3.139 \(\int \frac {1}{(a \cosh ^4(x))^{5/2}} \, dx\)

Optimal. Leaf size=117 \[ \frac {\sinh (x) \cosh (x)}{a^2 \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^7(x)}{9 a^2 \sqrt {a \cosh ^4(x)}}-\frac {4 \sinh ^2(x) \tanh ^5(x)}{7 a^2 \sqrt {a \cosh ^4(x)}}+\frac {6 \sinh ^2(x) \tanh ^3(x)}{5 a^2 \sqrt {a \cosh ^4(x)}}-\frac {4 \sinh ^2(x) \tanh (x)}{3 a^2 \sqrt {a \cosh ^4(x)}} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 117, 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 = {3207, 3767} \[ \frac {\sinh (x) \cosh (x)}{a^2 \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^7(x)}{9 a^2 \sqrt {a \cosh ^4(x)}}-\frac {4 \sinh ^2(x) \tanh ^5(x)}{7 a^2 \sqrt {a \cosh ^4(x)}}+\frac {6 \sinh ^2(x) \tanh ^3(x)}{5 a^2 \sqrt {a \cosh ^4(x)}}-\frac {4 \sinh ^2(x) \tanh (x)}{3 a^2 \sqrt {a \cosh ^4(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[x]*Sinh[x])/(a^2*Sqrt[a*Cosh[x]^4]) - (4*Sinh[x]^2*Tanh[x])/(3*a^2*Sqrt[a*Cosh[x]^4]) + (6*Sinh[x]^2*Tan
h[x]^3)/(5*a^2*Sqrt[a*Cosh[x]^4]) - (4*Sinh[x]^2*Tanh[x]^5)/(7*a^2*Sqrt[a*Cosh[x]^4]) + (Sinh[x]^2*Tanh[x]^7)/
(9*a^2*Sqrt[a*Cosh[x]^4])

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

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]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 47, normalized size = 0.40 \[ \frac {(130 \cosh (2 x)+46 \cosh (4 x)+10 \cosh (6 x)+\cosh (8 x)+128) \tanh (x) \text {sech}^6(x)}{315 a^2 \sqrt {a \cosh ^4(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((128 + 130*Cosh[2*x] + 46*Cosh[4*x] + 10*Cosh[6*x] + Cosh[8*x])*Sech[x]^6*Tanh[x])/(315*a^2*Sqrt[a*Cosh[x]^4]
)

