\(\int (a \text {csch}^4(x))^{5/2} \, dx\) [43]

Optimal result

Integrand size = 10, antiderivative size = 118 \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=\frac {4}{3} a^2 \cosh ^2(x) \coth (x) \sqrt {a \text {csch}^4(x)}-\frac {6}{5} a^2 \cosh ^2(x) \coth ^3(x) \sqrt {a \text {csch}^4(x)}+\frac {4}{7} a^2 \cosh ^2(x) \coth ^5(x) \sqrt {a \text {csch}^4(x)}-\frac {1}{9} a^2 \cosh ^2(x) \coth ^7(x) \sqrt {a \text {csch}^4(x)}-a^2 \cosh (x) \sqrt {a \text {csch}^4(x)} \sinh (x) \] Output:

4/3*a^2*cosh(x)^2*coth(x)*(a*csch(x)^4)^(1/2)-6/5*a^2*cosh(x)^2*coth(x)^3* 
(a*csch(x)^4)^(1/2)+4/7*a^2*cosh(x)^2*coth(x)^5*(a*csch(x)^4)^(1/2)-1/9*a^ 
2*cosh(x)^2*coth(x)^7*(a*csch(x)^4)^(1/2)-a^2*cosh(x)*(a*csch(x)^4)^(1/2)* 
sinh(x)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.40 \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=-\frac {1}{315} a^2 \cosh (x) \sqrt {a \text {csch}^4(x)} \left (128-64 \text {csch}^2(x)+48 \text {csch}^4(x)-40 \text {csch}^6(x)+35 \text {csch}^8(x)\right ) \sinh (x) \] Input:

Integrate[(a*Csch[x]^4)^(5/2),x]
 

Output:

-1/315*(a^2*Cosh[x]*Sqrt[a*Csch[x]^4]*(128 - 64*Csch[x]^2 + 48*Csch[x]^4 - 
 40*Csch[x]^6 + 35*Csch[x]^8)*Sinh[x])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.58, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3042, 4611, 25, 3042, 25, 4254, 2009}

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.

\(\displaystyle \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a \sec \left (\frac {\pi }{2}+i x\right )^4\right )^{5/2}dx\)

\(\Big \downarrow \) 4611

\(\displaystyle -a^2 \sinh ^2(x) \sqrt {a \text {csch}^4(x)} \int -\text {csch}^{10}(x)dx\)

\(\Big \downarrow \) 25

\(\displaystyle a^2 \sinh ^2(x) \sqrt {a \text {csch}^4(x)} \int \text {csch}^{10}(x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 \sinh ^2(x) \sqrt {a \text {csch}^4(x)} \int -\csc (i x)^{10}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -a^2 \sinh ^2(x) \sqrt {a \text {csch}^4(x)} \int \csc (i x)^{10}dx\)

\(\Big \downarrow \) 4254

\(\displaystyle -i a^2 \sinh ^2(x) \sqrt {a \text {csch}^4(x)} \int \left (\coth ^8(x)-4 \coth ^6(x)+6 \coth ^4(x)-4 \coth ^2(x)+1\right )d(-i \coth (x))\)

\(\Big \downarrow \) 2009

\(\displaystyle -i a^2 \sinh ^2(x) \left (-\frac {1}{9} i \coth ^9(x)+\frac {4}{7} i \coth ^7(x)-\frac {6}{5} i \coth ^5(x)+\frac {4}{3} i \coth ^3(x)-i \coth (x)\right ) \sqrt {a \text {csch}^4(x)}\)

Input:

Int[(a*Csch[x]^4)^(5/2),x]
 

Output:

(-I)*a^2*((-I)*Coth[x] + ((4*I)/3)*Coth[x]^3 - ((6*I)/5)*Coth[x]^5 + ((4*I 
)/7)*Coth[x]^7 - (I/9)*Coth[x]^9)*Sqrt[a*Csch[x]^4]*Sinh[x]^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4254

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, 
 d}, x] && IGtQ[n/2, 0]
 

rule 4611

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Simp[b^ 
IntPart[p]*((b*(c*Sec[e + f*x])^n)^FracPart[p]/(c*Sec[e + f*x])^(n*FracPart 
[p]))   Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p}, x] 
&&  !IntegerQ[p]
 

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.51

method	result	size
risch	\(-\frac {256 a^{2} {\mathrm e}^{-2 x} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}\, \left (126 \,{\mathrm e}^{8 x}-84 \,{\mathrm e}^{6 x}+36 \,{\mathrm e}^{4 x}-9 \,{\mathrm e}^{2 x}+1\right )}{315 \left ({\mathrm e}^{2 x}-1\right )^{7}}\)	\(60\)

Input:

int((a*csch(x)^4)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-256/315*a^2*exp(-2*x)/(exp(2*x)-1)^7*(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)*(1 
26*exp(8*x)-84*exp(6*x)+36*exp(4*x)-9*exp(2*x)+1)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1493 vs. \(2 (100) = 200\).

