3.219 \(\int \frac {\coth ^2(x)}{(a+b \text {sech}^2(x))^{5/2}} \, dx\)
Optimal. Leaf size=133 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{a^{5/2}}-\frac {(a-3 b) (3 a+b) \coth (x) \sqrt {a-b \tanh ^2(x)+b}}{3 a^2 (a+b)^3}-\frac {b (7 a+3 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a-b \tanh ^2(x)+b}}-\frac {b \coth (x)}{3 a (a+b) \left (a-b \tanh ^2(x)+b\right )^{3/2}} \]
[Out]
arctanh(a^(1/2)*tanh(x)/(a+b-b*tanh(x)^2)^(1/2))/a^(5/2)-1/3*b*(7*a+3*b)*coth(x)/a^2/(a+b)^2/(a+b-b*tanh(x)^2)
^(1/2)-1/3*(a-3*b)*(3*a+b)*coth(x)*(a+b-b*tanh(x)^2)^(1/2)/a^2/(a+b)^3-1/3*b*coth(x)/a/(a+b)/(a+b-b*tanh(x)^2)
^(3/2)
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Rubi [A] time = 0.38, antiderivative size = 133, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used =
{4141, 1975, 472, 579, 583, 12, 377, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{a^{5/2}}-\frac {(a-3 b) (3 a+b) \coth (x) \sqrt {a-b \tanh ^2(x)+b}}{3 a^2 (a+b)^3}-\frac {b (7 a+3 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a-b \tanh ^2(x)+b}}-\frac {b \coth (x)}{3 a (a+b) \left (a-b \tanh ^2(x)+b\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[Coth[x]^2/(a + b*Sech[x]^2)^(5/2),x]
[Out]
ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(5/2) - (b*Coth[x])/(3*a*(a + b)*(a + b - b*Tanh[x]^2)^
(3/2)) - (b*(7*a + 3*b)*Coth[x])/(3*a^2*(a + b)^2*Sqrt[a + b - b*Tanh[x]^2]) - ((a - 3*b)*(3*a + b)*Coth[x]*Sq
rt[a + b - b*Tanh[x]^2])/(3*a^2*(a + b)^3)
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 377
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 472
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 579
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +
1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(
m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
Rule 583
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]
Rule 1975
Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&& !BinomialMatchQ[{u, v}, x]
Rule 4141
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((d*ff*x)^m*(a + b*(1 + ff^2*x^2)^(n/2))^p)/(1 + ff^
2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] ||
EqQ[n, 2])
Rubi steps
\begin {align*} \int \frac {\coth ^2(x)}{\left (a+b \text {sech}^2(x)\right )^{5/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right ) \left (a+b \left (1-x^2\right )\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right ) \left (a+b-b x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=-\frac {b \coth (x)}{3 a (a+b) \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {-3 a+b-4 b x^2}{x^2 \left (1-x^2\right ) \left (a+b-b x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )}{3 a (a+b)}\\ &=-\frac {b \coth (x)}{3 a (a+b) \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {b (7 a+3 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a+b-b \tanh ^2(x)}}+\frac {\operatorname {Subst}\left (\int \frac {(a-3 b) (3 a+b)+2 b (7 a+3 b) x^2}{x^2 \left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\tanh (x)\right )}{3 a^2 (a+b)^2}\\ &=-\frac {b \coth (x)}{3 a (a+b) \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {b (7 a+3 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a+b-b \tanh ^2(x)}}-\frac {(a-3 b) (3 a+b) \coth (x) \sqrt {a+b-b \tanh ^2(x)}}{3 a^2 (a+b)^3}-\frac {\operatorname {Subst}\left (\int -\frac {3 (a+b)^3}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\tanh (x)\right )}{3 a^2 (a+b)^3}\\ &=-\frac {b \coth (x)}{3 a (a+b) \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {b (7 a+3 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a+b-b \tanh ^2(x)}}-\frac {(a-3 b) (3 a+b) \coth (x) \sqrt {a+b-b \tanh ^2(x)}}{3 a^2 (a+b)^3}+\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\tanh (x)\right )}{a^2}\\ &=-\frac {b \coth (x)}{3 a (a+b) \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {b (7 a+3 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a+b-b \tanh ^2(x)}}-\frac {(a-3 b) (3 a+b) \coth (x) \sqrt {a+b-b \tanh ^2(x)}}{3 a^2 (a+b)^3}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{a^2}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{a^{5/2}}-\frac {b \coth (x)}{3 a (a+b) \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {b (7 a+3 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a+b-b \tanh ^2(x)}}-\frac {(a-3 b) (3 a+b) \coth (x) \sqrt {a+b-b \tanh ^2(x)}}{3 a^2 (a+b)^3}\\ \end {align*}