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fricas [B]  time = 0.64, size = 3065, normalized size = 26.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-256/315*(1008*cosh(x)*e^(2*x)*sinh(x)^7 + 126*e^(2*x)*sinh(x)^8 + 84*(42*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^6 + 5
04*(14*cosh(x)^3 + cosh(x))*e^(2*x)*sinh(x)^5 + 36*(245*cosh(x)^4 + 35*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^4 + 48*(
147*cosh(x)^5 + 35*cosh(x)^3 + 3*cosh(x))*e^(2*x)*sinh(x)^3 + 9*(392*cosh(x)^6 + 140*cosh(x)^4 + 24*cosh(x)^2
+ 1)*e^(2*x)*sinh(x)^2 + 18*(56*cosh(x)^7 + 28*cosh(x)^5 + 8*cosh(x)^3 + cosh(x))*e^(2*x)*sinh(x) + (126*cosh(
x)^8 + 84*cosh(x)^6 + 36*cosh(x)^4 + 9*cosh(x)^2 + 1)*e^(2*x))*sqrt(a*e^(8*x) + 4*a*e^(6*x) + 6*a*e^(4*x) + 4*
a*e^(2*x) + a)*e^(-2*x)/(a^3*cosh(x)^18 + 9*a^3*cosh(x)^16 + (a^3*e^(4*x) + 2*a^3*e^(2*x) + a^3)*sinh(x)^18 +
18*(a^3*cosh(x)*e^(4*x) + 2*a^3*cosh(x)*e^(2*x) + a^3*cosh(x))*sinh(x)^17 + 36*a^3*cosh(x)^14 + 9*(17*a^3*cosh
(x)^2 + a^3 + (17*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(17*a^3*cosh(x)^2 + a^3)*e^(2*x))*sinh(x)^16 + 48*(17*a^3*c
osh(x)^3 + 3*a^3*cosh(x) + (17*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(4*x) + 2*(17*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e
^(2*x))*sinh(x)^15 + 84*a^3*cosh(x)^12 + 36*(85*a^3*cosh(x)^4 + 30*a^3*cosh(x)^2 + a^3 + (85*a^3*cosh(x)^4 + 3
0*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(85*a^3*cosh(x)^4 + 30*a^3*cosh(x)^2 + a^3)*e^(2*x))*sinh(x)^14 + 504*(17*a
^3*cosh(x)^5 + 10*a^3*cosh(x)^3 + a^3*cosh(x) + (17*a^3*cosh(x)^5 + 10*a^3*cosh(x)^3 + a^3*cosh(x))*e^(4*x) +
2*(17*a^3*cosh(x)^5 + 10*a^3*cosh(x)^3 + a^3*cosh(x))*e^(2*x))*sinh(x)^13 + 126*a^3*cosh(x)^10 + 84*(221*a^3*c
osh(x)^6 + 195*a^3*cosh(x)^4 + 39*a^3*cosh(x)^2 + a^3 + (221*a^3*cosh(x)^6 + 195*a^3*cosh(x)^4 + 39*a^3*cosh(x
)^2 + a^3)*e^(4*x) + 2*(221*a^3*cosh(x)^6 + 195*a^3*cosh(x)^4 + 39*a^3*cosh(x)^2 + a^3)*e^(2*x))*sinh(x)^12 +
144*(221*a^3*cosh(x)^7 + 273*a^3*cosh(x)^5 + 91*a^3*cosh(x)^3 + 7*a^3*cosh(x) + (221*a^3*cosh(x)^7 + 273*a^3*c
osh(x)^5 + 91*a^3*cosh(x)^3 + 7*a^3*cosh(x))*e^(4*x) + 2*(221*a^3*cosh(x)^7 + 273*a^3*cosh(x)^5 + 91*a^3*cosh(
x)^3 + 7*a^3*cosh(x))*e^(2*x))*sinh(x)^11 + 126*a^3*cosh(x)^8 + 18*(2431*a^3*cosh(x)^8 + 4004*a^3*cosh(x)^6 +
2002*a^3*cosh(x)^4 + 308*a^3*cosh(x)^2 + 7*a^3 + (2431*a^3*cosh(x)^8 + 4004*a^3*cosh(x)^6 + 2002*a^3*cosh(x)^4
+ 308*a^3*cosh(x)^2 + 7*a^3)*e^(4*x) + 2*(2431*a^3*cosh(x)^8 + 4004*a^3*cosh(x)^6 + 2002*a^3*cosh(x)^4 + 308*
a^3*cosh(x)^2 + 7*a^3)*e^(2*x))*sinh(x)^10 + 4*(12155*a^3*cosh(x)^9 + 25740*a^3*cosh(x)^7 + 18018*a^3*cosh(x)^
5 + 4620*a^3*cosh(x)^3 + 315*a^3*cosh(x) + (12155*a^3*cosh(x)^9 + 25740*a^3*cosh(x)^7 + 18018*a^3*cosh(x)^5 +
4620*a^3*cosh(x)^3 + 315*a^3*cosh(x))*e^(4*x) + 2*(12155*a^3*cosh(x)^9 + 25740*a^3*cosh(x)^7 + 18018*a^3*cosh(
x)^5 + 4620*a^3*cosh(x)^3 + 315*a^3*cosh(x))*e^(2*x))*sinh(x)^9 + 84*a^3*cosh(x)^6 + 18*(2431*a^3*cosh(x)^10 +
6435*a^3*cosh(x)^8 + 6006*a^3*cosh(x)^6 + 2310*a^3*cosh(x)^4 + 315*a^3*cosh(x)^2 + 7*a^3 + (2431*a^3*cosh(x)^
10 + 6435*a^3*cosh(x)^8 + 6006*a^3*cosh(x)^6 + 2310*a^3*cosh(x)^4 + 315*a^3*cosh(x)^2 + 7*a^3)*e^(4*x) + 2*(24
31*a^3*cosh(x)^10 + 6435*a^3*cosh(x)^8 + 6006*a^3*cosh(x)^6 + 2310*a^3*cosh(x)^4 + 315*a^3*cosh(x)^2 + 7*a^3)*
e^(2*x))*sinh(x)^8 + 144*(221*a^3*cosh(x)^11 + 715*a^3*cosh(x)^9 + 858*a^3*cosh(x)^7 + 462*a^3*cosh(x)^5 + 105
*a^3*cosh(x)^3 + 7*a^3*cosh(x) + (221*a^3*cosh(x)^11 + 715*a^3*cosh(x)^9 + 858*a^3*cosh(x)^7 + 462*a^3*cosh(x)
^5 + 105*a^3*cosh(x)^3 + 7*a^3*cosh(x))*e^(4*x) + 2*(221*a^3*cosh(x)^11 + 715*a^3*cosh(x)^9 + 858*a^3*cosh(x)^
