Integrand size = 10, antiderivative size = 132 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{5/2}} \, dx=\frac {63 \coth (x)}{256 a^2 \sqrt {a \text {csch}^4(x)}}-\frac {63 x \text {csch}^2(x)}{256 a^2 \sqrt {a \text {csch}^4(x)}}-\frac {21 \cosh (x) \sinh (x)}{128 a^2 \sqrt {a \text {csch}^4(x)}}+\frac {21 \cosh (x) \sinh ^3(x)}{160 a^2 \sqrt {a \text {csch}^4(x)}}-\frac {9 \cosh (x) \sinh ^5(x)}{80 a^2 \sqrt {a \text {csch}^4(x)}}+\frac {\cosh (x) \sinh ^7(x)}{10 a^2 \sqrt {a \text {csch}^4(x)}} \] Output:
63/256*coth(x)/a^2/(a*csch(x)^4)^(1/2)-63/256*x*csch(x)^2/a^2/(a*csch(x)^4 )^(1/2)-21/128*cosh(x)*sinh(x)/a^2/(a*csch(x)^4)^(1/2)+21/160*cosh(x)*sinh (x)^3/a^2/(a*csch(x)^4)^(1/2)-9/80*cosh(x)*sinh(x)^5/a^2/(a*csch(x)^4)^(1/ 2)+1/10*cosh(x)*sinh(x)^7/a^2/(a*csch(x)^4)^(1/2)
Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{5/2}} \, dx=\frac {\sqrt {a \text {csch}^4(x)} \sinh ^2(x) (-2520 x+2100 \sinh (2 x)-600 \sinh (4 x)+150 \sinh (6 x)-25 \sinh (8 x)+2 \sinh (10 x))}{10240 a^3} \] Input:
Integrate[(a*Csch[x]^4)^(-5/2),x]
Output:
(Sqrt[a*Csch[x]^4]*Sinh[x]^2*(-2520*x + 2100*Sinh[2*x] - 600*Sinh[4*x] + 1 50*Sinh[6*x] - 25*Sinh[8*x] + 2*Sinh[10*x]))/(10240*a^3)
Time = 0.49 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.70, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.900, Rules used = {3042, 4611, 25, 3042, 25, 3115, 3042, 3115, 25, 3042, 25, 3115, 3042, 3115, 25, 3042, 25, 3115, 24}
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.
| \(\displaystyle \int \frac {1}{\left (a \text {csch}^4(x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \sec \left (\frac {\pi }{2}+i x\right )^4\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4611 |
| \(\displaystyle -\frac {\text {csch}^2(x) \int -\sinh ^{10}(x)dx}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\text {csch}^2(x) \int \sinh ^{10}(x)dx}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {csch}^2(x) \int -\sin (i x)^{10}dx}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\text {csch}^2(x) \int \sin (i x)^{10}dx}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (\frac {9}{10} \int \sinh ^8(x)dx-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (-\frac {1}{10} \sinh ^9(x) \cosh (x)+\frac {9}{10} \int \sin (i x)^8dx\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \int -\sinh ^6(x)dx+\frac {1}{8} \sinh ^7(x) \cosh (x)\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)-\frac {7}{8} \int \sinh ^6(x)dx\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (-\frac {1}{10} \sinh ^9(x) \cosh (x)+\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)-\frac {7}{8} \int -\sin (i x)^6dx\right )\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (-\frac {1}{10} \sinh ^9(x) \cosh (x)+\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)+\frac {7}{8} \int \sin (i x)^6dx\right )\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \sinh ^4(x)dx-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )+\frac {1}{8} \sinh ^7(x) \cosh (x)\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (-\frac {1}{10} \sinh ^9(x) \cosh (x)+\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)+\frac {7}{8} \left (-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \int \sin (i x)^4dx\right )\right )\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int -\sinh ^2(x)dx+\frac {1}{4} \sinh ^3(x) \cosh (x)\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )+\frac {1}{8} \sinh ^7(x) \cosh (x)\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int \sinh ^2(x)dx\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )+\frac {1}{8} \sinh ^7(x) \cosh (x)\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (-\frac {1}{10} \sinh ^9(x) \cosh (x)+\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)+\frac {7}{8} \left (-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int -\sin (i x)^2dx\right )\right )\right )\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (-\frac {1}{10} \sinh ^9(x) \cosh (x)+\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)+\frac {7}{8} \left (-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \int \sin (i x)^2dx\right )\right )\right )\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {1}{4} \sinh ^3(x) \cosh (x)\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )+\frac {1}{8} \sinh ^7(x) \cosh (x)\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {\text {csch}^2(x) \left (\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)+\frac {7}{8} \left (\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )}{a^2 \sqrt {a \text {csch}^4(x)}}\) |
Input:
Int[(a*Csch[x]^4)^(-5/2),x]
Output:
-((Csch[x]^2*(-1/10*(Cosh[x]*Sinh[x]^9) + (9*((Cosh[x]*Sinh[x]^7)/8 + (7*( -1/6*(Cosh[x]*Sinh[x]^5) + (5*((Cosh[x]*Sinh[x]^3)/4 + (3*(x/2 - (Cosh[x]* Sinh[x])/2))/4))/6))/8))/10))/(a^2*Sqrt[a*Csch[x]^4]))
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(361\) vs. \(2(108)=216\).
