Integrand size = 10, antiderivative size = 132 \[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{5/2}} \, dx=\frac {63 x \text {sech}^2(x)}{256 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {21 \cosh (x) \sinh (x)}{128 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {21 \cosh ^3(x) \sinh (x)}{160 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {9 \cosh ^5(x) \sinh (x)}{80 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {\cosh ^7(x) \sinh (x)}{10 a^2 \sqrt {a \text {sech}^4(x)}}+\frac {63 \tanh (x)}{256 a^2 \sqrt {a \text {sech}^4(x)}} \]
63/256*x*sech(x)^2/a^2/(a*sech(x)^4)^(1/2)+21/128*cosh(x)*sinh(x)/a^2/(a*s ech(x)^4)^(1/2)+21/160*cosh(x)^3*sinh(x)/a^2/(a*sech(x)^4)^(1/2)+9/80*cosh (x)^5*sinh(x)/a^2/(a*sech(x)^4)^(1/2)+1/10*cosh(x)^7*sinh(x)/a^2/(a*sech(x )^4)^(1/2)+63/256*tanh(x)/a^2/(a*sech(x)^4)^(1/2)
Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{5/2}} \, dx=\frac {\cosh ^2(x) \sqrt {a \text {sech}^4(x)} (2520 x+2100 \sinh (2 x)+600 \sinh (4 x)+150 \sinh (6 x)+25 \sinh (8 x)+2 \sinh (10 x))}{10240 a^3} \]
(Cosh[x]^2*Sqrt[a*Sech[x]^4]*(2520*x + 2100*Sinh[2*x] + 600*Sinh[4*x] + 15 0*Sinh[6*x] + 25*Sinh[8*x] + 2*Sinh[10*x]))/(10240*a^3)
Time = 0.49 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.70, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {3042, 4611, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 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.
| \(\displaystyle \int \frac {1}{\left (a \text {sech}^4(x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \sec (i x)^4\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4611 |
| \(\displaystyle \frac {\text {sech}^2(x) \int \cosh ^{10}(x)dx}{a^2 \sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^2(x) \int \sin \left (i x+\frac {\pi }{2}\right )^{10}dx}{a^2 \sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\text {sech}^2(x) \left (\frac {9}{10} \int \cosh ^8(x)dx+\frac {1}{10} \sinh (x) \cosh ^9(x)\right )}{a^2 \sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^2(x) \left (\frac {1}{10} \sinh (x) \cosh ^9(x)+\frac {9}{10} \int \sin \left (i x+\frac {\pi }{2}\right )^8dx\right )}{a^2 \sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\text {sech}^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \int \cosh ^6(x)dx+\frac {1}{8} \sinh (x) \cosh ^7(x)\right )+\frac {1}{10} \sinh (x) \cosh ^9(x)\right )}{a^2 \sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^2(x) \left (\frac {1}{10} \sinh (x) \cosh ^9(x)+\frac {9}{10} \left (\frac {1}{8} \sinh (x) \cosh ^7(x)+\frac {7}{8} \int \sin \left (i x+\frac {\pi }{2}\right )^6dx\right )\right )}{a^2 \sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\text {sech}^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \cosh ^4(x)dx+\frac {1}{6} \sinh (x) \cosh ^5(x)\right )+\frac {1}{8} \sinh (x) \cosh ^7(x)\right )+\frac {1}{10} \sinh (x) \cosh ^9(x)\right )}{a^2 \sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^2(x) \left (\frac {1}{10} \sinh (x) \cosh ^9(x)+\frac {9}{10} \left (\frac {1}{8} \sinh (x) \cosh ^7(x)+\frac {7}{8} \left (\frac {1}{6} \sinh (x) \cosh ^5(x)+\frac {5}{6} \int \sin \left (i x+\frac {\pi }{2}\right )^4dx\right )\right )\right )}{a^2 \sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\text {sech}^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cosh ^2(x)dx+\frac {1}{4} \sinh (x) \cosh ^3(x)\right )+\frac {1}{6} \sinh (x) \cosh ^5(x)\right )+\frac {1}{8} \sinh (x) \cosh ^7(x)\right )+\frac {1}{10} \sinh (x) \cosh ^9(x)\right )}{a^2 \sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^2(x) \left (\frac {1}{10} \sinh (x) \cosh ^9(x)+\frac {9}{10} \left (\frac {1}{8} \sinh (x) \cosh ^7(x)+\frac {7}{8} \left (\frac {1}{6} \sinh (x) \cosh ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sinh (x) \cosh ^3(x)+\frac {3}{4} \int \sin \left (i x+\frac {\pi }{2}\right )^2dx\right )\right )\right )\right )}{a^2 \sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\text {sech}^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {1}{4} \sinh (x) \cosh ^3(x)\right )+\frac {1}{6} \sinh (x) \cosh ^5(x)\right )+\frac {1}{8} \sinh (x) \cosh ^7(x)\right )+\frac {1}{10} \sinh (x) \cosh ^9(x)\right )}{a^2 \sqrt {a \text {sech}^4(x)}}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\text {sech}^2(x) \left (\frac {1}{10} \sinh (x) \cosh ^9(x)+\frac {9}{10} \left (\frac {1}{8} \sinh (x) \cosh ^7(x)+\frac {7}{8} \left (\frac {1}{6} \sinh (x) \cosh ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sinh (x) \cosh ^3(x)+\frac {3}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\right )\right )\right )\right )}{a^2 \sqrt {a \text {sech}^4(x)}}\) |
