3.51 \(\int \frac{1}{(a \text{sech}^4(x))^{5/2}} \, dx\)

Optimal. Leaf size=132 \[ \frac{63 x \text{sech}^2(x)}{256 a^2 \sqrt{a \text{sech}^4(x)}}+\frac{63 \tanh (x)}{256 a^2 \sqrt{a \text{sech}^4(x)}}+\frac{\sinh (x) \cosh ^7(x)}{10 a^2 \sqrt{a \text{sech}^4(x)}}+\frac{9 \sinh (x) \cosh ^5(x)}{80 a^2 \sqrt{a \text{sech}^4(x)}}+\frac{21 \sinh (x) \cosh ^3(x)}{160 a^2 \sqrt{a \text{sech}^4(x)}}+\frac{21 \sinh (x) \cosh (x)}{128 a^2 \sqrt{a \text{sech}^4(x)}} \]

[Out]

(63*x*Sech[x]^2)/(256*a^2*Sqrt[a*Sech[x]^4]) + (21*Cosh[x]*Sinh[x])/(128*a^2*Sqrt[a*Sech[x]^4]) + (21*Cosh[x]^
3*Sinh[x])/(160*a^2*Sqrt[a*Sech[x]^4]) + (9*Cosh[x]^5*Sinh[x])/(80*a^2*Sqrt[a*Sech[x]^4]) + (Cosh[x]^7*Sinh[x]
)/(10*a^2*Sqrt[a*Sech[x]^4]) + (63*Tanh[x])/(256*a^2*Sqrt[a*Sech[x]^4])

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Rubi [A]  time = 0.0561511, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4123, 2635, 8} \[ \frac{63 x \text{sech}^2(x)}{256 a^2 \sqrt{a \text{sech}^4(x)}}+\frac{63 \tanh (x)}{256 a^2 \sqrt{a \text{sech}^4(x)}}+\frac{\sinh (x) \cosh ^7(x)}{10 a^2 \sqrt{a \text{sech}^4(x)}}+\frac{9 \sinh (x) \cosh ^5(x)}{80 a^2 \sqrt{a \text{sech}^4(x)}}+\frac{21 \sinh (x) \cosh ^3(x)}{160 a^2 \sqrt{a \text{sech}^4(x)}}+\frac{21 \sinh (x) \cosh (x)}{128 a^2 \sqrt{a \text{sech}^4(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(63*x*Sech[x]^2)/(256*a^2*Sqrt[a*Sech[x]^4]) + (21*Cosh[x]*Sinh[x])/(128*a^2*Sqrt[a*Sech[x]^4]) + (21*Cosh[x]^
3*Sinh[x])/(160*a^2*Sqrt[a*Sech[x]^4]) + (9*Cosh[x]^5*Sinh[x])/(80*a^2*Sqrt[a*Sech[x]^4]) + (Cosh[x]^7*Sinh[x]
)/(10*a^2*Sqrt[a*Sech[x]^4]) + (63*Tanh[x])/(256*a^2*Sqrt[a*Sech[x]^4])

Rule 4123

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(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]

Rule 2635

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] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (a \text{sech}^4(x)\right )^{5/2}} \, dx &=\frac{\text{sech}^2(x) \int \cosh ^{10}(x) \, dx}{a^2 \sqrt{a \text{sech}^4(x)}}\\ &=\frac{\cosh ^7(x) \sinh (x)}{10 a^2 \sqrt{a \text{sech}^4(x)}}+\frac{\left (9 \text{sech}^2(x)\right ) \int \cosh ^8(x) \, dx}{10 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{\left (63 \text{sech}^2(x)\right ) \int \cosh ^6(x) \, dx}{80 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{\left (21 \text{sech}^2(x)\right ) \int \cosh ^4(x) \, dx}{32 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{\left (63 \text{sech}^2(x)\right ) \int \cosh ^2(x) \, dx}{128 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)}}+\frac{\left (63 \text{sech}^2(x)\right ) \int 1 \, dx}{256 a^2 \sqrt{a \text{sech}^4(x)}}\\ &=\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)}}\\ \end{align*}

Mathematica [A]  time = 0.0762338, size = 55, normalized size = 0.42 \[ \frac{(2520 x+2100 \sinh (2 x)+600 \sinh (4 x)+150 \sinh (6 x)+25 \sinh (8 x)+2 \sinh (10 x)) \cosh ^2(x) \sqrt{a \text{sech}^4(x)}}{10240 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[x]^2*Sqrt[a*Sech[x]^4]*(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)