7 + 462*a^3*cosh(x)^5 + 105*a^3*cosh(x)^3 + 7*a^3*cosh(x))*e^(2*x))*sinh(x)^7 + 36*a^3*cosh(x)^4 + 84*(221*a^3
*cosh(x)^12 + 858*a^3*cosh(x)^10 + 1287*a^3*cosh(x)^8 + 924*a^3*cosh(x)^6 + 315*a^3*cosh(x)^4 + 42*a^3*cosh(x)
^2 + a^3 + (221*a^3*cosh(x)^12 + 858*a^3*cosh(x)^10 + 1287*a^3*cosh(x)^8 + 924*a^3*cosh(x)^6 + 315*a^3*cosh(x)
^4 + 42*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(221*a^3*cosh(x)^12 + 858*a^3*cosh(x)^10 + 1287*a^3*cosh(x)^8 + 924*a
^3*cosh(x)^6 + 315*a^3*cosh(x)^4 + 42*a^3*cosh(x)^2 + a^3)*e^(2*x))*sinh(x)^6 + 504*(17*a^3*cosh(x)^13 + 78*a^
3*cosh(x)^11 + 143*a^3*cosh(x)^9 + 132*a^3*cosh(x)^7 + 63*a^3*cosh(x)^5 + 14*a^3*cosh(x)^3 + a^3*cosh(x) + (17
*a^3*cosh(x)^13 + 78*a^3*cosh(x)^11 + 143*a^3*cosh(x)^9 + 132*a^3*cosh(x)^7 + 63*a^3*cosh(x)^5 + 14*a^3*cosh(x
)^3 + a^3*cosh(x))*e^(4*x) + 2*(17*a^3*cosh(x)^13 + 78*a^3*cosh(x)^11 + 143*a^3*cosh(x)^9 + 132*a^3*cosh(x)^7
+ 63*a^3*cosh(x)^5 + 14*a^3*cosh(x)^3 + a^3*cosh(x))*e^(2*x))*sinh(x)^5 + 9*a^3*cosh(x)^2 + 36*(85*a^3*cosh(x)
^14 + 455*a^3*cosh(x)^12 + 1001*a^3*cosh(x)^10 + 1155*a^3*cosh(x)^8 + 735*a^3*cosh(x)^6 + 245*a^3*cosh(x)^4 +
35*a^3*cosh(x)^2 + a^3 + (85*a^3*cosh(x)^14 + 455*a^3*cosh(x)^12 + 1001*a^3*cosh(x)^10 + 1155*a^3*cosh(x)^8 +
735*a^3*cosh(x)^6 + 245*a^3*cosh(x)^4 + 35*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(85*a^3*cosh(x)^14 + 455*a^3*cosh(
x)^12 + 1001*a^3*cosh(x)^10 + 1155*a^3*cosh(x)^8 + 735*a^3*cosh(x)^6 + 245*a^3*cosh(x)^4 + 35*a^3*cosh(x)^2 +
a^3)*e^(2*x))*sinh(x)^4 + 48*(17*a^3*cosh(x)^15 + 105*a^3*cosh(x)^13 + 273*a^3*cosh(x)^11 + 385*a^3*cosh(x)^9
+ 315*a^3*cosh(x)^7 + 147*a^3*cosh(x)^5 + 35*a^3*cosh(x)^3 + 3*a^3*cosh(x) + (17*a^3*cosh(x)^15 + 105*a^3*cosh
(x)^13 + 273*a^3*cosh(x)^11 + 385*a^3*cosh(x)^9 + 315*a^3*cosh(x)^7 + 147*a^3*cosh(x)^5 + 35*a^3*cosh(x)^3 + 3
*a^3*cosh(x))*e^(4*x) + 2*(17*a^3*cosh(x)^15 + 105*a^3*cosh(x)^13 + 273*a^3*cosh(x)^11 + 385*a^3*cosh(x)^9 + 3
15*a^3*cosh(x)^7 + 147*a^3*cosh(x)^5 + 35*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(2*x))*sinh(x)^3 + a^3 + 9*(17*a^3*
cosh(x)^16 + 120*a^3*cosh(x)^14 + 364*a^3*cosh(x)^12 + 616*a^3*cosh(x)^10 + 630*a^3*cosh(x)^8 + 392*a^3*cosh(x
)^6 + 140*a^3*cosh(x)^4 + 24*a^3*cosh(x)^2 + a^3 + (17*a^3*cosh(x)^16 + 120*a^3*cosh(x)^14 + 364*a^3*cosh(x)^1
2 + 616*a^3*cosh(x)^10 + 630*a^3*cosh(x)^8 + 392*a^3*cosh(x)^6 + 140*a^3*cosh(x)^4 + 24*a^3*cosh(x)^2 + a^3)*e
^(4*x) + 2*(17*a^3*cosh(x)^16 + 120*a^3*cosh(x)^14 + 364*a^3*cosh(x)^12 + 616*a^3*cosh(x)^10 + 630*a^3*cosh(x)
^8 + 392*a^3*cosh(x)^6 + 140*a^3*cosh(x)^4 + 24*a^3*cosh(x)^2 + a^3)*e^(2*x))*sinh(x)^2 + (a^3*cosh(x)^18 + 9*
a^3*cosh(x)^16 + 36*a^3*cosh(x)^14 + 84*a^3*cosh(x)^12 + 126*a^3*cosh(x)^10 + 126*a^3*cosh(x)^8 + 84*a^3*cosh(
x)^6 + 36*a^3*cosh(x)^4 + 9*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(a^3*cosh(x)^18 + 9*a^3*cosh(x)^16 + 36*a^3*cosh(
x)^14 + 84*a^3*cosh(x)^12 + 126*a^3*cosh(x)^10 + 126*a^3*cosh(x)^8 + 84*a^3*cosh(x)^6 + 36*a^3*cosh(x)^4 + 9*a
^3*cosh(x)^2 + a^3)*e^(2*x) + 18*(a^3*cosh(x)^17 + 8*a^3*cosh(x)^15 + 28*a^3*cosh(x)^13 + 56*a^3*cosh(x)^11 +
70*a^3*cosh(x)^9 + 56*a^3*cosh(x)^7 + 28*a^3*cosh(x)^5 + 8*a^3*cosh(x)^3 + a^3*cosh(x) + (a^3*cosh(x)^17 + 8*a
^3*cosh(x)^15 + 28*a^3*cosh(x)^13 + 56*a^3*cosh(x)^11 + 70*a^3*cosh(x)^9 + 56*a^3*cosh(x)^7 + 28*a^3*cosh(x)^5
+ 8*a^3*cosh(x)^3 + a^3*cosh(x))*e^(4*x) + 2*(a^3*cosh(x)^17 + 8*a^3*cosh(x)^15 + 28*a^3*cosh(x)^13 + 56*a^3*
cosh(x)^11 + 70*a^3*cosh(x)^9 + 56*a^3*cosh(x)^7 + 28*a^3*cosh(x)^5 + 8*a^3*cosh(x)^3 + a^3*cosh(x))*e^(2*x))*
sinh(x))

