3.45 \(\int (a \text {sech}^4(x))^{7/2} \, dx\)

Optimal. Leaf size=163 \[ a^3 \sinh (x) \cosh (x) \sqrt {a \text {sech}^4(x)}+\frac {1}{13} a^3 \sinh ^2(x) \tanh ^{11}(x) \sqrt {a \text {sech}^4(x)}-\frac {6}{11} a^3 \sinh ^2(x) \tanh ^9(x) \sqrt {a \text {sech}^4(x)}+\frac {5}{3} a^3 \sinh ^2(x) \tanh ^7(x) \sqrt {a \text {sech}^4(x)}-\frac {20}{7} a^3 \sinh ^2(x) \tanh ^5(x) \sqrt {a \text {sech}^4(x)}+3 a^3 \sinh ^2(x) \tanh ^3(x) \sqrt {a \text {sech}^4(x)}-2 a^3 \sinh ^2(x) \tanh (x) \sqrt {a \text {sech}^4(x)} \]

[Out]

a^3*cosh(x)*sinh(x)*(a*sech(x)^4)^(1/2)-2*a^3*sinh(x)^2*(a*sech(x)^4)^(1/2)*tanh(x)+3*a^3*sinh(x)^2*(a*sech(x)
^4)^(1/2)*tanh(x)^3-20/7*a^3*sinh(x)^2*(a*sech(x)^4)^(1/2)*tanh(x)^5+5/3*a^3*sinh(x)^2*(a*sech(x)^4)^(1/2)*tan
h(x)^7-6/11*a^3*sinh(x)^2*(a*sech(x)^4)^(1/2)*tanh(x)^9+1/13*a^3*sinh(x)^2*(a*sech(x)^4)^(1/2)*tanh(x)^11

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Rubi [A]  time = 0.04, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4123, 3767} \[ a^3 \sinh (x) \cosh (x) \sqrt {a \text {sech}^4(x)}+\frac {1}{13} a^3 \sinh ^2(x) \tanh ^{11}(x) \sqrt {a \text {sech}^4(x)}-\frac {6}{11} a^3 \sinh ^2(x) \tanh ^9(x) \sqrt {a \text {sech}^4(x)}+\frac {5}{3} a^3 \sinh ^2(x) \tanh ^7(x) \sqrt {a \text {sech}^4(x)}-\frac {20}{7} a^3 \sinh ^2(x) \tanh ^5(x) \sqrt {a \text {sech}^4(x)}+3 a^3 \sinh ^2(x) \tanh ^3(x) \sqrt {a \text {sech}^4(x)}-2 a^3 \sinh ^2(x) \tanh (x) \sqrt {a \text {sech}^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sech[x]^4)^(7/2),x]

[Out]

a^3*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x] - 2*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x] + 3*a^3*Sqrt[a*Sech[x]^4]*Si
nh[x]^2*Tanh[x]^3 - (20*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^5)/7 + (5*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh
[x]^7)/3 - (6*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^9)/11 + (a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^11)/13

Rule 3767

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

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]

Rubi steps

\begin {align*} \int \left (a \text {sech}^4(x)\right )^{7/2} \, dx &=\left (a^3 \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int \text {sech}^{14}(x) \, dx\\ &=\left (i a^3 \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \operatorname {Subst}\left (\int \left (1+6 x^2+15 x^4+20 x^6+15 x^8+6 x^{10}+x^{12}\right ) \, dx,x,-i \tanh (x)\right )\\ &=a^3 \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-2 a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh (x)+3 a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^3(x)-\frac {20}{7} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^5(x)+\frac {5}{3} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^7(x)-\frac {6}{11} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^9(x)+\frac {1}{13} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^{11}(x)\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 54, normalized size = 0.33 \[ \frac {\sinh (x) \cosh (x) (2380 \cosh (2 x)+1093 \cosh (4 x)+378 \cosh (6 x)+92 \cosh (8 x)+14 \cosh (10 x)+\cosh (12 x)+2048) \left (a \text {sech}^4(x)\right )^{7/2}}{6006} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sech[x]^4)^(7/2),x]

[Out]

(Cosh[x]*(2048 + 2380*Cosh[2*x] + 1093*Cosh[4*x] + 378*Cosh[6*x] + 92*Cosh[8*x] + 14*Cosh[10*x] + Cosh[12*x])*
(a*Sech[x]^4)^(7/2)*Sinh[x])/6006