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Maple [B]  time = 0.069, size = 362, normalized size = 2.7 \begin{align*}{\frac{63\,{{\rm e}^{2\,x}}x}{256\,{a}^{2} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+{\frac{{{\rm e}^{12\,x}}}{10240\,{a}^{2} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+{\frac{5\,{{\rm e}^{10\,x}}}{4096\,{a}^{2} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+{\frac{15\,{{\rm e}^{8\,x}}}{2048\,{a}^{2} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+{\frac{15\,{{\rm e}^{6\,x}}}{512\,{a}^{2} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+{\frac{105\,{{\rm e}^{4\,x}}}{1024\,{a}^{2} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}-{\frac{105}{1024\,{a}^{2} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}-{\frac{15\,{{\rm e}^{-2\,x}}}{512\,{a}^{2} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}-{\frac{15\,{{\rm e}^{-4\,x}}}{2048\,{a}^{2} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}-{\frac{5\,{{\rm e}^{-6\,x}}}{4096\,{a}^{2} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}-{\frac{{{\rm e}^{-8\,x}}}{10240\,{a}^{2} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sech(x)^4)^(5/2),x)

[Out]

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)+5/4096/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*exp(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^2/(exp(
2*x)+1)^2/(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)-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*exp(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)

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Maxima [A]  time = 1.74461, size = 139, normalized size = 1.05 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)^4)^(5/2),x, algorithm="maxima")

[Out]

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
*sqrt(a))*e^(10*x)/a^3 + 63/256*x/a^(5/2)

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Fricas [B]  time = 2.761, size = 9443, normalized size = 71.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)^4)^(5/2),x, algorithm="fricas")

[Out]