Time = 0.12 (sec) , antiderivative size = 1493, normalized size of antiderivative = 12.65 \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((a*csch(x)^4)^(5/2),x, algorithm="fricas")
 

Output:

-256/315*(126*a^2*cosh(x)^8 + 126*(a^2*e^(4*x) - 2*a^2*e^(2*x) + a^2)*sinh 
(x)^8 - 84*a^2*cosh(x)^6 + 1008*(a^2*cosh(x)*e^(4*x) - 2*a^2*cosh(x)*e^(2* 
x) + a^2*cosh(x))*sinh(x)^7 + 84*(42*a^2*cosh(x)^2 - a^2 + (42*a^2*cosh(x) 
^2 - a^2)*e^(4*x) - 2*(42*a^2*cosh(x)^2 - a^2)*e^(2*x))*sinh(x)^6 + 36*a^2 
*cosh(x)^4 + 504*(14*a^2*cosh(x)^3 - a^2*cosh(x) + (14*a^2*cosh(x)^3 - a^2 
*cosh(x))*e^(4*x) - 2*(14*a^2*cosh(x)^3 - a^2*cosh(x))*e^(2*x))*sinh(x)^5 
+ 36*(245*a^2*cosh(x)^4 - 35*a^2*cosh(x)^2 + a^2 + (245*a^2*cosh(x)^4 - 35 
*a^2*cosh(x)^2 + a^2)*e^(4*x) - 2*(245*a^2*cosh(x)^4 - 35*a^2*cosh(x)^2 + 
a^2)*e^(2*x))*sinh(x)^4 - 9*a^2*cosh(x)^2 + 48*(147*a^2*cosh(x)^5 - 35*a^2 
*cosh(x)^3 + 3*a^2*cosh(x) + (147*a^2*cosh(x)^5 - 35*a^2*cosh(x)^3 + 3*a^2 
*cosh(x))*e^(4*x) - 2*(147*a^2*cosh(x)^5 - 35*a^2*cosh(x)^3 + 3*a^2*cosh(x 
))*e^(2*x))*sinh(x)^3 + 9*(392*a^2*cosh(x)^6 - 140*a^2*cosh(x)^4 + 24*a^2* 
cosh(x)^2 - a^2 + (392*a^2*cosh(x)^6 - 140*a^2*cosh(x)^4 + 24*a^2*cosh(x)^ 
2 - a^2)*e^(4*x) - 2*(392*a^2*cosh(x)^6 - 140*a^2*cosh(x)^4 + 24*a^2*cosh( 
x)^2 - a^2)*e^(2*x))*sinh(x)^2 + a^2 + (126*a^2*cosh(x)^8 - 84*a^2*cosh(x) 
^6 + 36*a^2*cosh(x)^4 - 9*a^2*cosh(x)^2 + a^2)*e^(4*x) - 2*(126*a^2*cosh(x 
)^8 - 84*a^2*cosh(x)^6 + 36*a^2*cosh(x)^4 - 9*a^2*cosh(x)^2 + a^2)*e^(2*x) 
 + 18*(56*a^2*cosh(x)^7 - 28*a^2*cosh(x)^5 + 8*a^2*cosh(x)^3 - a^2*cosh(x) 
 + (56*a^2*cosh(x)^7 - 28*a^2*cosh(x)^5 + 8*a^2*cosh(x)^3 - a^2*cosh(x))*e 
^(4*x) - 2*(56*a^2*cosh(x)^7 - 28*a^2*cosh(x)^5 + 8*a^2*cosh(x)^3 - a^2...
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=\int \left (a \operatorname {csch}^{4}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \] Input:

integrate((a*csch(x)**4)**(5/2),x)
 

Output:

Integral((a*csch(x)**4)**(5/2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (100) = 200\).

