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]
4/3*cosh(x)^2*coth(x)/a^2/(a*sinh(x)^4)^(1/2)-6/5*cosh(x)^2*coth(x)^3/a^2/(a*sinh(x)^4)^(1/2)+4/7*cosh(x)^2*co
th(x)^5/a^2/(a*sinh(x)^4)^(1/2)-1/9*cosh(x)^2*coth(x)^7/a^2/(a*sinh(x)^4)^(1/2)-cosh(x)*sinh(x)/a^2/(a*sinh(x)
^4)^(1/2)
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Rubi [A] time = 0.03, antiderivative size = 118, 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} \[ -\frac {\sinh (x) \cosh (x)}{a^2 \sqrt {a \sinh ^4(x)}}-\frac {\cosh ^2(x) \coth ^7(x)}{9 a^2 \sqrt {a \sinh ^4(x)}}+\frac {4 \cosh ^2(x) \coth ^5(x)}{7 a^2 \sqrt {a \sinh ^4(x)}}-\frac {6 \cosh ^2(x) \coth ^3(x)}{5 a^2 \sqrt {a \sinh ^4(x)}}+\frac {4 \cosh ^2(x) \coth (x)}{3 a^2 \sqrt {a \sinh ^4(x)}} \]
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*}
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Mathematica [A] time = 0.06, size = 47, normalized size = 0.40 \[ -\frac {\sinh (x) \cosh (x) \left (35 \text {csch}^8(x)-40 \text {csch}^6(x)+48 \text {csch}^4(x)-64 \text {csch}^2(x)+128\right )}{315 a^2 \sqrt {a \sinh ^4(x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a*Sinh[x]^4)^(-5/2),x]
[Out]
-1/315*(Cosh[x]*(128 - 64*Csch[x]^2 + 48*Csch[x]^4 - 40*Csch[x]^6 + 35*Csch[x]^8)*Sinh[x])/(a^2*Sqrt[a*Sinh[x]
^4])
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fricas [B] time = 0.82, size = 3093, normalized size = 26.21 \[ \text {result too large to display} \]
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))
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giac [A] time = 0.36, 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*sinh(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 = 0.13, size = 90, normalized size = 0.76 \[ -\frac {16 \left (8 \left (\cosh ^{4}\left (2 x \right )\right )-40 \left (\cosh ^{3}\left (2 x \right )\right )+84 \left (\cosh ^{2}\left (2 x \right )\right )-100 \cosh \left (2 x \right )+83\right ) \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (\cosh \left (2 x \right )+1\right )}}{315 a^{3} \left (-1+\cosh \left (2 x \right )\right )^{4} \sinh \left (2 x \right ) \sqrt {\left (-1+\cosh \left (2 x \right )\right )^{2} a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a*sinh(x)^4)^(5/2),x)
[Out]
-16/315/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)/((-1+cosh(2*x))^2*a)^(1/2)
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maxima [B] time = 0.42, size = 467, normalized size = 3.96 \[ -\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*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))
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mupad [B] time = 0.54, size = 256, normalized size = 2.17 \[ -\frac {2048\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}{5\,a^3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^5\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {4096\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}{3\,a^3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^6\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {12288\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}{7\,a^3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^7\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {1024\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}{a^3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^8\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {2048\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}{9\,a^3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^9\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a*sinh(x)^4)^(5/2),x)
[Out]
- (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)
)) - (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))) - (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) + e
xp(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) +
exp(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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \sinh ^{4}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
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)
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