c \right ) b \right )^{\frac {7}{2}}}d x\]

Input:

int(1/(csch(d*x+c)*b)^(7/2),x)
 

Output:

int(1/(csch(d*x+c)*b)^(7/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (92) = 184\).

Time = 0.10 (sec) , antiderivative size = 483, normalized size of antiderivative = 4.09 \[ \int \frac {1}{(b \text {csch}(c+d x))^{7/2}} \, dx=\frac {80 \, \sqrt {2} {\left (\cosh \left (d x + c\right )^{4} + 4 \, \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + \sinh \left (d x + c\right )^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + \sqrt {2} {\left (3 \, \cosh \left (d x + c\right )^{8} + 24 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \, \sinh \left (d x + c\right )^{8} + 2 \, {\left (42 \, \cosh \left (d x + c\right )^{2} - 13\right )} \sinh \left (d x + c\right )^{6} - 26 \, \cosh \left (d x + c\right )^{6} + 12 \, {\left (14 \, \cosh \left (d x + c\right )^{3} - 13 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 30 \, {\left (7 \, \cosh \left (d x + c\right )^{4} - 13 \, \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (21 \, \cosh \left (d x + c\right )^{5} - 65 \, \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (42 \, \cosh \left (d x + c\right )^{6} - 195 \, \cosh \left (d x + c\right )^{4} + 13\right )} \sinh \left (d x + c\right )^{2} + 26 \, \cosh \left (d x + c\right )^{2} + 4 \, {\left (6 \, \cosh \left (d x + c\right )^{7} - 39 \, \cosh \left (d x + c\right )^{5} + 13 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 3\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}}}{168 \, {\left (b^{4} d \cosh \left (d x + c\right )^{4} + 4 \, b^{4} d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, b^{4} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, b^{4} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{4} d \sinh \left (d x + c\right )^{4}\right )}} \] Input:

integrate(1/(b*csch(d*x+c))^(7/2),x, algorithm="fricas")
 

Output:

1/168*(80*sqrt(2)*(cosh(d*x + c)^4 + 4*cosh(d*x + c)^3*sinh(d*x + c) + 6*c 
osh(d*x + c)^2*sinh(d*x + c)^2 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d* 
x + c)^4)*sqrt(b)*weierstrassPInverse(4, 0, cosh(d*x + c) + sinh(d*x + c)) 
 + sqrt(2)*(3*cosh(d*x + c)^8 + 24*cosh(d*x + c)*sinh(d*x + c)^7 + 3*sinh( 
d*x + c)^8 + 2*(42*cosh(d*x + c)^2 - 13)*sinh(d*x + c)^6 - 26*cosh(d*x + c 
)^6 + 12*(14*cosh(d*x + c)^3 - 13*cosh(d*x + c))*sinh(d*x + c)^5 + 30*(7*c 
osh(d*x + c)^4 - 13*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(21*cosh(d*x + c) 
^5 - 65*cosh(d*x + c)^3)*sinh(d*x + c)^3 + 2*(42*cosh(d*x + c)^6 - 195*cos 
h(d*x + c)^4 + 13)*sinh(d*x + c)^2 + 26*cosh(d*x + c)^2 + 4*(6*cosh(d*x + 
c)^7 - 39*cosh(d*x + c)^5 + 13*cosh(d*x + c))*sinh(d*x + c) - 3)*sqrt((b*c 
osh(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)))/(b^4*d*cosh(d*x + c)^4 + 4*b^4*d*cosh(d*x 
+ c)^3*sinh(d*x + c) + 6*b^4*d*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b^4*d*c 
osh(d*x + c)*sinh(d*x + c)^3 + b^4*d*sinh(d*x + c)^4)
 

Sympy [F]

\[ \int \frac {1}{(b \text {csch}(c+d x))^{7/2}} \, dx=\int \frac {1}{\left (b \operatorname {csch}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \] Input:

integrate(1/(b*csch(d*x+c))**(7/2),x)
 

Output:

Integral((b*csch(c + d*x))**(-7/2), x)
 

Maxima [F]

\[ \int \frac {1}{(b \text {csch}(c+d x))^{7/2}} \, dx=\int { \frac {1}{\left (b \operatorname {csch}\left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(b*csch(d*x+c))^(7/2),x, algorithm="maxima")
 

Output:

integrate((b*csch(d*x + c))^(-7/2), x)
 

Giac [F]

\[ \int \frac {1}{(b \text {csch}(c+d x))^{7/2}} \, dx=\int { \frac {1}{\left (b \operatorname {csch}\left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(b*csch(d*x+c))^(7/2),x, algorithm="giac")
 

Output:

integrate((b*csch(d*x + c))^(-7/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(b \text {csch}(c+d x))^{7/2}} \, dx=\int \frac {1}{{\left (\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}\right )}^{7/2}} \,d x \] Input:

int(1/(b/sinh(c + d*x))^(7/2),x)
 

Output:

int(1/(b/sinh(c + d*x))^(7/2), x)
 

Reduce [F]

\[ \int \frac {1}{(b \text {csch}(c+d x))^{7/2}} \, dx=\frac {\sqrt {b}\, \left (\int \frac {\sqrt {\mathrm {csch}\left (d x +c \right )}}{\mathrm {csch}\left (d x +c \right )^{4}}d x \right )}{b^{4}} \] Input:

int(1/(b*csch(d*x+c))^(7/2),x)
 

Output:

(sqrt(b)*int(sqrt(csch(c + d*x))/csch(c + d*x)**4,x))/b**4