3.456 \(\int \frac{\coth ^8(e+f x)}{(a+a \sinh ^2(e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{\coth (e+f x) \text{csch}^6(e+f x)}{7 a f \sqrt{a \cosh ^2(e+f x)}}-\frac{2 \coth (e+f x) \text{csch}^4(e+f x)}{5 a f \sqrt{a \cosh ^2(e+f x)}}-\frac{\coth (e+f x) \text{csch}^2(e+f x)}{3 a f \sqrt{a \cosh ^2(e+f x)}} \]

[Out]

-(Coth[e + f*x]*Csch[e + f*x]^2)/(3*a*f*Sqrt[a*Cosh[e + f*x]^2]) - (2*Coth[e + f*x]*Csch[e + f*x]^4)/(5*a*f*Sq
rt[a*Cosh[e + f*x]^2]) - (Coth[e + f*x]*Csch[e + f*x]^6)/(7*a*f*Sqrt[a*Cosh[e + f*x]^2])

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Rubi [A]  time = 0.149823, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3176, 3207, 2606, 270} \[ -\frac{\coth (e+f x) \text{csch}^6(e+f x)}{7 a f \sqrt{a \cosh ^2(e+f x)}}-\frac{2 \coth (e+f x) \text{csch}^4(e+f x)}{5 a f \sqrt{a \cosh ^2(e+f x)}}-\frac{\coth (e+f x) \text{csch}^2(e+f x)}{3 a f \sqrt{a \cosh ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[Coth[e + f*x]^8/(a + a*Sinh[e + f*x]^2)^(3/2),x]

[Out]

-(Coth[e + f*x]*Csch[e + f*x]^2)/(3*a*f*Sqrt[a*Cosh[e + f*x]^2]) - (2*Coth[e + f*x]*Csch[e + f*x]^4)/(5*a*f*Sq
rt[a*Cosh[e + f*x]^2]) - (Coth[e + f*x]*Csch[e + f*x]^6)/(7*a*f*Sqrt[a*Cosh[e + f*x]^2])

Rule 3176

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

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 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.136423, size = 51, normalized size = 0.44 \[ -\frac{\coth ^3(e+f x) \left (15 \text{csch}^4(e+f x)+42 \text{csch}^2(e+f x)+35\right )}{105 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[e + f*x]^8/(a + a*Sinh[e + f*x]^2)^(3/2),x]

[Out]

-(Coth[e + f*x]^3*(35 + 42*Csch[e + f*x]^2 + 15*Csch[e + f*x]^4))/(105*f*(a*Cosh[e + f*x]^2)^(3/2))

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Maple [A]  time = 0.105, size = 57, normalized size = 0.5 \begin{align*} -{\frac{\cosh \left ( fx+e \right ) \left ( 35\, \left ( \cosh \left ( fx+e \right ) \right ) ^{4}-28\, \left ( \cosh \left ( fx+e \right ) \right ) ^{2}+8 \right ) }{105\,a \left ( \sinh \left ( fx+e \right ) \right ) ^{7}f}{\frac{1}{\sqrt{a \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(f*x+e)^8/(a+a*sinh(f*x+e)^2)^(3/2),x)

[Out]

-1/105*cosh(f*x+e)*(35*cosh(f*x+e)^4-28*cosh(f*x+e)^2+8)/a/sinh(f*x+e)^7/(a*cosh(f*x+e)^2)^(1/2)/f

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Maxima [B]  time = 2.60066, size = 2992, normalized size = 26.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^8/(a+a*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