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Mathematica [A] time = 0.83, size = 155, normalized size = 1.17 \[ \frac {\text {sech}^5(x) \left (\frac {\sqrt {2} (a \cosh (2 x)+a+2 b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sinh (x)}{\sqrt {a \cosh (2 x)+a+2 b}}\right )}{a^{5/2}}-\frac {(a \cosh (2 x)+a+2 b) \left (3 a^2 \text {csch}(x) (a \cosh (2 x)+a+2 b)^2-4 b^3 (a+b) \sinh (x)+2 b^2 (9 a+4 b) \sinh (x) (a \cosh (2 x)+a+2 b)\right )}{3 a^2 (a+b)^3}\right )}{8 \left (a+b \text {sech}^2(x)\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[x]^2/(a + b*Sech[x]^2)^(5/2),x]
[Out]
(Sech[x]^5*((Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*(a + 2*b + a*Cosh[2*x])^(5
/2))/a^(5/2) - ((a + 2*b + a*Cosh[2*x])*(3*a^2*(a + 2*b + a*Cosh[2*x])^2*Csch[x] - 4*b^3*(a + b)*Sinh[x] + 2*b
^2*(9*a + 4*b)*(a + 2*b + a*Cosh[2*x])*Sinh[x]))/(3*a^2*(a + b)^3)))/(8*(a + b*Sech[x]^2)^(5/2))
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fricas [B] time = 4.18, size = 11205, normalized size = 84.25 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^2/(a+b*sech(x)^2)^(5/2),x, algorithm="fricas")
[Out]
[1/12*(3*((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^10 + 10*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)*
sinh(x)^9 + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sinh(x)^10 + (3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8
*a*b^4)*cosh(x)^8 + (3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4 + 45*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^
2*b^3)*cosh(x)^2)*sinh(x)^8 + 8*(15*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^3 + (3*a^5 + 17*a^4*b + 33*a
^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x))*sinh(x)^7 + 2*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*
b^5)*cosh(x)^6 + 2*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5 + 105*(a^5 + 3*a^4*b + 3*a^3*b^
2 + a^2*b^3)*cosh(x)^4 + 14*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^2)*sinh(x)^6 + 4*(6
3*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^5 + 14*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*
cosh(x)^3 + 3*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x))*sinh(x)^5 - a^5 - 3*a^4*b
- 3*a^3*b^2 - a^2*b^3 - 2*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^4 + 2*(105*(a^5
+ 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^6 - a^5 - 7*a^4*b - 23*a^3*b^2 - 37*a^2*b^3 - 28*a*b^4 - 8*b^5 + 35*
(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^4 + 15*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3
+ 28*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^4 + 8*(15*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^7 + 7*(3*a^5 +
17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^5 + 5*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b
^4 + 8*b^5)*cosh(x)^3 - (a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x))*sinh(x)^3 - (3*a
^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^2 + (45*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(
x)^8 + 28*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^6 - 3*a^5 - 17*a^4*b - 33*a^3*b^2 - 2
7*a^2*b^3 - 8*a*b^4 + 30*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^4 - 12*(a^5 + 7*
a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^2 + 2*(5*(a^5 + 3*a^4*b + 3*a^3*b^2 + a
^2*b^3)*cosh(x)^9 + 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^7 + 6*(a^5 + 7*a^4*b + 23
*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^5 - 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 +
8*b^5)*cosh(x)^3 - (3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x))*sinh(x))*sqrt(a)*log((a*b^2
*cosh(x)^8 + 8*a*b^2*cosh(x)*sinh(x)^7 + a*b^2*sinh(x)^8 - 2*(a*b^2 - b^3)*cosh(x)^6 + 2*(14*a*b^2*cosh(x)^2 -
a*b^2 + b^3)*sinh(x)^6 + 4*(14*a*b^2*cosh(x)^3 - 3*(a*b^2 - b^3)*cosh(x))*sinh(x)^5 + (a^3 + 4*a^2*b + 9*a*b^
2)*cosh(x)^4 + (70*a*b^2*cosh(x)^4 + a^3 + 4*a^2*b + 9*a*b^2 - 30*(a*b^2 - b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*a