Time = 0.14 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.74
| method | result | size |
| risch | \(-\frac {63 \,{\mathrm e}^{2 x} x}{256 a^{2} \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}+\frac {{\mathrm e}^{12 x}}{10240 a^{2} \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}-\frac {5 \,{\mathrm e}^{10 x}}{4096 a^{2} \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}+\frac {15 \,{\mathrm e}^{8 x}}{2048 a^{2} \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}-\frac {15 \,{\mathrm e}^{6 x}}{512 a^{2} \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}+\frac {105 \,{\mathrm e}^{4 x}}{1024 a^{2} \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}-\frac {105}{1024 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 x}-1\right )^{2} a^{2}}+\frac {15 \,{\mathrm e}^{-2 x}}{512 a^{2} \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}-\frac {15 \,{\mathrm e}^{-4 x}}{2048 a^{2} \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}+\frac {5 \,{\mathrm e}^{-6 x}}{4096 a^{2} \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}-\frac {{\mathrm e}^{-8 x}}{10240 a^{2} \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}\) | \(362\) |
Input:
int(1/(a*csch(x)^4)^(5/2),x,method=_RETURNVERBOSE)
Output:
-63/256/a^2*exp(2*x)/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)*x+1/ 10240/a^2*exp(12*x)/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)-5/409 6/a^2*exp(10*x)/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)+15/2048/a ^2*exp(8*x)/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)-15/512/a^2*ex p(6*x)/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)+105/1024/a^2*exp(4 *x)/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)-105/1024/(a*exp(4*x)/ (exp(2*x)-1)^4)^(1/2)/(exp(2*x)-1)^2/a^2+15/512/a^2*exp(-2*x)/(exp(2*x)-1) ^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)-15/2048/a^2*exp(-4*x)/(exp(2*x)-1)^2/ (a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)+5/4096/a^2*exp(-6*x)/(exp(2*x)-1)^2/(a*e xp(4*x)/(exp(2*x)-1)^4)^(1/2)-1/10240/a^2*exp(-8*x)/(exp(2*x)-1)^2/(a*exp( 4*x)/(exp(2*x)-1)^4)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 2600 vs. \(2 (108) = 216\).
Time = 0.13 (sec) , antiderivative size = 2600, normalized size of antiderivative = 19.70 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(a*csch(x)^4)^(5/2),x, algorithm="fricas")
Output:
1/20480*(2*(e^(4*x) - 2*e^(2*x) + 1)*sinh(x)^20 + 2*cosh(x)^20 + 40*(cosh( x)*e^(4*x) - 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^19 + 5*(76*cosh(x)^2 + ( 76*cosh(x)^2 - 5)*e^(4*x) - 2*(76*cosh(x)^2 - 5)*e^(2*x) - 5)*sinh(x)^18 - 25*cosh(x)^18 + 30*(76*cosh(x)^3 + (76*cosh(x)^3 - 15*cosh(x))*e^(4*x) - 2*(76*cosh(x)^3 - 15*cosh(x))*e^(2*x) - 15*cosh(x))*sinh(x)^17 + 15*(646*c osh(x)^4 - 255*cosh(x)^2 + (646*cosh(x)^4 - 255*cosh(x)^2 + 10)*e^(4*x) - 2*(646*cosh(x)^4 - 255*cosh(x)^2 + 10)*e^(2*x) + 10)*sinh(x)^16 + 150*cosh (x)^16 + 48*(646*cosh(x)^5 - 425*cosh(x)^3 + (646*cosh(x)^5 - 425*cosh(x)^ 3 + 50*cosh(x))*e^(4*x) - 2*(646*cosh(x)^5 - 425*cosh(x)^3 + 50*cosh(x))*e ^(2*x) + 50*cosh(x))*sinh(x)^15 + 60*(1292*cosh(x)^6 - 1275*cosh(x)^4 + 30 0*cosh(x)^2 + (1292*cosh(x)^6 - 1275*cosh(x)^4 + 300*cosh(x)^2 - 10)*e^(4* x) - 2*(1292*cosh(x)^6 - 1275*cosh(x)^4 + 300*cosh(x)^2 - 10)*e^(2*x) - 10 )*sinh(x)^14 - 600*cosh(x)^14 + 120*(1292*cosh(x)^7 - 1785*cosh(x)^5 + 700 *cosh(x)^3 + (1292*cosh(x)^7 - 1785*cosh(x)^5 + 700*cosh(x)^3 - 70*cosh(x) )*e^(4*x) - 2*(1292*cosh(x)^7 - 1785*cosh(x)^5 + 700*cosh(x)^3 - 70*cosh(x ))*e^(2*x) - 70*cosh(x))*sinh(x)^13 + 60*(4199*cosh(x)^8 - 7735*cosh(x)^6 + 4550*cosh(x)^4 - 910*cosh(x)^2 + (4199*cosh(x)^8 - 7735*cosh(x)^6 + 4550 *cosh(x)^4 - 910*cosh(x)^2 + 35)*e^(4*x) - 2*(4199*cosh(x)^8 - 7735*cosh(x )^6 + 4550*cosh(x)^4 - 910*cosh(x)^2 + 35)*e^(2*x) + 35)*sinh(x)^12 + 2100 *cosh(x)^12 + 80*(4199*cosh(x)^9 - 9945*cosh(x)^7 + 8190*cosh(x)^5 - 27...