1/3840*(2*(4095*e^(-f*x - e) - 20090*e^(-3*f*x - 3*e) + 31654*e^(-5*f*x - 5*e) - 850*e^(-7*f*x - 7*e) - 51148*
e^(-9*f*x - 9*e) + 51090*e^(-11*f*x - 11*e) - 2646*e^(-13*f*x - 13*e) + 4410*e^(-15*f*x - 15*e) - 1155*e^(-17*
f*x - 17*e))/(5*a^(3/2)*e^(-2*f*x - 2*e) - 8*a^(3/2)*e^(-4*f*x - 4*e) + 14*a^(3/2)*e^(-8*f*x - 8*e) - 14*a^(3/
2)*e^(-10*f*x - 10*e) + 8*a^(3/2)*e^(-14*f*x - 14*e) - 5*a^(3/2)*e^(-16*f*x - 16*e) + a^(3/2)*e^(-18*f*x - 18*
e) - a^(3/2)) + 2940*arctan(e^(-f*x - e))/a^(3/2) + 2625*log(e^(-f*x - e) + 1)/a^(3/2) - 2625*log(e^(-f*x - e)
- 1)/a^(3/2))/f - 1/8960*(2*(4095*e^(-f*x - e) - 21630*e^(-3*f*x - 3*e) + 39354*e^(-5*f*x - 5*e) - 13830*e^(-
7*f*x - 7*e) - 47848*e^(-9*f*x - 9*e) + 66950*e^(-11*f*x - 11*e) - 22106*e^(-13*f*x - 13*e) - 18690*e^(-15*f*x
- 15*e) + 3465*e^(-17*f*x - 17*e))/(5*a^(3/2)*e^(-2*f*x - 2*e) - 8*a^(3/2)*e^(-4*f*x - 4*e) + 14*a^(3/2)*e^(-
8*f*x - 8*e) - 14*a^(3/2)*e^(-10*f*x - 10*e) + 8*a^(3/2)*e^(-14*f*x - 14*e) - 5*a^(3/2)*e^(-16*f*x - 16*e) + a
^(3/2)*e^(-18*f*x - 18*e) - a^(3/2)) + 7560*arctan(e^(-f*x - e))/a^(3/2) + 315*log(e^(-f*x - e) + 1)/a^(3/2) -
315*log(e^(-f*x - e) - 1)/a^(3/2))/f - 1/8960*(2*(3465*e^(-f*x - e) - 18690*e^(-3*f*x - 3*e) - 22106*e^(-5*f*
x - 5*e) + 66950*e^(-7*f*x - 7*e) - 47848*e^(-9*f*x - 9*e) - 13830*e^(-11*f*x - 11*e) + 39354*e^(-13*f*x - 13*
e) - 21630*e^(-15*f*x - 15*e) + 4095*e^(-17*f*x - 17*e))/(5*a^(3/2)*e^(-2*f*x - 2*e) - 8*a^(3/2)*e^(-4*f*x - 4
*e) + 14*a^(3/2)*e^(-8*f*x - 8*e) - 14*a^(3/2)*e^(-10*f*x - 10*e) + 8*a^(3/2)*e^(-14*f*x - 14*e) - 5*a^(3/2)*e
^(-16*f*x - 16*e) + a^(3/2)*e^(-18*f*x - 18*e) - a^(3/2)) + 7560*arctan(e^(-f*x - e))/a^(3/2) - 315*log(e^(-f*
x - e) + 1)/a^(3/2) + 315*log(e^(-f*x - e) - 1)/a^(3/2))/f - 1/3840*(2*(1155*e^(-f*x - e) - 4410*e^(-3*f*x - 3
*e) + 2646*e^(-5*f*x - 5*e) - 51090*e^(-7*f*x - 7*e) + 51148*e^(-9*f*x - 9*e) + 850*e^(-11*f*x - 11*e) - 31654
*e^(-13*f*x - 13*e) + 20090*e^(-15*f*x - 15*e) - 4095*e^(-17*f*x - 17*e))/(5*a^(3/2)*e^(-2*f*x - 2*e) - 8*a^(3
/2)*e^(-4*f*x - 4*e) + 14*a^(3/2)*e^(-8*f*x - 8*e) - 14*a^(3/2)*e^(-10*f*x - 10*e) + 8*a^(3/2)*e^(-14*f*x - 14
*e) - 5*a^(3/2)*e^(-16*f*x - 16*e) + a^(3/2)*e^(-18*f*x - 18*e) - a^(3/2)) - 2940*arctan(e^(-f*x - e))/a^(3/2)
+ 2625*log(e^(-f*x - e) + 1)/a^(3/2) - 2625*log(e^(-f*x - e) - 1)/a^(3/2))/f + 1/768*(2*(1155*e^(-f*x - e) -
5670*e^(-3*f*x - 3*e) + 8946*e^(-5*f*x - 5*e) - 270*e^(-7*f*x - 7*e) + 4696*e^(-9*f*x - 9*e) - 2930*e^(-11*f*x
