\(\int (b \text {csch}(c+d x))^{7/2} \, dx\) [13]

Optimal result

Integrand size = 12, antiderivative size = 116 \[ \int (b \text {csch}(c+d x))^{7/2} \, dx=\frac {6 b^3 \cosh (c+d x) \sqrt {b \text {csch}(c+d x)}}{5 d}-\frac {2 b \cosh (c+d x) (b \text {csch}(c+d x))^{5/2}}{5 d}+\frac {6 i b^4 E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{5 d \sqrt {b \text {csch}(c+d x)} \sqrt {i \sinh (c+d x)}} \] Output:

6/5*b^3*cosh(d*x+c)*(b*csch(d*x+c))^(1/2)/d-2/5*b*cosh(d*x+c)*(b*csch(d*x+ 
c))^(5/2)/d-6/5*I*b^4*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)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68 \[ \int (b \text {csch}(c+d x))^{7/2} \, dx=-\frac {2 b^3 \sqrt {b \text {csch}(c+d x)} \left (-3 \cosh (c+d x)+\coth (c+d x) \text {csch}(c+d x)+3 E\left (\left .\frac {1}{4} (-2 i c+\pi -2 i d x)\right |2\right ) \sqrt {i \sinh (c+d x)}\right )}{5 d} \] Input:

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

Output:

(-2*b^3*Sqrt[b*Csch[c + d*x]]*(-3*Cosh[c + d*x] + Coth[c + d*x]*Csch[c + d 
*x] + 3*EllipticE[((-2*I)*c + Pi - (2*I)*d*x)/4, 2]*Sqrt[I*Sinh[c + d*x]]) 
)/(5*d)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4255, 3042, 4255, 3042, 4258, 3042, 3119}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4255

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4255

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

Input:

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

Output:

(-2*b*Cosh[c + d*x]*(b*Csch[c + d*x])^(5/2))/(5*d) - (3*b^2*((-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]])))/5
 

Defintions of rubi rules used

rule 3042

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

rule 3119

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 4255

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] + Simp[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 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]

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

Input:

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

Output:

int((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. 484 vs. \(2 (91) = 182\).

Time = 0.09 (sec) , antiderivative size = 484, normalized size of antiderivative = 4.17 \[ \int (b \text {csch}(c+d x))^{7/2} \, dx=\frac {2 \, {\left (3 \, \sqrt {2} {\left (b^{3} \cosh \left (d x + c\right )^{4} + 4 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{3} \sinh \left (d x + c\right )^{4} - 2 \, b^{3} \cosh \left (d x + c\right )^{2} + b^{3} + 2 \, {\left (3 \, b^{3} \cosh \left (d x + c\right )^{2} - b^{3}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b^{3} \cosh \left (d x + c\right )^{3} - b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {b} {\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 (3 \, b^{3} \cosh \left (d x + c\right )^{5} + 15 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 3 \, b^{3} \sinh \left (d x + c\right )^{5} - 8 \, b^{3} \cosh \left (d x + c\right )^{3} + b^{3} \cosh \left (d x + c\right ) + 2 \, {\left (15 \, b^{3} \cosh \left (d x + c\right )^{2} - 4 \, b^{3}\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{3} - 4 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + {\left (15 \, b^{3} \cosh \left (d x + c\right )^{4} - 24 \, b^{3} \cosh \left (d x + c\right )^{2} + b^{3}\right )} \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 )}}{5 \, {\left (d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} - 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d\right )}} \] Input:

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

Output:

2/5*(3*sqrt(2)*(b^3*cosh(d*x + c)^4 + 4*b^3*cosh(d*x + c)*sinh(d*x + c)^3 
+ b^3*sinh(d*x + c)^4 - 2*b^3*cosh(d*x + c)^2 + b^3 + 2*(3*b^3*cosh(d*x + 
c)^2 - b^3)*sinh(d*x + c)^2 + 4*(b^3*cosh(d*x + c)^3 - b^3*cosh(d*x + c))* 
sinh(d*x + c))*sqrt(b)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos 
h(d*x + c) + sinh(d*x + c))) + sqrt(2)*(3*b^3*cosh(d*x + c)^5 + 15*b^3*cos 
h(d*x + c)*sinh(d*x + c)^4 + 3*b^3*sinh(d*x + c)^5 - 8*b^3*cosh(d*x + c)^3 
 + b^3*cosh(d*x + c) + 2*(15*b^3*cosh(d*x + c)^2 - 4*b^3)*sinh(d*x + c)^3 
+ 6*(5*b^3*cosh(d*x + c)^3 - 4*b^3*cosh(d*x + c))*sinh(d*x + c)^2 + (15*b^ 
3*cosh(d*x + c)^4 - 24*b^3*cosh(d*x + c)^2 + b^3)*sinh(d*x + c))*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)))/(d*cosh(d*x + c)^4 + 4*d*cosh(d*x + c)*sin 
h(d*x + c)^3 + d*sinh(d*x + c)^4 - 2*d*cosh(d*x + c)^2 + 2*(3*d*cosh(d*x + 
 c)^2 - d)*sinh(d*x + c)^2 + 4*(d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh( 
d*x + c) + d)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

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

Output:

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

Reduce [F]

\[ \int (b \text {csch}(c+d x))^{7/2} \, dx=\sqrt {b}\, \left (\int \sqrt {\mathrm {csch}\left (d x +c \right )}\, \mathrm {csch}\left (d x +c \right )^{3}d x \right ) b^{3} \] Input:

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

Output:

sqrt(b)*int(sqrt(csch(c + d*x))*csch(c + d*x)**3,x)*b**3