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*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])
________________________________________________________________________________________
Rubi [A] time = 0.0351632, antiderivative size = 117, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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*}
Mathematica [A] time = 0.0481742, size = 47, normalized size = 0.4 \[ \frac{(130 \cosh (2 x)+46 \cosh (4 x)+10 \cosh (6 x)+\cosh (8 x)+128) \tanh (x) \text{sech}^6(x)}{315 a^2 \sqrt{a \cosh ^4(x)}} \]
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]
)
________________________________________________________________________________________
Maple [A] time = 0.095, size = 96, normalized size = 0.8 \begin{align*}{\frac{4\,\sqrt{8}\sqrt{2} \left ( 8\, \left ( \cosh \left ( 2\,x \right ) \right ) ^{4}+40\, \left ( \cosh \left ( 2\,x \right ) \right ) ^{3}+84\, \left ( \cosh \left ( 2\,x \right ) \right ) ^{2}+100\,\cosh \left ( 2\,x \right ) +83 \right ) }{315\,{a}^{3} \left ( \cosh \left ( 2\,x \right ) +1 \right ) ^{4}\sinh \left ( 2\,x \right ) }\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a \left ( -1+\cosh \left ( 2\,x \right ) \right ) \left ( \cosh \left ( 2\,x \right ) +1 \right ) }{\frac{1}{\sqrt{a \left ( \cosh \left ( 2\,x \right ) +1 \right ) ^{2}}}}} \end{align*}
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)/(a*(cosh(2*x)+1)^2)^(1/2)
________________________________________________________________________________________
Maxima [B] time = 1.65212, size = 617, normalized size = 5.27 \begin{align*} \text{result too large to display} \end{align*}
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))
________________________________________________________________________________________
Fricas [B] time = 2.49368, size = 9072, normalized size = 77.54 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a*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))
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a*cosh(x)**4)**(5/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [A] time = 1.15315, size = 72, normalized size = 0.62 \begin{align*} -\frac{256 \,{\left (126 \, \sqrt{a} e^{\left (8 \, x\right )} + 84 \, \sqrt{a} e^{\left (6 \, x\right )} + 36 \, \sqrt{a} e^{\left (4 \, x\right )} + 9 \, \sqrt{a} e^{\left (2 \, x\right )} + \sqrt{a}\right )}}{315 \, a^{3}{\left (e^{\left (2 \, x\right )} + 1\right )}^{9}} \end{align*}
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*sqrt(a)*e^(8*x) + 84*sqrt(a)*e^(6*x) + 36*sqrt(a)*e^(4*x) + 9*sqrt(a)*e^(2*x) + sqrt(a))/(a^3*(e
^(2*x) + 1)^9)