*b^2*cosh(x)^5 - 10*(a*b^2 - b^3)*cosh(x)^3 + (a^3 + 4*a^2*b + 9*a*b^2)*cosh(x))*sinh(x)^3 + a^3 + 2*(a^3 + 3*
a^2*b)*cosh(x)^2 + 2*(14*a*b^2*cosh(x)^6 - 15*(a*b^2 - b^3)*cosh(x)^4 + a^3 + 3*a^2*b + 3*(a^3 + 4*a^2*b + 9*a
*b^2)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(b^2*cosh(x)^6 + 6*b^2*cosh(x)*sinh(x)^5 + b^2*sinh(x)^6 - 3*b^2*cosh(x)^
4 + 3*(5*b^2*cosh(x)^2 - b^2)*sinh(x)^4 + 4*(5*b^2*cosh(x)^3 - 3*b^2*cosh(x))*sinh(x)^3 - (a^2 + 4*a*b)*cosh(x
)^2 + (15*b^2*cosh(x)^4 - 18*b^2*cosh(x)^2 - a^2 - 4*a*b)*sinh(x)^2 - a^2 + 2*(3*b^2*cosh(x)^5 - 6*b^2*cosh(x)
^3 - (a^2 + 4*a*b)*cosh(x))*sinh(x))*sqrt(a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)
*sinh(x) + sinh(x)^2)) + 4*(2*a*b^2*cosh(x)^7 - 3*(a*b^2 - b^3)*cosh(x)^5 + (a^3 + 4*a^2*b + 9*a*b^2)*cosh(x)^
3 + (a^3 + 3*a^2*b)*cosh(x))*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3
*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + 3*((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*
b^3)*cosh(x)^10 + 10*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)*sinh(x)^9 + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^
2*b^3)*sinh(x)^10 + (3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^8 + (3*a^5 + 17*a^4*b + 33*
a^3*b^2 + 27*a^2*b^3 + 8*a*b^4 + 45*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^2)*sinh(x)^8 + 8*(15*(a^5 +
3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^3 + (3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x))*sin
h(x)^7 + 2*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^6 + 2*(a^5 + 7*a^4*b + 23*a^3*
b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5 + 105*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^4 + 14*(3*a^5 + 17*a^4
*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^2)*sinh(x)^6 + 4*(63*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cos
h(x)^5 + 14*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^3 + 3*(a^5 + 7*a^4*b + 23*a^3*b^2 +
37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x))*sinh(x)^5 - a^5 - 3*a^4*b - 3*a^3*b^2 - a^2*b^3 - 2*(a^5 + 7*a^4*b +
23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^4 + 2*(105*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^6
- a^5 - 7*a^4*b - 23*a^3*b^2 - 37*a^2*b^3 - 28*a*b^4 - 8*b^5 + 35*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3
+ 8*a*b^4)*cosh(x)^4 + 15*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^4 +
8*(15*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^7 + 7*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b
^4)*cosh(x)^5 + 5*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^3 - (a^5 + 7*a^4*b + 23
*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x))*sinh(x)^3 - (3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8
*a*b^4)*cosh(x)^2 + (45*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^8 + 28*(3*a^5 + 17*a^4*b + 33*a^3*b^2 +
27*a^2*b^3 + 8*a*b^4)*cosh(x)^6 - 3*a^5 - 17*a^4*b - 33*a^3*b^2 - 27*a^2*b^3 - 8*a*b^4 + 30*(a^5 + 7*a^4*b + 2
3*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^4 - 12*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4
+ 8*b^5)*cosh(x)^2)*sinh(x)^2 + 2*(5*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^9 + 4*(3*a^5 + 17*a^4*b + 3
3*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^7 + 6*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*c
osh(x)^5 - 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^3 - (3*a^5 + 17*a^4*b + 33*a
^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x))*sinh(x))*sqrt(a)*log(-(a*cosh(x)^4 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)
^4 + 2*(a + b)*cosh(x)^2 + 2*(3*a*cosh(x)^2 + a + b)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh
(x)^2 + 1)*sqrt(a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4
*(a*cosh(x)^3 + (a + b)*cosh(x))*sinh(x) + a)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) - 4*sqrt(2)*((3*a^5
+ 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^8 + 8*(3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)*sinh(x)^7 + (3*a^5 + 9*a^3*b^2
+ 4*a^2*b^3)*sinh(x)^8 + 4*(3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4)*cosh(x)^6 + 4*(3*a^5 + 6*a^4*b + 8*a^2*b^3