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fricas [B]  time = 0.53, size = 2804, normalized size = 17.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^4)^(7/2),x, algorithm="fricas")

[Out]

-2048/3003*(1716*a^3*cosh(x)^12 + 1287*a^3*cosh(x)^10 + 1716*(a^3*e^(4*x) + 2*a^3*e^(2*x) + a^3)*sinh(x)^12 +
20592*(a^3*cosh(x)*e^(4*x) + 2*a^3*cosh(x)*e^(2*x) + a^3*cosh(x))*sinh(x)^11 + 715*a^3*cosh(x)^8 + 1287*(88*a^
3*cosh(x)^2 + a^3 + (88*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(88*a^3*cosh(x)^2 + a^3)*e^(2*x))*sinh(x)^10 + 4290*(
88*a^3*cosh(x)^3 + 3*a^3*cosh(x) + (88*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(4*x) + 2*(88*a^3*cosh(x)^3 + 3*a^3*co
sh(x))*e^(2*x))*sinh(x)^9 + 286*a^3*cosh(x)^6 + 715*(1188*a^3*cosh(x)^4 + 81*a^3*cosh(x)^2 + a^3 + (1188*a^3*c
osh(x)^4 + 81*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(1188*a^3*cosh(x)^4 + 81*a^3*cosh(x)^2 + a^3)*e^(2*x))*sinh(x)^
8 + 1144*(1188*a^3*cosh(x)^5 + 135*a^3*cosh(x)^3 + 5*a^3*cosh(x) + (1188*a^3*cosh(x)^5 + 135*a^3*cosh(x)^3 + 5
*a^3*cosh(x))*e^(4*x) + 2*(1188*a^3*cosh(x)^5 + 135*a^3*cosh(x)^3 + 5*a^3*cosh(x))*e^(2*x))*sinh(x)^7 + 78*a^3
*cosh(x)^4 + 286*(5544*a^3*cosh(x)^6 + 945*a^3*cosh(x)^4 + 70*a^3*cosh(x)^2 + a^3 + (5544*a^3*cosh(x)^6 + 945*
a^3*cosh(x)^4 + 70*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(5544*a^3*cosh(x)^6 + 945*a^3*cosh(x)^4 + 70*a^3*cosh(x)^2
+ a^3)*e^(2*x))*sinh(x)^6 + 572*(2376*a^3*cosh(x)^7 + 567*a^3*cosh(x)^5 + 70*a^3*cosh(x)^3 + 3*a^3*cosh(x) +
(2376*a^3*cosh(x)^7 + 567*a^3*cosh(x)^5 + 70*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(4*x) + 2*(2376*a^3*cosh(x)^7 +
567*a^3*cosh(x)^5 + 70*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(2*x))*sinh(x)^5 + 13*a^3*cosh(x)^2 + 26*(32670*a^3*co
sh(x)^8 + 10395*a^3*cosh(x)^6 + 1925*a^3*cosh(x)^4 + 165*a^3*cosh(x)^2 + 3*a^3 + (32670*a^3*cosh(x)^8 + 10395*
a^3*cosh(x)^6 + 1925*a^3*cosh(x)^4 + 165*a^3*cosh(x)^2 + 3*a^3)*e^(4*x) + 2*(32670*a^3*cosh(x)^8 + 10395*a^3*c
osh(x)^6 + 1925*a^3*cosh(x)^4 + 165*a^3*cosh(x)^2 + 3*a^3)*e^(2*x))*sinh(x)^4 + 104*(3630*a^3*cosh(x)^9 + 1485
*a^3*cosh(x)^7 + 385*a^3*cosh(x)^5 + 55*a^3*cosh(x)^3 + 3*a^3*cosh(x) + (3630*a^3*cosh(x)^9 + 1485*a^3*cosh(x)
^7 + 385*a^3*cosh(x)^5 + 55*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(4*x) + 2*(3630*a^3*cosh(x)^9 + 1485*a^3*cosh(x)^
7 + 385*a^3*cosh(x)^5 + 55*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(2*x))*sinh(x)^3 + a^3 + 13*(8712*a^3*cosh(x)^10 +