- 11*e) - 658*e^(-13*f*x - 13*e) + 1190*e^(-15*f*x - 15*e) - 315*e^(-17*f*x - 17*e))/(5*a^(3/2)*e^(-2*f*x - 2
*e) - 8*a^(3/2)*e^(-4*f*x - 4*e) + 14*a^(3/2)*e^(-8*f*x - 8*e) - 14*a^(3/2)*e^(-10*f*x - 10*e) + 8*a^(3/2)*e^(
-14*f*x - 14*e) - 5*a^(3/2)*e^(-16*f*x - 16*e) + a^(3/2)*e^(-18*f*x - 18*e) - a^(3/2)) + 840*arctan(e^(-f*x -
e))/a^(3/2) + 735*log(e^(-f*x - e) + 1)/a^(3/2) - 735*log(e^(-f*x - e) - 1)/a^(3/2))/f - 1/768*(2*(315*e^(-f*x
- e) - 1190*e^(-3*f*x - 3*e) + 658*e^(-5*f*x - 5*e) + 2930*e^(-7*f*x - 7*e) - 4696*e^(-9*f*x - 9*e) + 270*e^(
-11*f*x - 11*e) - 8946*e^(-13*f*x - 13*e) + 5670*e^(-15*f*x - 15*e) - 1155*e^(-17*f*x - 17*e))/(5*a^(3/2)*e^(-
2*f*x - 2*e) - 8*a^(3/2)*e^(-4*f*x - 4*e) + 14*a^(3/2)*e^(-8*f*x - 8*e) - 14*a^(3/2)*e^(-10*f*x - 10*e) + 8*a^
(3/2)*e^(-14*f*x - 14*e) - 5*a^(3/2)*e^(-16*f*x - 16*e) + a^(3/2)*e^(-18*f*x - 18*e) - a^(3/2)) - 840*arctan(e
^(-f*x - e))/a^(3/2) + 735*log(e^(-f*x - e) + 1)/a^(3/2) - 735*log(e^(-f*x - e) - 1)/a^(3/2))/f - 1/128*((315*
e^(-f*x - e) - 1680*e^(-3*f*x - 3*e) + 3108*e^(-5*f*x - 5*e) - 1200*e^(-7*f*x - 7*e) - 3646*e^(-9*f*x - 9*e) -
1200*e^(-11*f*x - 11*e) + 3108*e^(-13*f*x - 13*e) - 1680*e^(-15*f*x - 15*e) + 315*e^(-17*f*x - 17*e))/(5*a^(3
/2)*e^(-2*f*x - 2*e) - 8*a^(3/2)*e^(-4*f*x - 4*e) + 14*a^(3/2)*e^(-8*f*x - 8*e) - 14*a^(3/2)*e^(-10*f*x - 10*e
) + 8*a^(3/2)*e^(-14*f*x - 14*e) - 5*a^(3/2)*e^(-16*f*x - 16*e) + a^(3/2)*e^(-18*f*x - 18*e) - a^(3/2)) + 315*
arctan(e^(-f*x - e))/a^(3/2))/f + 1/2688*(1155*e^(-f*x - e) + 1393*e^(-3*f*x - 3*e) - 4865*e^(-5*f*x - 5*e) +
3965*e^(-7*f*x - 7*e) + 825*e^(-9*f*x - 9*e) - 3245*e^(-11*f*x - 11*e) + 1925*e^(-13*f*x - 13*e) - 385*e^(-15*
f*x - 15*e))/((5*a^(3/2)*e^(-2*f*x - 2*e) - 8*a^(3/2)*e^(-4*f*x - 4*e) + 14*a^(3/2)*e^(-8*f*x - 8*e) - 14*a^(3
/2)*e^(-10*f*x - 10*e) + 8*a^(3/2)*e^(-14*f*x - 14*e) - 5*a^(3/2)*e^(-16*f*x - 16*e) + a^(3/2)*e^(-18*f*x - 18
*e) - a^(3/2))*f) - 1/2688*(385*e^(-3*f*x - 3*e) - 1925*e^(-5*f*x - 5*e) + 3245*e^(-7*f*x - 7*e) - 825*e^(-9*f
*x - 9*e) - 3965*e^(-11*f*x - 11*e) + 4865*e^(-13*f*x - 13*e) - 1393*e^(-15*f*x - 15*e) - 1155*e^(-17*f*x - 17
*e))/((5*a^(3/2)*e^(-2*f*x - 2*e) - 8*a^(3/2)*e^(-4*f*x - 4*e) + 14*a^(3/2)*e^(-8*f*x - 8*e) - 14*a^(3/2)*e^(-
10*f*x - 10*e) + 8*a^(3/2)*e^(-14*f*x - 14*e) - 5*a^(3/2)*e^(-16*f*x - 16*e) + a^(3/2)*e^(-18*f*x - 18*e) - a^
(3/2))*f) + 55/128*arctan(e^(-f*x - e))/(a^(3/2)*f)