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) + cos
h(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*cosh(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 + 300*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*cos
h(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*c
osh(x)^9 + 9945*cosh(x)^7 + 8190*cosh(x)^5 + 2730*cosh(x)^3 + (4199*cosh(x)^9 + 9945*cosh(x)^7 + 8190*cosh(x)^
5 + 2730*cosh(x)^3 + 315*cosh(x))*e^(4*x) + 2*(4199*cosh(x)^9 + 9945*cosh(x)^7 + 8190*cosh(x)^5 + 2730*cosh(x)
^3 + 315*cosh(x))*e^(2*x) + 315*cosh(x))*sinh(x)^11 + 5040*x*cosh(x)^10 + 2*(184756*cosh(x)^10 + 546975*cosh(x
)^8 + 600600*cosh(x)^6 + 300300*cosh(x)^4 + 69300*cosh(x)^2 + (184756*cosh(x)^10 + 546975*cosh(x)^8 + 600600*c
osh(x)^6 + 300300*cosh(x)^4 + 69300*cosh(x)^2 + 2520*x)*e^(4*x) + 2*(184756*cosh(x)^10 + 546975*cosh(x)^8 + 60
0600*cosh(x)^6 + 300300*cosh(x)^4 + 69300*cosh(x)^2 + 2520*x)*e^(2*x) + 2520*x)*sinh(x)^10 + 20*(16796*cosh(x)
^11 + 60775*cosh(x)^9 + 85800*cosh(x)^7 + 60060*cosh(x)^5 + 23100*cosh(x)^3 + 2520*x*cosh(x) + (16796*cosh(x)^
11 + 60775*cosh(x)^9 + 85800*cosh(x)^7 + 60060*cosh(x)^5 + 23100*cosh(x)^3 + 2520*x*cosh(x))*e^(4*x) + 2*(1679
6*cosh(x)^11 + 60775*cosh(x)^9 + 85800*cosh(x)^7 + 60060*cosh(x)^5 + 23100*cosh(x)^3 + 2520*x*cosh(x))*e^(2*x)
)*sinh(x)^9 + 30*(8398*cosh(x)^12 + 36465*cosh(x)^10 + 64350*cosh(x)^8 + 60060*cosh(x)^6 + 34650*cosh(x)^4 + 7
560*x*cosh(x)^2 + (8398*cosh(x)^12 + 36465*cosh(x)^10 + 64350*cosh(x)^8 + 60060*cosh(x)^6 + 34650*cosh(x)^4 +
7560*x*cosh(x)^2 - 70)*e^(4*x) + 2*(8398*cosh(x)^12 + 36465*cosh(x)^10 + 64350*cosh(x)^8 + 60060*cosh(x)^6 + 3
4650*cosh(x)^4 + 7560*x*cosh(x)^2 - 70)*e^(2*x) - 70)*sinh(x)^8 - 2100*cosh(x)^8 + 240*(646*cosh(x)^13 + 3315*
cosh(x)^11 + 7150*cosh(x)^9 + 8580*cosh(x)^7 + 6930*cosh(x)^5 + 2520*x*cosh(x)^3 + (646*cosh(x)^13 + 3315*cosh
(x)^11 + 7150*cosh(x)^9 + 8580*cosh(x)^7 + 6930*cosh(x)^5 + 2520*x*cosh(x)^3 - 70*cosh(x))*e^(4*x) + 2*(646*co
sh(x)^13 + 3315*cosh(x)^11 + 7150*cosh(x)^9 + 8580*cosh(x)^7 + 6930*cosh(x)^5 + 2520*x*cosh(x)^3 - 70*cosh(x))
*e^(2*x) - 70*cosh(x))*sinh(x)^7 + 60*(1292*cosh(x)^14 + 7735*cosh(x)^12 + 20020*cosh(x)^10 + 30030*cosh(x)^8
+ 32340*cosh(x)^6 + 17640*x*cosh(x)^4 - 980*cosh(x)^2 + (1292*cosh(x)^14 + 7735*cosh(x)^12 + 20020*cosh(x)^10
+ 30030*cosh(x)^8 + 32340*cosh(x)^6 + 17640*x*cosh(x)^4 - 980*cosh(x)^2 - 10)*e^(4*x) + 2*(1292*cosh(x)^14 + 7
735*cosh(x)^12 + 20020*cosh(x)^10 + 30030*cosh(x)^8 + 32340*cosh(x)^6 + 17640*x*cosh(x)^4 - 980*cosh(x)^2 - 10
)*e^(2*x) - 10)*sinh(x)^6 - 600*cosh(x)^6 + 24*(1292*cosh(x)^15 + 8925*cosh(x)^13 + 27300*cosh(x)^11 + 50050*c
osh(x)^9 + 69300*cosh(x)^7 + 52920*x*cosh(x)^5 - 4900*cosh(x)^3 + (1292*cosh(x)^15 + 8925*cosh(x)^13 + 27300*c
osh(x)^11 + 50050*cosh(x)^9 + 69300*cosh(x)^7 + 52920*x*cosh(x)^5 - 4900*cosh(x)^3 - 150*cosh(x))*e^(4*x) + 2*
(1292*cosh(x)^15 + 8925*cosh(x)^13 + 27300*cosh(x)^11 + 50050*cosh(x)^9 + 69300*cosh(x)^7 + 52920*x*cosh(x)^5
- 4900*cosh(x)^3 - 150*cosh(x))*e^(2*x) - 150*cosh(x))*sinh(x)^5 + 30*(323*cosh(x)^16 + 2550*cosh(x)^14 + 9100
*cosh(x)^12 + 20020*cosh(x)^10 + 34650*cosh(x)^8 + 35280*x*cosh(x)^6 - 4900*cosh(x)^4 - 300*cosh(x)^2 + (323*c