4455*a^3*cosh(x)^8 + 1540*a^3*cosh(x)^6 + 330*a^3*cosh(x)^4 + 36*a^3*cosh(x)^2 + a^3 + (8712*a^3*cosh(x)^10 +
4455*a^3*cosh(x)^8 + 1540*a^3*cosh(x)^6 + 330*a^3*cosh(x)^4 + 36*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(8712*a^3*c
osh(x)^10 + 4455*a^3*cosh(x)^8 + 1540*a^3*cosh(x)^6 + 330*a^3*cosh(x)^4 + 36*a^3*cosh(x)^2 + a^3)*e^(2*x))*sin
h(x)^2 + (1716*a^3*cosh(x)^12 + 1287*a^3*cosh(x)^10 + 715*a^3*cosh(x)^8 + 286*a^3*cosh(x)^6 + 78*a^3*cosh(x)^4
+ 13*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(1716*a^3*cosh(x)^12 + 1287*a^3*cosh(x)^10 + 715*a^3*cosh(x)^8 + 286*a^
3*cosh(x)^6 + 78*a^3*cosh(x)^4 + 13*a^3*cosh(x)^2 + a^3)*e^(2*x) + 26*(792*a^3*cosh(x)^11 + 495*a^3*cosh(x)^9
+ 220*a^3*cosh(x)^7 + 66*a^3*cosh(x)^5 + 12*a^3*cosh(x)^3 + a^3*cosh(x) + (792*a^3*cosh(x)^11 + 495*a^3*cosh(x
)^9 + 220*a^3*cosh(x)^7 + 66*a^3*cosh(x)^5 + 12*a^3*cosh(x)^3 + a^3*cosh(x))*e^(4*x) + 2*(792*a^3*cosh(x)^11 +
495*a^3*cosh(x)^9 + 220*a^3*cosh(x)^7 + 66*a^3*cosh(x)^5 + 12*a^3*cosh(x)^3 + a^3*cosh(x))*e^(2*x))*sinh(x))*
sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)/(26*cosh(x)*e^(2*x)*sinh(x)^25 + e^(2*x)*sin
h(x)^26 + 13*(25*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^24 + 104*(25*cosh(x)^3 + 3*cosh(x))*e^(2*x)*sinh(x)^23 + 26*(5
75*cosh(x)^4 + 138*cosh(x)^2 + 3)*e^(2*x)*sinh(x)^22 + 572*(115*cosh(x)^5 + 46*cosh(x)^3 + 3*cosh(x))*e^(2*x)*
sinh(x)^21 + 286*(805*cosh(x)^6 + 483*cosh(x)^4 + 63*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^20 + 1144*(575*cosh(x)^7 +
483*cosh(x)^5 + 105*cosh(x)^3 + 5*cosh(x))*e^(2*x)*sinh(x)^19 + 143*(10925*cosh(x)^8 + 12236*cosh(x)^6 + 3990
*cosh(x)^4 + 380*cosh(x)^2 + 5)*e^(2*x)*sinh(x)^18 + 286*(10925*cosh(x)^9 + 15732*cosh(x)^7 + 7182*cosh(x)^5 +
1140*cosh(x)^3 + 45*cosh(x))*e^(2*x)*sinh(x)^17 + 143*(37145*cosh(x)^10 + 66861*cosh(x)^8 + 40698*cosh(x)^6 +
9690*cosh(x)^4 + 765*cosh(x)^2 + 9)*e^(2*x)*sinh(x)^16 + 208*(37145*cosh(x)^11 + 81719*cosh(x)^9 + 63954*cosh
(x)^7 + 21318*cosh(x)^5 + 2805*cosh(x)^3 + 99*cosh(x))*e^(2*x)*sinh(x)^15 + 52*(185725*cosh(x)^12 + 490314*cos
h(x)^10 + 479655*cosh(x)^8 + 213180*cosh(x)^6 + 42075*cosh(x)^4 + 2970*cosh(x)^2 + 33)*e^(2*x)*sinh(x)^14 + 8*
(1300075*cosh(x)^13 + 4056234*cosh(x)^11 + 4849845*cosh(x)^9 + 2771340*cosh(x)^7 + 765765*cosh(x)^5 + 90090*co
sh(x)^3 + 3003*cosh(x))*e^(2*x)*sinh(x)^13 + 52*(185725*cosh(x)^14 + 676039*cosh(x)^12 + 969969*cosh(x)^10 + 6
92835*cosh(x)^8 + 255255*cosh(x)^6 + 45045*cosh(x)^4 + 3003*cosh(x)^2 + 33)*e^(2*x)*sinh(x)^12 + 208*(37145*co
