\(\int (a \text {csch}^4(x))^{7/2} \, dx\) [42]

Optimal result

Integrand size = 10, antiderivative size = 164 \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=2 a^3 \cosh ^2(x) \coth (x) \sqrt {a \text {csch}^4(x)}-3 a^3 \cosh ^2(x) \coth ^3(x) \sqrt {a \text {csch}^4(x)}+\frac {20}{7} a^3 \cosh ^2(x) \coth ^5(x) \sqrt {a \text {csch}^4(x)}-\frac {5}{3} a^3 \cosh ^2(x) \coth ^7(x) \sqrt {a \text {csch}^4(x)}+\frac {6}{11} a^3 \cosh ^2(x) \coth ^9(x) \sqrt {a \text {csch}^4(x)}-\frac {1}{13} a^3 \cosh ^2(x) \coth ^{11}(x) \sqrt {a \text {csch}^4(x)}-a^3 \cosh (x) \sqrt {a \text {csch}^4(x)} \sinh (x) \]

[Out]

2*a^3*cosh(x)^2*coth(x)*(a*csch(x)^4)^(1/2)-3*a^3*cosh(x)^2*coth(x)^3*(a*csch(x)^4)^(1/2)+20/7*a^3*cosh(x)^2*c
oth(x)^5*(a*csch(x)^4)^(1/2)-5/3*a^3*cosh(x)^2*coth(x)^7*(a*csch(x)^4)^(1/2)+6/11*a^3*cosh(x)^2*coth(x)^9*(a*c
sch(x)^4)^(1/2)-1/13*a^3*cosh(x)^2*coth(x)^11*(a*csch(x)^4)^(1/2)-a^3*cosh(x)*sinh(x)*(a*csch(x)^4)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4208, 3852} \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=-\frac {1}{13} a^3 \cosh ^2(x) \coth ^{11}(x) \sqrt {a \text {csch}^4(x)}+\frac {6}{11} a^3 \cosh ^2(x) \coth ^9(x) \sqrt {a \text {csch}^4(x)}-\frac {5}{3} a^3 \cosh ^2(x) \coth ^7(x) \sqrt {a \text {csch}^4(x)}+\frac {20}{7} a^3 \cosh ^2(x) \coth ^5(x) \sqrt {a \text {csch}^4(x)}-3 a^3 \cosh ^2(x) \coth ^3(x) \sqrt {a \text {csch}^4(x)}+2 a^3 \cosh ^2(x) \coth (x) \sqrt {a \text {csch}^4(x)}-a^3 \sinh (x) \cosh (x) \sqrt {a \text {csch}^4(x)} \]

[In]

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

[Out]

2*a^3*Cosh[x]^2*Coth[x]*Sqrt[a*Csch[x]^4] - 3*a^3*Cosh[x]^2*Coth[x]^3*Sqrt[a*Csch[x]^4] + (20*a^3*Cosh[x]^2*Co
th[x]^5*Sqrt[a*Csch[x]^4])/7 - (5*a^3*Cosh[x]^2*Coth[x]^7*Sqrt[a*Csch[x]^4])/3 + (6*a^3*Cosh[x]^2*Coth[x]^9*Sq
rt[a*Csch[x]^4])/11 - (a^3*Cosh[x]^2*Coth[x]^11*Sqrt[a*Csch[x]^4])/13 - a^3*Cosh[x]*Sqrt[a*Csch[x]^4]*Sinh[x]

Rule 3852

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 4208

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*} \text {integral}& = \left (a^3 \sqrt {a \text {csch}^4(x)} \sinh ^2(x)\right ) \int \text {csch}^{14}(x) \, dx \\ & = -\left (\left (i a^3 \sqrt {a \text {csch}^4(x)} \sinh ^2(x)\right ) \text {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 \coth (x)\right )\right ) \\ & = 2 a^3 \cosh ^2(x) \coth (x) \sqrt {a \text {csch}^4(x)}-3 a^3 \cosh ^2(x) \coth ^3(x) \sqrt {a \text {csch}^4(x)}+\frac {20}{7} a^3 \cosh ^2(x) \coth ^5(x) \sqrt {a \text {csch}^4(x)}-\frac {5}{3} a^3 \cosh ^2(x) \coth ^7(x) \sqrt {a \text {csch}^4(x)}+\frac {6}{11} a^3 \cosh ^2(x) \coth ^9(x) \sqrt {a \text {csch}^4(x)}-\frac {1}{13} a^3 \cosh ^2(x) \coth ^{11}(x) \sqrt {a \text {csch}^4(x)}-a^3 \cosh (x) \sqrt {a \text {csch}^4(x)} \sinh (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.36 \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=-\frac {a^3 \cosh (x) \sqrt {a \text {csch}^4(x)} \left (1024-512 \text {csch}^2(x)+384 \text {csch}^4(x)-320 \text {csch}^6(x)+280 \text {csch}^8(x)-252 \text {csch}^{10}(x)+231 \text {csch}^{12}(x)\right ) \sinh (x)}{3003} \]

[In]

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

[Out]

-1/3003*(a^3*Cosh[x]*Sqrt[a*Csch[x]^4]*(1024 - 512*Csch[x]^2 + 384*Csch[x]^4 - 320*Csch[x]^6 + 280*Csch[x]^8 -
252*Csch[x]^10 + 231*Csch[x]^12)*Sinh[x])

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.44

method	result	size
risch	\(-\frac {2048 a^{3} {\mathrm e}^{-2 x} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}\, \left (1716 \,{\mathrm e}^{12 x}-1287 \,{\mathrm e}^{10 x}+715 \,{\mathrm e}^{8 x}-286 \,{\mathrm e}^{6 x}+78 \,{\mathrm e}^{4 x}-13 \,{\mathrm e}^{2 x}+1\right )}{3003 \left ({\mathrm e}^{2 x}-1\right )^{11}}\)	\(72\)

[In]

int((a*csch(x)^4)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2048/3003*a^3*exp(-2*x)/(exp(2*x)-1)^11*(a*exp(4*x)/(exp(2*x)-1)^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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2825 vs. \(2 (142) = 284\).

Time = 0.40 (sec) , antiderivative size = 2825, normalized size of antiderivative = 17.23 \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=\text {Too large to display} \]

[In]

integrate((a*csch(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))

Sympy [F]

\[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=\int \left (a \operatorname {csch}^{4}{\left (x \right )}\right )^{\frac {7}{2}}\, dx \]

[In]

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

[Out]

Integral((a*csch(x)**4)**(7/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (142) = 284\).

Time = 0.28 (sec) , antiderivative size = 620, normalized size of antiderivative = 3.78 \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=\text {Too large to display} \]

[In]

integrate((a*csch(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^(-
26*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^(-2
4*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) + 12
87*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) +
78*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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.31 \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=-\frac {2048 \, a^{\frac {7}{2}} {\left (1716 \, e^{\left (12 \, x\right )} - 1287 \, e^{\left (10 \, x\right )} + 715 \, e^{\left (8 \, x\right )} - 286 \, e^{\left (6 \, x\right )} + 78 \, e^{\left (4 \, x\right )} - 13 \, e^{\left (2 \, x\right )} + 1\right )}}{3003 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{13}} \]

[In]

integrate((a*csch(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

Mupad [B] (verification not implemented)

Time = 2.24 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.04 \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=-\frac {2048\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,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 {1536\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,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 {10240\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,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 (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,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 (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,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 (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,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 (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,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 )} \]

[In]

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

[Out]

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