osh(x)^16 + 2550*cosh(x)^14 + 9100*cosh(x)^12 + 20020*cosh(x)^10 + 34650*cosh(x)^8 + 35280*x*cosh(x)^6 - 4900*
cosh(x)^4 - 300*cosh(x)^2 - 5)*e^(4*x) + 2*(323*cosh(x)^16 + 2550*cosh(x)^14 + 9100*cosh(x)^12 + 20020*cosh(x)
^10 + 34650*cosh(x)^8 + 35280*x*cosh(x)^6 - 4900*cosh(x)^4 - 300*cosh(x)^2 - 5)*e^(2*x) - 5)*sinh(x)^4 - 150*c
osh(x)^4 + 120*(19*cosh(x)^17 + 170*cosh(x)^15 + 700*cosh(x)^13 + 1820*cosh(x)^11 + 3850*cosh(x)^9 + 5040*x*co
sh(x)^7 - 980*cosh(x)^5 - 100*cosh(x)^3 + (19*cosh(x)^17 + 170*cosh(x)^15 + 700*cosh(x)^13 + 1820*cosh(x)^11 +
3850*cosh(x)^9 + 5040*x*cosh(x)^7 - 980*cosh(x)^5 - 100*cosh(x)^3 - 5*cosh(x))*e^(4*x) + 2*(19*cosh(x)^17 + 1
70*cosh(x)^15 + 700*cosh(x)^13 + 1820*cosh(x)^11 + 3850*cosh(x)^9 + 5040*x*cosh(x)^7 - 980*cosh(x)^5 - 100*cos
h(x)^3 - 5*cosh(x))*e^(2*x) - 5*cosh(x))*sinh(x)^3 + 5*(76*cosh(x)^18 + 765*cosh(x)^16 + 3600*cosh(x)^14 + 109
20*cosh(x)^12 + 27720*cosh(x)^10 + 45360*x*cosh(x)^8 - 11760*cosh(x)^6 - 1800*cosh(x)^4 - 180*cosh(x)^2 + (76*
cosh(x)^18 + 765*cosh(x)^16 + 3600*cosh(x)^14 + 10920*cosh(x)^12 + 27720*cosh(x)^10 + 45360*x*cosh(x)^8 - 1176
0*cosh(x)^6 - 1800*cosh(x)^4 - 180*cosh(x)^2 - 5)*e^(4*x) + 2*(76*cosh(x)^18 + 765*cosh(x)^16 + 3600*cosh(x)^1
4 + 10920*cosh(x)^12 + 27720*cosh(x)^10 + 45360*x*cosh(x)^8 - 11760*cosh(x)^6 - 1800*cosh(x)^4 - 180*cosh(x)^2
- 5)*e^(2*x) - 5)*sinh(x)^2 - 25*cosh(x)^2 + (2*cosh(x)^20 + 25*cosh(x)^18 + 150*cosh(x)^16 + 600*cosh(x)^14
+ 2100*cosh(x)^12 + 5040*x*cosh(x)^10 - 2100*cosh(x)^8 - 600*cosh(x)^6 - 150*cosh(x)^4 - 25*cosh(x)^2 - 2)*e^(
4*x) + 2*(2*cosh(x)^20 + 25*cosh(x)^18 + 150*cosh(x)^16 + 600*cosh(x)^14 + 2100*cosh(x)^12 + 5040*x*cosh(x)^10
- 2100*cosh(x)^8 - 600*cosh(x)^6 - 150*cosh(x)^4 - 25*cosh(x)^2 - 2)*e^(2*x) + 10*(4*cosh(x)^19 + 45*cosh(x)^
17 + 240*cosh(x)^15 + 840*cosh(x)^13 + 2520*cosh(x)^11 + 5040*x*cosh(x)^9 - 1680*cosh(x)^7 - 360*cosh(x)^5 - 6
0*cosh(x)^3 + (4*cosh(x)^19 + 45*cosh(x)^17 + 240*cosh(x)^15 + 840*cosh(x)^13 + 2520*cosh(x)^11 + 5040*x*cosh(
x)^9 - 1680*cosh(x)^7 - 360*cosh(x)^5 - 60*cosh(x)^3 - 5*cosh(x))*e^(4*x) + 2*(4*cosh(x)^19 + 45*cosh(x)^17 +
240*cosh(x)^15 + 840*cosh(x)^13 + 2520*cosh(x)^11 + 5040*x*cosh(x)^9 - 1680*cosh(x)^7 - 360*cosh(x)^5 - 60*cos
h(x)^3 - 5*cosh(x))*e^(2*x) - 5*cosh(x))*sinh(x) - 2)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1)
)*e^(2*x)/(a^3*cosh(x)^10*e^(2*x) + 10*a^3*cosh(x)^9*e^(2*x)*sinh(x) + 45*a^3*cosh(x)^8*e^(2*x)*sinh(x)^2 + 12
0*a^3*cosh(x)^7*e^(2*x)*sinh(x)^3 + 210*a^3*cosh(x)^6*e^(2*x)*sinh(x)^4 + 252*a^3*cosh(x)^5*e^(2*x)*sinh(x)^5
+ 210*a^3*cosh(x)^4*e^(2*x)*sinh(x)^6 + 120*a^3*cosh(x)^3*e^(2*x)*sinh(x)^7 + 45*a^3*cosh(x)^2*e^(2*x)*sinh(x)
^8 + 10*a^3*cosh(x)*e^(2*x)*sinh(x)^9 + a^3*e^(2*x)*sinh(x)^10)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{sech}^{4}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)**4)**(5/2),x)

[Out]

Integral((a*sech(x)**4)**(-5/2), x)

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Giac [A]  time = 1.12396, size = 103, normalized size = 0.78 \begin{align*} -\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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)^4)^(5/2),x, algorithm="giac")

[Out]

-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) - 600*e^(4*x) - 2100*e^(2*x))/a^(5/2)