3.1.22 \(\int \frac {1}{(b \sinh (c+d x))^{7/2}} \, dx\) [22]

3.1.22.1 Optimal result

Integrand size = 12, antiderivative size = 118 \[ \int \frac {1}{(b \sinh (c+d x))^{7/2}} \, dx=-\frac {2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}+\frac {6 \cosh (c+d x)}{5 b^3 d \sqrt {b \sinh (c+d x)}}+\frac {6 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{5 b^4 d \sqrt {i \sinh (c+d x)}} \]

output

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

3.1.22.2 Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(b \sinh (c+d x))^{7/2}} \, dx=-\frac {2 \left (-3 \cosh (c+d x)+\coth (c+d x) \text {csch}(c+d x)+3 E\left (\left .\frac {1}{4} (-2 i c+\pi -2 i d x)\right |2\right ) \sqrt {i \sinh (c+d x)}\right )}{5 b^3 d \sqrt {b \sinh (c+d x)}} \]

input

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

output

(-2*(-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*b^3*d*Sqrt[b*Sinh[c + 
 d*x]])
 

3.1.22.3 Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3116, 3042, 3116, 3042, 3121, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3121

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

input

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

output

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

3.1.22.3.1 Defintions of rubi rules used

rule 3042

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

rule 3116

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1))   I 
nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && 
 IntegerQ[2*n]
 

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 3121

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) 
^n/Sin[c + d*x]^n   Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt 
Q[-1, n, 1] && IntegerQ[2*n]
 

3.1.22.4 Maple [A] (verified)

Time = 0.93 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.74

method	result	size
default	\(-\frac {6 \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (i-\sinh \left (d x +c \right )\right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \sinh \left (d x +c \right )^{2} \operatorname {EllipticE}\left (\sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (i-\sinh \left (d x +c \right )\right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \sinh \left (d x +c \right )^{2} \operatorname {EllipticF}\left (\sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-6 \sinh \left (d x +c \right )^{4}-4 \sinh \left (d x +c \right )^{2}+2}{5 b^{3} \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right ) \sqrt {b \sinh \left (d x +c \right )}\, d}\)	\(205\)

input

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

output

-1/5/b^3/sinh(d*x+c)^2*(6*(-I*(sinh(d*x+c)+I))^(1/2)*2^(1/2)*(-I*(I-sinh(d 
*x+c)))^(1/2)*(I*sinh(d*x+c))^(1/2)*sinh(d*x+c)^2*EllipticE((-I*(sinh(d*x+ 
c)+I))^(1/2),1/2*2^(1/2))-3*(-I*(sinh(d*x+c)+I))^(1/2)*2^(1/2)*(-I*(I-sinh 
(d*x+c)))^(1/2)*(I*sinh(d*x+c))^(1/2)*sinh(d*x+c)^2*EllipticF((-I*(sinh(d* 
x+c)+I))^(1/2),1/2*2^(1/2))-6*sinh(d*x+c)^4-4*sinh(d*x+c)^2+2)/cosh(d*x+c) 
/(b*sinh(d*x+c))^(1/2)/d
 

3.1.22.5 Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 675, normalized size of antiderivative = 5.72 \[ \int \frac {1}{(b \sinh (c+d x))^{7/2}} \, dx=\frac {2 \, {\left (3 \, {\left (\sqrt {2} \cosh \left (d x + c\right )^{6} + 6 \, \sqrt {2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + \sqrt {2} \sinh \left (d x + c\right )^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (d x + c\right )^{2} - \sqrt {2}\right )} \sinh \left (d x + c\right )^{4} - 3 \, \sqrt {2} \cosh \left (d x + c\right )^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \left (d x + c\right )^{3} - 3 \, \sqrt {2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (d x + c\right )^{4} - 6 \, \sqrt {2} \cosh \left (d x + c\right )^{2} + \sqrt {2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, \sqrt {2} \cosh \left (d x + c\right )^{2} + 6 \, {\left (\sqrt {2} \cosh \left (d x + c\right )^{5} - 2 \, \sqrt {2} \cosh \left (d x + c\right )^{3} + \sqrt {2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - \sqrt {2}\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 ) + 2 \, {\left (3 \, \cosh \left (d x + c\right )^{6} + 18 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 3 \, \sinh \left (d x + c\right )^{6} + {\left (45 \, \cosh \left (d x + c\right )^{2} - 8\right )} \sinh \left (d x + c\right )^{4} - 8 \, \cosh \left (d x + c\right )^{4} + 4 \, {\left (15 \, \cosh \left (d x + c\right )^{3} - 8 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + {\left (45 \, \cosh \left (d x + c\right )^{4} - 48 \, \cosh \left (d x + c\right )^{2} + 1\right )} \sinh \left (d x + c\right )^{2} + \cosh \left (d x + c\right )^{2} + 2 \, {\left (9 \, \cosh \left (d x + c\right )^{5} - 16 \, \cosh \left (d x + c\right )^{3} + \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {b \sinh \left (d x + c\right )}\right )}}{5 \, {\left (b^{4} d \cosh \left (d x + c\right )^{6} + 6 \, b^{4} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b^{4} d \sinh \left (d x + c\right )^{6} - 3 \, b^{4} d \cosh \left (d x + c\right )^{4} + 3 \, b^{4} d \cosh \left (d x + c\right )^{2} - b^{4} d + 3 \, {\left (5 \, b^{4} d \cosh \left (d x + c\right )^{2} - b^{4} d\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b^{4} d \cosh \left (d x + c\right )^{3} - 3 \, b^{4} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (5 \, b^{4} d \cosh \left (d x + c\right )^{4} - 6 \, b^{4} d \cosh \left (d x + c\right )^{2} + b^{4} d\right )} \sinh \left (d x + c\right )^{2} + 6 \, {\left (b^{4} d \cosh \left (d x + c\right )^{5} - 2 \, b^{4} d \cosh \left (d x + c\right )^{3} + b^{4} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \]