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Fricas [B]  time = 2.14295, size = 6649, normalized size = 57.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^8/(a+a*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

-8/105*(385*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^10 + 35*e^(f*x + e)*sinh(f*x + e)^11 + 7*(275*cosh(f*x + e
)^2 + 4)*e^(f*x + e)*sinh(f*x + e)^9 + 21*(275*cosh(f*x + e)^3 + 12*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^8
+ 6*(1925*cosh(f*x + e)^4 + 168*cosh(f*x + e)^2 + 19)*e^(f*x + e)*sinh(f*x + e)^7 + 42*(385*cosh(f*x + e)^5 +
56*cosh(f*x + e)^3 + 19*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^6 + 14*(1155*cosh(f*x + e)^6 + 252*cosh(f*x
+ e)^4 + 171*cosh(f*x + e)^2 + 2)*e^(f*x + e)*sinh(f*x + e)^5 + 14*(825*cosh(f*x + e)^7 + 252*cosh(f*x + e)^5
+ 285*cosh(f*x + e)^3 + 10*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^4 + 7*(825*cosh(f*x + e)^8 + 336*cosh(f*x
+ e)^6 + 570*cosh(f*x + e)^4 + 40*cosh(f*x + e)^2 + 5)*e^(f*x + e)*sinh(f*x + e)^3 + 7*(275*cosh(f*x + e)^9 +
144*cosh(f*x + e)^7 + 342*cosh(f*x + e)^5 + 40*cosh(f*x + e)^3 + 15*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^2
+ 7*(55*cosh(f*x + e)^10 + 36*cosh(f*x + e)^8 + 114*cosh(f*x + e)^6 + 20*cosh(f*x + e)^4 + 15*cosh(f*x + e)^2
)*e^(f*x + e)*sinh(f*x + e) + (35*cosh(f*x + e)^11 + 28*cosh(f*x + e)^9 + 114*cosh(f*x + e)^7 + 28*cosh(f*x +
e)^5 + 35*cosh(f*x + e)^3)*e^(f*x + e))*sqrt(a*e^(4*f*x + 4*e) + 2*a*e^(2*f*x + 2*e) + a)*e^(-f*x - e)/(a^2*f*
cosh(f*x + e)^14 - 7*a^2*f*cosh(f*x + e)^12 + (a^2*f*e^(2*f*x + 2*e) + a^2*f)*sinh(f*x + e)^14 + 14*(a^2*f*cos
h(f*x + e)*e^(2*f*x + 2*e) + a^2*f*cosh(f*x + e))*sinh(f*x + e)^13 + 21*a^2*f*cosh(f*x + e)^10 + 7*(13*a^2*f*c
osh(f*x + e)^2 - a^2*f + (13*a^2*f*cosh(f*x + e)^2 - a^2*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^12 + 28*(13*a^2*f*c
osh(f*x + e)^3 - 3*a^2*f*cosh(f*x + e) + (13*a^2*f*cosh(f*x + e)^3 - 3*a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*s
inh(f*x + e)^11 - 35*a^2*f*cosh(f*x + e)^8 + 7*(143*a^2*f*cosh(f*x + e)^4 - 66*a^2*f*cosh(f*x + e)^2 + 3*a^2*f