(Sech[x]^2*((Cosh[x]^9*Sinh[x])/10 + (9*((Cosh[x]^7*Sinh[x])/8 + (7*((Cosh [x]^5*Sinh[x])/6 + (5*((Cosh[x]^3*Sinh[x])/4 + (3*(x/2 + (Cosh[x]*Sinh[x]) /2))/4))/6))/8))/10))/(a^2*Sqrt[a*Sech[x]^4])
3.1.51.3.1 Defintions of rubi rules used
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 = 0.16 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.74
| method | result | size |
| risch | \(\frac {63 \,{\mathrm e}^{2 x} x}{256 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}+\frac {{\mathrm e}^{12 x}}{10240 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}+\frac {5 \,{\mathrm e}^{10 x}}{4096 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}+\frac {15 \,{\mathrm e}^{8 x}}{2048 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}+\frac {15 \,{\mathrm e}^{6 x}}{512 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}+\frac {105 \,{\mathrm e}^{4 x}}{1024 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}-\frac {105}{1024 \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2} a^{2}}-\frac {15 \,{\mathrm e}^{-2 x}}{512 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}-\frac {15 \,{\mathrm e}^{-4 x}}{2048 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}-\frac {5 \,{\mathrm e}^{-6 x}}{4096 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}-\frac {{\mathrm e}^{-8 x}}{10240 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}\) | \(362\) |
63/256/a^2*exp(2*x)/(1+exp(2*x))^2/(exp(4*x)*a/(1+exp(2*x))^4)^(1/2)*x+1/1 0240/a^2*exp(12*x)/(1+exp(2*x))^2/(exp(4*x)*a/(1+exp(2*x))^4)^(1/2)+5/4096 /a^2*exp(10*x)/(1+exp(2*x))^2/(exp(4*x)*a/(1+exp(2*x))^4)^(1/2)+15/2048/a^ 2*exp(8*x)/(1+exp(2*x))^2/(exp(4*x)*a/(1+exp(2*x))^4)^(1/2)+15/512/a^2*exp (6*x)/(1+exp(2*x))^2/(exp(4*x)*a/(1+exp(2*x))^4)^(1/2)+105/1024/a^2*exp(4* x)/(1+exp(2*x))^2/(exp(4*x)*a/(1+exp(2*x))^4)^(1/2)-105/1024/(exp(4*x)*a/( 1+exp(2*x))^4)^(1/2)/(1+exp(2*x))^2/a^2-15/512/a^2*exp(-2*x)/(1+exp(2*x))^ 2/(exp(4*x)*a/(1+exp(2*x))^4)^(1/2)-15/2048/a^2*exp(-4*x)/(1+exp(2*x))^2/( exp(4*x)*a/(1+exp(2*x))^4)^(1/2)-5/4096/a^2*exp(-6*x)/(1+exp(2*x))^2/(exp( 4*x)*a/(1+exp(2*x))^4)^(1/2)-1/10240/a^2*exp(-8*x)/(1+exp(2*x))^2/(exp(4*x )*a/(1+exp(2*x))^4)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 2600 vs. \(2 (108) = 216\).
Time = 0.32 (sec) , antiderivative size = 2600, normalized size of antiderivative = 19.70 \[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
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...
\[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \operatorname {sech}^{4}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{5/2}} \, dx=\frac {{\left (25 \, \sqrt {a} e^{\left (-2 \, x\right )} + 150 \, \sqrt {a} e^{\left (-4 \, x\right )} + 600 \, \sqrt {a} e^{\left (-6 \, x\right )} + 2100 \, \sqrt {a} e^{\left (-8 \, x\right )} - 2100 \, \sqrt {a} e^{\left (-12 \, x\right )} - 600 \, \sqrt {a} e^{\left (-14 \, x\right )} - 150 \, \sqrt {a} e^{\left (-16 \, x\right )} - 25 \, \sqrt {a} e^{\left (-18 \, x\right )} - 2 \, \sqrt {a} e^{\left (-20 \, x\right )} + 2 \, \sqrt {a}\right )} e^{\left (10 \, x\right )}}{20480 \, a^{3}} + \frac {63 \, x}{256 \, a^{\frac {5}{2}}} \]
1/20480*(25*sqrt(a)*e^(-2*x) + 150*sqrt(a)*e^(-4*x) + 600*sqrt(a)*e^(-6*x) + 2100*sqrt(a)*e^(-8*x) - 2100*sqrt(a)*e^(-12*x) - 600*sqrt(a)*e^(-14*x) - 150*sqrt(a)*e^(-16*x) - 25*sqrt(a)*e^(-18*x) - 2*sqrt(a)*e^(-20*x) + 2*s qrt(a))*e^(10*x)/a^3 + 63/256*x/a^(5/2)
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{5/2}} \, dx=-\frac {{\left (5754 \, e^{\left (10 \, x\right )} + 2100 \, e^{\left (8 \, x\right )} + 600 \, e^{\left (6 \, x\right )} + 150 \, e^{\left (4 \, x\right )} + 25 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-10 \, x\right )} - 5040 \, x - 2 \, e^{\left (10 \, x\right )} - 25 \, e^{\left (8 \, x\right )} - 150 \, e^{\left (6 \, x\right )} - 600 \, e^{\left (4 \, x\right )} - 2100 \, e^{\left (2 \, x\right )}}{20480 \, a^{\frac {5}{2}}} \]
-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) - 6 00*e^(4*x) - 2100*e^(2*x))/a^(5/2)
Timed out. \[ \int \frac {1}{\left (a \text {sech}^4(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^4}\right )}^{5/2}} \,d x \]