+ 3*a*b^4 + 7*(3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^2)*sinh(x)^6 + 8*(7*(3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cos
h(x)^3 + 3*(3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4)*cosh(x))*sinh(x)^5 + 3*a^5 + 9*a^3*b^2 + 4*a^2*b^3 + 6*(3*a
^5 + 8*a^4*b + 5*a^3*b^2 - 12*a^2*b^3 - 4*a*b^4)*cosh(x)^4 + 2*(9*a^5 + 24*a^4*b + 15*a^3*b^2 - 36*a^2*b^3 - 1
2*a*b^4 + 35*(3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^4 + 30*(3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4)*cosh(x)^2)
*sinh(x)^4 + 8*(7*(3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^5 + 10*(3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4)*cosh(
x)^3 + 3*(3*a^5 + 8*a^4*b + 5*a^3*b^2 - 12*a^2*b^3 - 4*a*b^4)*cosh(x))*sinh(x)^3 + 4*(3*a^5 + 6*a^4*b + 8*a^2*
b^3 + 3*a*b^4)*cosh(x)^2 + 4*(7*(3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^6 + 3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*
b^4 + 15*(3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4)*cosh(x)^4 + 9*(3*a^5 + 8*a^4*b + 5*a^3*b^2 - 12*a^2*b^3 - 4*a
*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^7 + 3*(3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3
*a*b^4)*cosh(x)^5 + 3*(3*a^5 + 8*a^4*b + 5*a^3*b^2 - 12*a^2*b^3 - 4*a*b^4)*cosh(x)^3 + (3*a^5 + 6*a^4*b + 8*a^
2*b^3 + 3*a*b^4)*cosh(x))*sinh(x))*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) +
sinh(x)^2)))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)^10 + 10*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cos
h(x)*sinh(x)^9 + (a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*sinh(x)^10 + (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^
3 + 8*a^4*b^4)*cosh(x)^8 + (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4 + 45*(a^8 + 3*a^7*b + 3*a^6
*b^2 + a^5*b^3)*cosh(x)^2)*sinh(x)^8 - a^8 - 3*a^7*b - 3*a^6*b^2 - a^5*b^3 + 8*(15*(a^8 + 3*a^7*b + 3*a^6*b^2
+ a^5*b^3)*cosh(x)^3 + (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(x))*sinh(x)^7 + 2*(a^8 +
7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh(x)^6 + 2*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5
*b^3 + 28*a^4*b^4 + 8*a^3*b^5 + 105*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)^4 + 14*(3*a^8 + 17*a^7*b + 3
3*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(x)^2)*sinh(x)^6 + 4*(63*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)
^5 + 14*(3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(x)^3 + 3*(a^8 + 7*a^7*b + 23*a^6*b^2 + 3
7*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh(x))*sinh(x)^5 - 2*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b
^4 + 8*a^3*b^5)*cosh(x)^4 - 2*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5 - 105*(a^8 + 3
*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)^6 - 35*(3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(x)^
4 - 15*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh(x)^2)*sinh(x)^4 + 8*(15*(a^8 +
3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)^7 + 7*(3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(x)^
5 + 5*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh(x)^3 - (a^8 + 7*a^7*b + 23*a^6*b
^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh(x))*sinh(x)^3 - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8
*a^4*b^4)*cosh(x)^2 + (45*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)^8 - 3*a^8 - 17*a^7*b - 33*a^6*b^2 - 27
*a^5*b^3 - 8*a^4*b^4 + 28*(3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(x)^6 + 30*(a^8 + 7*a^7
*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh(x)^4 - 12*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3
+ 28*a^4*b^4 + 8*a^3*b^5)*cosh(x)^2)*sinh(x)^2 + 2*(5*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)^9 + 4*(3*
a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(x)^7 + 6*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 +
28*a^4*b^4 + 8*a^3*b^5)*cosh(x)^5 - 4*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh
(x)^3 - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(x))*sinh(x)), -1/6*(3*((a^5 + 3*a^4*b +
3*a^3*b^2 + a^2*b^3)*cosh(x)^10 + 10*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)*sinh(x)^9 + (a^5 + 3*a^4*b