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giac [A]  time = 0.21, size = 39, normalized size = 0.33 \[ -\frac {256 \, {\left (126 \, e^{\left (8 \, x\right )} + 84 \, e^{\left (6 \, x\right )} + 36 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} + 1\right )}}{315 \, a^{\frac {5}{2}} {\left (e^{\left (2 \, x\right )} + 1\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.36, size = 96, normalized size = 0.82 \[ \frac {4 \sqrt {8}\, \sqrt {2}\, \left (8 \left (\cosh ^{4}\left (2 x \right )\right )+40 \left (\cosh ^{3}\left (2 x \right )\right )+84 \left (\cosh ^{2}\left (2 x \right )\right )+100 \cosh \left (2 x \right )+83\right ) \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (\cosh \left (2 x \right )+1\right )}}{315 a^{3} \left (\cosh \left (2 x \right )+1\right )^{4} \sinh \left (2 x \right ) \sqrt {\left (\cosh \left (2 x \right )+1\right )^{2} a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/315*8^(1/2)*2^(1/2)/a^3*(8*cosh(2*x)^4+40*cosh(2*x)^3+84*cosh(2*x)^2+100*cosh(2*x)+83)*(a*sinh(2*x)^2)^(1/2)
*(a*(-1+cosh(2*x))*(cosh(2*x)+1))^(1/2)/(cosh(2*x)+1)^4/sinh(2*x)/((cosh(2*x)+1)^2*a)^(1/2)