sh(x)^15 + 156009*cosh(x)^13 + 264537*cosh(x)^11 + 230945*cosh(x)^9 + 109395*cosh(x)^7 + 27027*cosh(x)^5 + 300
3*cosh(x)^3 + 99*cosh(x))*e^(2*x)*sinh(x)^11 + 143*(37145*cosh(x)^16 + 178296*cosh(x)^14 + 352716*cosh(x)^12 +
369512*cosh(x)^10 + 218790*cosh(x)^8 + 72072*cosh(x)^6 + 12012*cosh(x)^4 + 792*cosh(x)^2 + 9)*e^(2*x)*sinh(x)
^10 + 286*(10925*cosh(x)^17 + 59432*cosh(x)^15 + 135660*cosh(x)^13 + 167960*cosh(x)^11 + 121550*cosh(x)^9 + 51
480*cosh(x)^7 + 12012*cosh(x)^5 + 1320*cosh(x)^3 + 45*cosh(x))*e^(2*x)*sinh(x)^9 + 143*(10925*cosh(x)^18 + 668
61*cosh(x)^16 + 174420*cosh(x)^14 + 251940*cosh(x)^12 + 218790*cosh(x)^10 + 115830*cosh(x)^8 + 36036*cosh(x)^6
+ 5940*cosh(x)^4 + 405*cosh(x)^2 + 5)*e^(2*x)*sinh(x)^8 + 1144*(575*cosh(x)^19 + 3933*cosh(x)^17 + 11628*cosh
(x)^15 + 19380*cosh(x)^13 + 19890*cosh(x)^11 + 12870*cosh(x)^9 + 5148*cosh(x)^7 + 1188*cosh(x)^5 + 135*cosh(x)
^3 + 5*cosh(x))*e^(2*x)*sinh(x)^7 + 286*(805*cosh(x)^20 + 6118*cosh(x)^18 + 20349*cosh(x)^16 + 38760*cosh(x)^1
4 + 46410*cosh(x)^12 + 36036*cosh(x)^10 + 18018*cosh(x)^8 + 5544*cosh(x)^6 + 945*cosh(x)^4 + 70*cosh(x)^2 + 1)
*e^(2*x)*sinh(x)^6 + 572*(115*cosh(x)^21 + 966*cosh(x)^19 + 3591*cosh(x)^17 + 7752*cosh(x)^15 + 10710*cosh(x)^
13 + 9828*cosh(x)^11 + 6006*cosh(x)^9 + 2376*cosh(x)^7 + 567*cosh(x)^5 + 70*cosh(x)^3 + 3*cosh(x))*e^(2*x)*sin
h(x)^5 + 26*(575*cosh(x)^22 + 5313*cosh(x)^20 + 21945*cosh(x)^18 + 53295*cosh(x)^16 + 84150*cosh(x)^14 + 90090
*cosh(x)^12 + 66066*cosh(x)^10 + 32670*cosh(x)^8 + 10395*cosh(x)^6 + 1925*cosh(x)^4 + 165*cosh(x)^2 + 3)*e^(2*
x)*sinh(x)^4 + 104*(25*cosh(x)^23 + 253*cosh(x)^21 + 1155*cosh(x)^19 + 3135*cosh(x)^17 + 5610*cosh(x)^15 + 693
0*cosh(x)^13 + 6006*cosh(x)^11 + 3630*cosh(x)^9 + 1485*cosh(x)^7 + 385*cosh(x)^5 + 55*cosh(x)^3 + 3*cosh(x))*e
^(2*x)*sinh(x)^3 + 13*(25*cosh(x)^24 + 276*cosh(x)^22 + 1386*cosh(x)^20 + 4180*cosh(x)^18 + 8415*cosh(x)^16 +
11880*cosh(x)^14 + 12012*cosh(x)^12 + 8712*cosh(x)^10 + 4455*cosh(x)^8 + 1540*cosh(x)^6 + 330*cosh(x)^4 + 36*c
osh(x)^2 + 1)*e^(2*x)*sinh(x)^2 + 26*(cosh(x)^25 + 12*cosh(x)^23 + 66*cosh(x)^21 + 220*cosh(x)^19 + 495*cosh(x
)^17 + 792*cosh(x)^15 + 924*cosh(x)^13 + 792*cosh(x)^11 + 495*cosh(x)^9 + 220*cosh(x)^7 + 66*cosh(x)^5 + 12*co
sh(x)^3 + cosh(x))*e^(2*x)*sinh(x) + (cosh(x)^26 + 13*cosh(x)^24 + 78*cosh(x)^22 + 286*cosh(x)^20 + 715*cosh(x
)^18 + 1287*cosh(x)^16 + 1716*cosh(x)^14 + 1716*cosh(x)^12 + 1287*cosh(x)^10 + 715*cosh(x)^8 + 286*cosh(x)^6 +
78*cosh(x)^4 + 13*cosh(x)^2 + 1)*e^(2*x))