+ (143*a^2*f*cosh(f*x + e)^4 - 66*a^2*f*cosh(f*x + e)^2 + 3*a^2*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^10 + 14*(14
3*a^2*f*cosh(f*x + e)^5 - 110*a^2*f*cosh(f*x + e)^3 + 15*a^2*f*cosh(f*x + e) + (143*a^2*f*cosh(f*x + e)^5 - 11
0*a^2*f*cosh(f*x + e)^3 + 15*a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^9 + 35*a^2*f*cosh(f*x + e)^6
+ 7*(429*a^2*f*cosh(f*x + e)^6 - 495*a^2*f*cosh(f*x + e)^4 + 135*a^2*f*cosh(f*x + e)^2 - 5*a^2*f + (429*a^2*f*
cosh(f*x + e)^6 - 495*a^2*f*cosh(f*x + e)^4 + 135*a^2*f*cosh(f*x + e)^2 - 5*a^2*f)*e^(2*f*x + 2*e))*sinh(f*x +
e)^8 + 8*(429*a^2*f*cosh(f*x + e)^7 - 693*a^2*f*cosh(f*x + e)^5 + 315*a^2*f*cosh(f*x + e)^3 - 35*a^2*f*cosh(f
*x + e) + (429*a^2*f*cosh(f*x + e)^7 - 693*a^2*f*cosh(f*x + e)^5 + 315*a^2*f*cosh(f*x + e)^3 - 35*a^2*f*cosh(f
*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^7 - 21*a^2*f*cosh(f*x + e)^4 + 7*(429*a^2*f*cosh(f*x + e)^8 - 924*a^2*
f*cosh(f*x + e)^6 + 630*a^2*f*cosh(f*x + e)^4 - 140*a^2*f*cosh(f*x + e)^2 + 5*a^2*f + (429*a^2*f*cosh(f*x + e)
^8 - 924*a^2*f*cosh(f*x + e)^6 + 630*a^2*f*cosh(f*x + e)^4 - 140*a^2*f*cosh(f*x + e)^2 + 5*a^2*f)*e^(2*f*x + 2
*e))*sinh(f*x + e)^6 + 14*(143*a^2*f*cosh(f*x + e)^9 - 396*a^2*f*cosh(f*x + e)^7 + 378*a^2*f*cosh(f*x + e)^5 -
140*a^2*f*cosh(f*x + e)^3 + 15*a^2*f*cosh(f*x + e) + (143*a^2*f*cosh(f*x + e)^9 - 396*a^2*f*cosh(f*x + e)^7 +
378*a^2*f*cosh(f*x + e)^5 - 140*a^2*f*cosh(f*x + e)^3 + 15*a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e
)^5 + 7*a^2*f*cosh(f*x + e)^2 + 7*(143*a^2*f*cosh(f*x + e)^10 - 495*a^2*f*cosh(f*x + e)^8 + 630*a^2*f*cosh(f*x
+ e)^6 - 350*a^2*f*cosh(f*x + e)^4 + 75*a^2*f*cosh(f*x + e)^2 - 3*a^2*f + (143*a^2*f*cosh(f*x + e)^10 - 495*a
^2*f*cosh(f*x + e)^8 + 630*a^2*f*cosh(f*x + e)^6 - 350*a^2*f*cosh(f*x + e)^4 + 75*a^2*f*cosh(f*x + e)^2 - 3*a^
2*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^4 + 28*(13*a^2*f*cosh(f*x + e)^11 - 55*a^2*f*cosh(f*x + e)^9 + 90*a^2*f*co
sh(f*x + e)^7 - 70*a^2*f*cosh(f*x + e)^5 + 25*a^2*f*cosh(f*x + e)^3 - 3*a^2*f*cosh(f*x + e) + (13*a^2*f*cosh(f