\[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \operatorname {csch}^{4}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(a*csch(x)**4)**(5/2),x)
Output:
Integral((a*csch(x)**4)**(-5/2), x)
Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{5/2}} \, dx=-\frac {{\left (25 \, e^{\left (-2 \, x\right )} - 150 \, e^{\left (-4 \, x\right )} + 600 \, e^{\left (-6 \, x\right )} - 2100 \, e^{\left (-8 \, x\right )} + 2100 \, e^{\left (-12 \, x\right )} - 600 \, e^{\left (-14 \, x\right )} + 150 \, e^{\left (-16 \, x\right )} - 25 \, e^{\left (-18 \, x\right )} + 2 \, e^{\left (-20 \, x\right )} - 2\right )} e^{\left (10 \, x\right )}}{20480 \, a^{\frac {5}{2}}} - \frac {63 \, x}{256 \, a^{\frac {5}{2}}} \] Input:
integrate(1/(a*csch(x)^4)^(5/2),x, algorithm="maxima")
Output:
-1/20480*(25*e^(-2*x) - 150*e^(-4*x) + 600*e^(-6*x) - 2100*e^(-8*x) + 2100 *e^(-12*x) - 600*e^(-14*x) + 150*e^(-16*x) - 25*e^(-18*x) + 2*e^(-20*x) - 2)*e^(10*x)/a^(5/2) - 63/256*x/a^(5/2)
Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{5/2}} \, dx=\frac {{\left (5754 \, e^{\left (10 \, x\right )} - 2100 \, e^{\left (8 \, x\right )} + 600 \, e^{\left (6 \, x\right )} - 150 \, e^{\left (4 \, x\right )} + 25 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-10 \, x\right )} - 5040 \, x + 2 \, e^{\left (10 \, x\right )} - 25 \, e^{\left (8 \, x\right )} + 150 \, e^{\left (6 \, x\right )} - 600 \, e^{\left (4 \, x\right )} + 2100 \, e^{\left (2 \, x\right )}}{20480 \, a^{\frac {5}{2}}} \] Input:
integrate(1/(a*csch(x)^4)^(5/2),x, algorithm="giac")
Output:
1/20480*((5754*e^(10*x) - 2100*e^(8*x) + 600*e^(6*x) - 150*e^(4*x) + 25*e^ (2*x) - 2)*e^(-10*x) - 5040*x + 2*e^(10*x) - 25*e^(8*x) + 150*e^(6*x) - 60 0*e^(4*x) + 2100*e^(2*x))/a^(5/2)
Timed out. \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {sinh}\left (x\right )}^4}\right )}^{5/2}} \,d x \] Input:
int(1/(a/sinh(x)^4)^(5/2),x)
Output:
int(1/(a/sinh(x)^4)^(5/2), x)
Time = 0.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{5/2}} \, dx=\frac {\sqrt {a}\, \left (2 e^{20 x}-25 e^{18 x}+150 e^{16 x}-600 e^{14 x}+2100 e^{12 x}-5040 e^{10 x} x -2100 e^{8 x}+600 e^{6 x}-150 e^{4 x}+25 e^{2 x}-2\right )}{20480 e^{10 x} a^{3}} \] Input:
int(1/(a*csch(x)^4)^(5/2),x)
Output:
(sqrt(a)*(2*e**(20*x) - 25*e**(18*x) + 150*e**(16*x) - 600*e**(14*x) + 210 0*e**(12*x) - 5040*e**(10*x)*x - 2100*e**(8*x) + 600*e**(6*x) - 150*e**(4* x) + 25*e**(2*x) - 2))/(20480*e**(10*x)*a**3)