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maxima [B]  time = 0.43, size = 457, normalized size = 3.91 \[ \frac {256 \, e^{\left (-2 \, x\right )}}{35 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} + \frac {1024 \, e^{\left (-4 \, x\right )}}{35 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} + \frac {1024 \, e^{\left (-6 \, x\right )}}{15 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} + \frac {512 \, e^{\left (-8 \, x\right )}}{5 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} + \frac {256}{315 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.99, size = 256, normalized size = 2.19 \[ \frac {4096\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{3\,a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^6\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {2048\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{5\,a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^5\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {12288\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{7\,a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^7\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}+\frac {1024\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^8\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {2048\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{9\,a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^9\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4096*exp(4*x)*(a*(exp(-x)/2 + exp(x)/2)^4)^(1/2))/(3*a^3*(exp(2*x) + 1)^6*(exp(2*x) + 2*exp(4*x) + exp(6*x)))
- (2048*exp(4*x)*(a*(exp(-x)/2 + exp(x)/2)^4)^(1/2))/(5*a^3*(exp(2*x) + 1)^5*(exp(2*x) + 2*exp(4*x) + exp(6*x
))) - (12288*exp(4*x)*(a*(exp(-x)/2 + exp(x)/2)^4)^(1/2))/(7*a^3*(exp(2*x) + 1)^7*(exp(2*x) + 2*exp(4*x) + exp
(6*x))) + (1024*exp(4*x)*(a*(exp(-x)/2 + exp(x)/2)^4)^(1/2))/(a^3*(exp(2*x) + 1)^8*(exp(2*x) + 2*exp(4*x) + ex
p(6*x))) - (2048*exp(4*x)*(a*(exp(-x)/2 + exp(x)/2)^4)^(1/2))/(9*a^3*(exp(2*x) + 1)^9*(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(1/(a*cosh(x)**4)**(5/2),x)

[Out]

Timed out

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