*x + e)^11 - 55*a^2*f*cosh(f*x + e)^9 + 90*a^2*f*cosh(f*x + e)^7 - 70*a^2*f*cosh(f*x + e)^5 + 25*a^2*f*cosh(f*
x + e)^3 - 3*a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^3 - a^2*f + 7*(13*a^2*f*cosh(f*x + e)^12 - 66
*a^2*f*cosh(f*x + e)^10 + 135*a^2*f*cosh(f*x + e)^8 - 140*a^2*f*cosh(f*x + e)^6 + 75*a^2*f*cosh(f*x + e)^4 - 1
8*a^2*f*cosh(f*x + e)^2 + a^2*f + (13*a^2*f*cosh(f*x + e)^12 - 66*a^2*f*cosh(f*x + e)^10 + 135*a^2*f*cosh(f*x
+ e)^8 - 140*a^2*f*cosh(f*x + e)^6 + 75*a^2*f*cosh(f*x + e)^4 - 18*a^2*f*cosh(f*x + e)^2 + a^2*f)*e^(2*f*x + 2
*e))*sinh(f*x + e)^2 + (a^2*f*cosh(f*x + e)^14 - 7*a^2*f*cosh(f*x + e)^12 + 21*a^2*f*cosh(f*x + e)^10 - 35*a^2
*f*cosh(f*x + e)^8 + 35*a^2*f*cosh(f*x + e)^6 - 21*a^2*f*cosh(f*x + e)^4 + 7*a^2*f*cosh(f*x + e)^2 - a^2*f)*e^
(2*f*x + 2*e) + 14*(a^2*f*cosh(f*x + e)^13 - 6*a^2*f*cosh(f*x + e)^11 + 15*a^2*f*cosh(f*x + e)^9 - 20*a^2*f*co
sh(f*x + e)^7 + 15*a^2*f*cosh(f*x + e)^5 - 6*a^2*f*cosh(f*x + e)^3 + a^2*f*cosh(f*x + e) + (a^2*f*cosh(f*x + e
)^13 - 6*a^2*f*cosh(f*x + e)^11 + 15*a^2*f*cosh(f*x + e)^9 - 20*a^2*f*cosh(f*x + e)^7 + 15*a^2*f*cosh(f*x + e)
^5 - 6*a^2*f*cosh(f*x + e)^3 + a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e))

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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(coth(f*x+e)**8/(a+a*sinh(f*x+e)**2)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.94312, size = 132, normalized size = 1.15 \begin{align*} -\frac{8 \,{\left (35 \, \sqrt{a} e^{\left (11 \, f x + 11 \, e\right )} + 28 \, \sqrt{a} e^{\left (9 \, f x + 9 \, e\right )} + 114 \, \sqrt{a} e^{\left (7 \, f x + 7 \, e\right )} + 28 \, \sqrt{a} e^{\left (5 \, f x + 5 \, e\right )} + 35 \, \sqrt{a} e^{\left (3 \, f x + 3 \, e\right )}\right )}}{105 \, a^{2} f{\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^8/(a+a*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

-8/105*(35*sqrt(a)*e^(11*f*x + 11*e) + 28*sqrt(a)*e^(9*f*x + 9*e) + 114*sqrt(a)*e^(7*f*x + 7*e) + 28*sqrt(a)*e
^(5*f*x + 5*e) + 35*sqrt(a)*e^(3*f*x + 3*e))/(a^2*f*(e^(2*f*x + 2*e) - 1)^7)