input

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

output

2/5*(3*(sqrt(2)*cosh(d*x + c)^6 + 6*sqrt(2)*cosh(d*x + c)*sinh(d*x + c)^5 
+ sqrt(2)*sinh(d*x + c)^6 + 3*(5*sqrt(2)*cosh(d*x + c)^2 - sqrt(2))*sinh(d 
*x + c)^4 - 3*sqrt(2)*cosh(d*x + c)^4 + 4*(5*sqrt(2)*cosh(d*x + c)^3 - 3*s 
qrt(2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*sqrt(2)*cosh(d*x + c)^4 - 6*s 
qrt(2)*cosh(d*x + c)^2 + sqrt(2))*sinh(d*x + c)^2 + 3*sqrt(2)*cosh(d*x + c 
)^2 + 6*(sqrt(2)*cosh(d*x + c)^5 - 2*sqrt(2)*cosh(d*x + c)^3 + sqrt(2)*cos 
h(d*x + c))*sinh(d*x + c) - sqrt(2))*sqrt(b)*weierstrassZeta(4, 0, weierst 
rassPInverse(4, 0, cosh(d*x + c) + sinh(d*x + c))) + 2*(3*cosh(d*x + c)^6 
+ 18*cosh(d*x + c)*sinh(d*x + c)^5 + 3*sinh(d*x + c)^6 + (45*cosh(d*x + c) 
^2 - 8)*sinh(d*x + c)^4 - 8*cosh(d*x + c)^4 + 4*(15*cosh(d*x + c)^3 - 8*co 
sh(d*x + c))*sinh(d*x + c)^3 + (45*cosh(d*x + c)^4 - 48*cosh(d*x + c)^2 + 
1)*sinh(d*x + c)^2 + cosh(d*x + c)^2 + 2*(9*cosh(d*x + c)^5 - 16*cosh(d*x 
+ c)^3 + cosh(d*x + c))*sinh(d*x + c))*sqrt(b*sinh(d*x + c)))/(b^4*d*cosh( 
d*x + c)^6 + 6*b^4*d*cosh(d*x + c)*sinh(d*x + c)^5 + b^4*d*sinh(d*x + c)^6 
 - 3*b^4*d*cosh(d*x + c)^4 + 3*b^4*d*cosh(d*x + c)^2 - b^4*d + 3*(5*b^4*d* 
cosh(d*x + c)^2 - b^4*d)*sinh(d*x + c)^4 + 4*(5*b^4*d*cosh(d*x + c)^3 - 3* 
b^4*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*b^4*d*cosh(d*x + c)^4 - 6*b^4* 
d*cosh(d*x + c)^2 + b^4*d)*sinh(d*x + c)^2 + 6*(b^4*d*cosh(d*x + c)^5 - 2* 
b^4*d*cosh(d*x + c)^3 + b^4*d*cosh(d*x + c))*sinh(d*x + c))
 

3.1.22.6 Sympy [F]

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

input

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

output

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

3.1.22.7 Maxima [F]

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

input

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

output

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

3.1.22.8 Giac [F]

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

input

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

output

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

3.1.22.9 Mupad [F(-1)]

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

input

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

output

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