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giac [A]  time = 0.12, size = 51, normalized size = 0.31 \[ -\frac {2048 \, a^{\frac {7}{2}} {\left (1716 \, e^{\left (12 \, x\right )} + 1287 \, e^{\left (10 \, x\right )} + 715 \, e^{\left (8 \, x\right )} + 286 \, e^{\left (6 \, x\right )} + 78 \, e^{\left (4 \, x\right )} + 13 \, e^{\left (2 \, x\right )} + 1\right )}}{3003 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^4)^(7/2),x, algorithm="giac")

[Out]

-2048/3003*a^(7/2)*(1716*e^(12*x) + 1287*e^(10*x) + 715*e^(8*x) + 286*e^(6*x) + 78*e^(4*x) + 13*e^(2*x) + 1)/(
e^(2*x) + 1)^13

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maple [A]  time = 0.26, size = 72, normalized size = 0.44 \[ -\frac {2048 a^{3} {\mathrm e}^{-2 x} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1716 \,{\mathrm e}^{12 x}+1287 \,{\mathrm e}^{10 x}+715 \,{\mathrm e}^{8 x}+286 \,{\mathrm e}^{6 x}+78 \,{\mathrm e}^{4 x}+13 \,{\mathrm e}^{2 x}+1\right )}{3003 \left (1+{\mathrm e}^{2 x}\right )^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sech(x)^4)^(7/2),x)

[Out]

-2048/3003*a^3*exp(-2*x)/(1+exp(2*x))^11*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*(1716*exp(12*x)+1287*exp(10*x)+715*
exp(8*x)+286*exp(6*x)+78*exp(4*x)+13*exp(2*x)+1)

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maxima [B]  time = 0.46, size = 620, normalized size = 3.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^4)^(7/2),x, algorithm="maxima")

[Out]

2048/231*a^(7/2)*e^(-2*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(
-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-2
6*x) + 1) + 4096/77*a^(7/2)*e^(-4*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x)
+ 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24
*x) + e^(-26*x) + 1) + 4096/21*a^(7/2)*e^(-6*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 128
7*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^(-20*x) + 78*e^(-22*x)
+ 13*e^(-24*x) + e^(-26*x) + 1) + 10240/21*a^(7/2)*e^(-8*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^
(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^(-20*x) + 7
8*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) + 6144/7*a^(7/2)*e^(-10*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*
x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^
(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) + 8192/7*a^(7/2)*e^(-12*x)/(13*e^(-2*x) + 78*e^(-4*x) +
286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*
x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) + 2048/3003*a^(7/2)/(13*e^(-2*x) + 78*e^(-4*
x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(
-18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1)

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mupad [B]  time = 1.45, size = 498, normalized size = 3.06 \[ \frac {1536\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,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 {2048\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,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 {10240\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^9\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}+\frac {4096\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^{10}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {30720\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{11\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^{11}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}+\frac {1024\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^{12}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {2048\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{13\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^{13}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/cosh(x)^4)^(7/2),x)

[Out]

(1536*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1))/((exp(2*x)
+ 1)^8*(exp(2*x) + 2*exp(4*x) + exp(6*x))) - (2048*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp
(4*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))) - (10240*a^3*(a/(e
xp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1))/(3*(exp(2*x) + 1)^9*(exp(
2*x) + 2*exp(4*x) + exp(6*x))) + (4096*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp
(6*x) + exp(8*x) + 1))/((exp(2*x) + 1)^10*(exp(2*x) + 2*exp(4*x) + exp(6*x))) - (30720*a^3*(a/(exp(-x)/2 + exp
(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1))/(11*(exp(2*x) + 1)^11*(exp(2*x) + 2*exp
(4*x) + exp(6*x))) + (1024*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(
8*x) + 1))/((exp(2*x) + 1)^12*(exp(2*x) + 2*exp(4*x) + exp(6*x))) - (2048*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/
2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1))/(13*(exp(2*x) + 1)^13*(exp(2*x) + 2*exp(4*x) + exp(6
*x)))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)**4)**(7/2),x)

[Out]

Timed out

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