3.157 \(\int \frac{1}{(a \sinh ^4(x))^{5/2}} \, dx\)
Optimal. Leaf size=118 \[ -\frac{\sinh (x) \cosh (x)}{a^2 \sqrt{a \sinh ^4(x)}}-\frac{\cosh ^2(x) \coth ^7(x)}{9 a^2 \sqrt{a \sinh ^4(x)}}+\frac{4 \cosh ^2(x) \coth ^5(x)}{7 a^2 \sqrt{a \sinh ^4(x)}}-\frac{6 \cosh ^2(x) \coth ^3(x)}{5 a^2 \sqrt{a \sinh ^4(x)}}+\frac{4 \cosh ^2(x) \coth (x)}{3 a^2 \sqrt{a \sinh ^4(x)}} \]
[Out]
(4*Cosh[x]^2*Coth[x])/(3*a^2*Sqrt[a*Sinh[x]^4]) - (6*Cosh[x]^2*Coth[x]^3)/(5*a^2*Sqrt[a*Sinh[x]^4]) + (4*Cosh[
x]^2*Coth[x]^5)/(7*a^2*Sqrt[a*Sinh[x]^4]) - (Cosh[x]^2*Coth[x]^7)/(9*a^2*Sqrt[a*Sinh[x]^4]) - (Cosh[x]*Sinh[x]
)/(a^2*Sqrt[a*Sinh[x]^4])
________________________________________________________________________________________
Rubi [A] time = 0.0326209, antiderivative size = 118, 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} \[ -\frac{\sinh (x) \cosh (x)}{a^2 \sqrt{a \sinh ^4(x)}}-\frac{\cosh ^2(x) \coth ^7(x)}{9 a^2 \sqrt{a \sinh ^4(x)}}+\frac{4 \cosh ^2(x) \coth ^5(x)}{7 a^2 \sqrt{a \sinh ^4(x)}}-\frac{6 \cosh ^2(x) \coth ^3(x)}{5 a^2 \sqrt{a \sinh ^4(x)}}+\frac{4 \cosh ^2(x) \coth (x)}{3 a^2 \sqrt{a \sinh ^4(x)}} \]
Antiderivative was successfully verified.
[In]
Int[(a*Sinh[x]^4)^(-5/2),x]
[Out]
(4*Cosh[x]^2*Coth[x])/(3*a^2*Sqrt[a*Sinh[x]^4]) - (6*Cosh[x]^2*Coth[x]^3)/(5*a^2*Sqrt[a*Sinh[x]^4]) + (4*Cosh[
x]^2*Coth[x]^5)/(7*a^2*Sqrt[a*Sinh[x]^4]) - (Cosh[x]^2*Coth[x]^7)/(9*a^2*Sqrt[a*Sinh[x]^4]) - (Cosh[x]*Sinh[x]
)/(a^2*Sqrt[a*Sinh[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 \sinh ^4(x)\right )^{5/2}} \, dx &=\frac{\sinh ^2(x) \int \text{csch}^{10}(x) \, dx}{a^2 \sqrt{a \sinh ^4(x)}}\\ &=-\frac{\left (i \sinh ^2(x)\right ) \operatorname{Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,-i \coth (x)\right )}{a^2 \sqrt{a \sinh ^4(x)}}\\ &=\frac{4 \cosh ^2(x) \coth (x)}{3 a^2 \sqrt{a \sinh ^4(x)}}-\frac{6 \cosh ^2(x) \coth ^3(x)}{5 a^2 \sqrt{a \sinh ^4(x)}}+\frac{4 \cosh ^2(x) \coth ^5(x)}{7 a^2 \sqrt{a \sinh ^4(x)}}-\frac{\cosh ^2(x) \coth ^7(x)}{9 a^2 \sqrt{a \sinh ^4(x)}}-\frac{\cosh (x) \sinh (x)}{a^2 \sqrt{a \sinh ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.058047, size = 47, normalized size = 0.4 \[ -\frac{\sinh (x) \cosh (x) \left (35 \text{csch}^8(x)-40 \text{csch}^6(x)+48 \text{csch}^4(x)-64 \text{csch}^2(x)+128\right )}{315 a^2 \sqrt{a \sinh ^4(x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a*Sinh[x]^4)^(-5/2),x]
[Out]
-(Cosh[x]*(128 - 64*Csch[x]^2 + 48*Csch[x]^4 - 40*Csch[x]^6 + 35*Csch[x]^8)*Sinh[x])/(315*a^2*Sqrt[a*Sinh[x]^4
])
________________________________________________________________________________________
Maple [A] time = 0.086, 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 ( -1+\cosh \left ( 2\,x \right ) \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 ( -1+\cosh \left ( 2\,x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a*sinh(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)/(-1+cosh(2*x))^4/sinh(2*x)/(a*(-1+cosh(2*x))^2)^(1/2)
________________________________________________________________________________________
Maxima [B] time = 2.01625, size = 630, normalized size = 5.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a*sinh(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^(-1
8*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) - 12
6*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) - 12
6*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 = 2.56442, size = 9072, normalized size = 76.88 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a*sinh(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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sinh ^{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*sinh(x)**4)**(5/2),x)
[Out]
Integral((a*sinh(x)**4)**(-5/2), x)
________________________________________________________________________________________
Giac [A] time = 1.39216, size = 72, normalized size = 0.61 \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*sinh(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)