3.5.56 \(\int \frac {\coth ^8(e+f x)}{(a+a \sinh ^2(e+f x))^{3/2}} \, dx\) [456]

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

[Out]

-1/3*coth(f*x+e)*csch(f*x+e)^2/a/f/(a*cosh(f*x+e)^2)^(1/2)-2/5*coth(f*x+e)*csch(f*x+e)^4/a/f/(a*cosh(f*x+e)^2)
^(1/2)-1/7*coth(f*x+e)*csch(f*x+e)^6/a/f/(a*cosh(f*x+e)^2)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3255, 3286, 2686, 276} \begin {gather*} -\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)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(Coth[e + f*x]*Csch[e + f*x]^2)/(a*f*Sqrt[a*Cosh[e + f*x]^2]) - (2*Coth[e + f*x]*Csch[e + f*x]^4)/(5*a*f*
Sqrt[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 276

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]

Rule 2686

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 3255

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 3286

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
]])

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)) \text {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)) \text {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*}

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Mathematica [A]
time = 0.10, size = 51, normalized size = 0.44 \begin {gather*} -\frac {\coth ^3(e+f x) \left (35+42 \text {csch}^2(e+f x)+15 \text {csch}^4(e+f x)\right )}{105 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/105*(Coth[e + f*x]^3*(35 + 42*Csch[e + f*x]^2 + 15*Csch[e + f*x]^4))/(f*(a*Cosh[e + f*x]^2)^(3/2))

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Maple [A]
time = 1.78, size = 57, normalized size = 0.50

method	result	size
default	\(-\frac {\cosh \left (f x +e \right ) \left (35 \left (\cosh ^{4}\left (f x +e \right )\right )-28 \left (\cosh ^{2}\left (f x +e \right )\right )+8\right )}{105 a \sinh \left (f x +e \right )^{7} \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}\, f}\)	\(57\)
risch	\(-\frac {8 \left (35 \,{\mathrm e}^{8 f x +8 e}+28 \,{\mathrm e}^{6 f x +6 e}+114 \,{\mathrm e}^{4 f x +4 e}+28 \,{\mathrm e}^{2 f x +2 e}+35\right ) \left ({\mathrm e}^{2 f x +2 e}+1\right ) {\mathrm e}^{2 f x +2 e}}{105 \left ({\mathrm e}^{2 f x +2 e}-1\right )^{7} f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, a}\)	\(114\)

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,method=_RETURNVERBOSE)

[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] Leaf count of result is larger than twice the leaf count of optimal. 2378 vs. \(2 (112) = 224\).
time = 0.66, size = 2378, normalized size = 20.68 \begin {gather*} \text {Too large to display} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 2511 vs. \(2 (103) = 206\).
time = 0.45, size = 2511, normalized size = 21.83 \begin {gather*} \text {Too large to display} \end {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 4370 deep

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [B]
time = 0.95, size = 457, normalized size = 3.97 \begin {gather*} -\frac {16\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^2\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {464\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{15\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^3\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {3072\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{35\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^4\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {4736\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{35\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^5\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {768\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^6\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {256\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^7\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (16*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(3*a^2*f*(exp(2*e + 2*f*x) - 1)^2*
(exp(e + f*x) + exp(3*e + 3*f*x))) - (464*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2)
)/(15*a^2*f*(exp(2*e + 2*f*x) - 1)^3*(exp(e + f*x) + exp(3*e + 3*f*x))) - (3072*exp(3*e + 3*f*x)*(a + a*(exp(e
+ f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(35*a^2*f*(exp(2*e + 2*f*x) - 1)^4*(exp(e + f*x) + exp(3*e + 3*f*x)))
- (4736*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(35*a^2*f*(exp(2*e + 2*f*x) - 1)
^5*(exp(e + f*x) + exp(3*e + 3*f*x))) - (768*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1
/2))/(7*a^2*f*(exp(2*e + 2*f*x) - 1)^6*(exp(e + f*x) + exp(3*e + 3*f*x))) - (256*exp(3*e + 3*f*x)*(a + a*(exp(
e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(7*a^2*f*(exp(2*e + 2*f*x) - 1)^7*(exp(e + f*x) + exp(3*e + 3*f*x)))

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