Time = 0.13 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.73 \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=-\frac {256 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )}}{35 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac {1024 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )}}{35 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} - \frac {1024 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )}}{15 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac {512 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )}}{5 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac {256 \, a^{\frac {5}{2}}}{315 \, {\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} \] Input:

integrate((a*csch(x)^4)^(5/2),x, algorithm="maxima")
 

Output:

-256/35*a^(5/2)*e^(-2*x)/(9*e^(-2*x) - 36*e^(-4*x) + 84*e^(-6*x) - 126*e^( 
-8*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36*e^(-14*x) - 9*e^(-16*x) + e^(-18 
*x) - 1) + 1024/35*a^(5/2)*e^(-4*x)/(9*e^(-2*x) - 36*e^(-4*x) + 84*e^(-6*x 
) - 126*e^(-8*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36*e^(-14*x) - 9*e^(-16* 
x) + e^(-18*x) - 1) - 1024/15*a^(5/2)*e^(-6*x)/(9*e^(-2*x) - 36*e^(-4*x) + 
 84*e^(-6*x) - 126*e^(-8*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36*e^(-14*x) 
- 9*e^(-16*x) + e^(-18*x) - 1) + 512/5*a^(5/2)*e^(-8*x)/(9*e^(-2*x) - 36*e 
^(-4*x) + 84*e^(-6*x) - 126*e^(-8*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36*e 
^(-14*x) - 9*e^(-16*x) + e^(-18*x) - 1) + 256/315*a^(5/2)/(9*e^(-2*x) - 36 
*e^(-4*x) + 84*e^(-6*x) - 126*e^(-8*x) + 126*e^(-10*x) - 84*e^(-12*x) + 36 
*e^(-14*x) - 9*e^(-16*x) + e^(-18*x) - 1)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.33 \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=-\frac {256 \, a^{\frac {5}{2}} {\left (126 \, e^{\left (8 \, x\right )} - 84 \, e^{\left (6 \, x\right )} + 36 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} + 1\right )}}{315 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{9}} \] Input:

integrate((a*csch(x)^4)^(5/2),x, algorithm="giac")
 

Output:

-256/315*a^(5/2)*(126*e^(8*x) - 84*e^(6*x) + 36*e^(4*x) - 9*e^(2*x) + 1)/( 
e^(2*x) - 1)^9
 

Mupad [B] (verification not implemented)

Time = 2.34 (sec) , antiderivative size = 356, normalized size of antiderivative = 3.02 \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=-\frac {128\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{5\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^5\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {256\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^6\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {768\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{7\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^7\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {64\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^8\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {128\,a^2\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{9\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^9\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \] Input:

int((a/sinh(x)^4)^(5/2),x)
 

Output:

- (128*a^2*(a/(exp(-x)/2 - exp(x)/2)^4)^(1/2)*(6*exp(4*x) - 4*exp(2*x) - 4 
*exp(6*x) + exp(8*x) + 1))/(5*(exp(2*x) - 1)^5*(exp(2*x) - 2*exp(4*x) + ex 
p(6*x))) - (256*a^2*(a/(exp(-x)/2 - exp(x)/2)^4)^(1/2)*(6*exp(4*x) - 4*exp 
(2*x) - 4*exp(6*x) + exp(8*x) + 1))/(3*(exp(2*x) - 1)^6*(exp(2*x) - 2*exp( 
4*x) + exp(6*x))) - (768*a^2*(a/(exp(-x)/2 - exp(x)/2)^4)^(1/2)*(6*exp(4*x 
) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1))/(7*(exp(2*x) - 1)^7*(exp(2*x) 
 - 2*exp(4*x) + exp(6*x))) - (64*a^2*(a/(exp(-x)/2 - exp(x)/2)^4)^(1/2)*(6 
*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1))/((exp(2*x) - 1)^8*(ex 
p(2*x) - 2*exp(4*x) + exp(6*x))) - (128*a^2*(a/(exp(-x)/2 - exp(x)/2)^4)^( 
1/2)*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1))/(9*(exp(2*x) - 
 1)^9*(exp(2*x) - 2*exp(4*x) + exp(6*x)))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.88 \[ \int \left (a \text {csch}^4(x)\right )^{5/2} \, dx=\frac {256 \sqrt {a}\, a^{2} \left (-126 e^{8 x}+84 e^{6 x}-36 e^{4 x}+9 e^{2 x}-1\right )}{315 e^{18 x}-2835 e^{16 x}+11340 e^{14 x}-26460 e^{12 x}+39690 e^{10 x}-39690 e^{8 x}+26460 e^{6 x}-11340 e^{4 x}+2835 e^{2 x}-315} \] Input:

int((a*csch(x)^4)^(5/2),x)
 

Output:

(256*sqrt(a)*a**2*( - 126*e**(8*x) + 84*e**(6*x) - 36*e**(4*x) + 9*e**(2*x 
) - 1))/(315*(e**(18*x) - 9*e**(16*x) + 36*e**(14*x) - 84*e**(12*x) + 126* 
e**(10*x) - 126*e**(8*x) + 84*e**(6*x) - 36*e**(4*x) + 9*e**(2*x) - 1))