+ 3*a^3*b^2 + a^2*b^3)*sinh(x)^10 + (3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^8 + (3*a^5
+ 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4 + 45*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^2)*sinh(x)^8
+ 8*(15*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^3 + (3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b
^4)*cosh(x))*sinh(x)^7 + 2*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^6 + 2*(a^5 + 7
*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5 + 105*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^4 + 14
*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^2)*sinh(x)^6 + 4*(63*(a^5 + 3*a^4*b + 3*a^3*b^
2 + a^2*b^3)*cosh(x)^5 + 14*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^3 + 3*(a^5 + 7*a^4*
b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x))*sinh(x)^5 - a^5 - 3*a^4*b - 3*a^3*b^2 - a^2*b^3 - 2*(
a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^4 + 2*(105*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^
2*b^3)*cosh(x)^6 - a^5 - 7*a^4*b - 23*a^3*b^2 - 37*a^2*b^3 - 28*a*b^4 - 8*b^5 + 35*(3*a^5 + 17*a^4*b + 33*a^3*
b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^4 + 15*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x
)^2)*sinh(x)^4 + 8*(15*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^7 + 7*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27
*a^2*b^3 + 8*a*b^4)*cosh(x)^5 + 5*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^3 - (a^
5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x))*sinh(x)^3 - (3*a^5 + 17*a^4*b + 33*a^3*b^2
+ 27*a^2*b^3 + 8*a*b^4)*cosh(x)^2 + (45*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^8 + 28*(3*a^5 + 17*a^4*b
+ 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^6 - 3*a^5 - 17*a^4*b - 33*a^3*b^2 - 27*a^2*b^3 - 8*a*b^4 + 30*(a
^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^4 - 12*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2
*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^2 + 2*(5*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^9 + 4*(3*a^
5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^7 + 6*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*
a*b^4 + 8*b^5)*cosh(x)^5 - 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^3 - (3*a^5 +
17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x))*sinh(x))*sqrt(-a)*arctan(sqrt(2)*(b*cosh(x)^2 + 2*b*co
sh(x)*sinh(x) + b*sinh(x)^2 + a)*sqrt(-a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*si
nh(x) + sinh(x)^2))/(a*b*cosh(x)^4 + 4*a*b*cosh(x)*sinh(x)^3 + a*b*sinh(x)^4 - (a^2 + 3*a*b)*cosh(x)^2 + (6*a*
b*cosh(x)^2 - a^2 - 3*a*b)*sinh(x)^2 - a^2 + 2*(2*a*b*cosh(x)^3 - (a^2 + 3*a*b)*cosh(x))*sinh(x))) + 3*((a^5 +
3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^10 + 10*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)*sinh(x)^9 + (a^5
+ 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sinh(x)^10 + (3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^
8 + (3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4 + 45*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^2
)*sinh(x)^8 + 8*(15*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^3 + (3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*
b^3 + 8*a*b^4)*cosh(x))*sinh(x)^7 + 2*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^6 +
2*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5 + 105*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cos
h(x)^4 + 14*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^2)*sinh(x)^6 + 4*(63*(a^5 + 3*a^4*b
+ 3*a^3*b^2 + a^2*b^3)*cosh(x)^5 + 14*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^3 + 3*(a
^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x))*sinh(x)^5 - a^5 - 3*a^4*b - 3*a^3*b^2 - a^
2*b^3 - 2*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^4 + 2*(105*(a^5 + 3*a^4*b + 3*a
^3*b^2 + a^2*b^3)*cosh(x)^6 - a^5 - 7*a^4*b - 23*a^3*b^2 - 37*a^2*b^3 - 28*a*b^4 - 8*b^5 + 35*(3*a^5 + 17*a^4*
b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^4 + 15*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*
b^5)*cosh(x)^2)*sinh(x)^4 + 8*(15*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^7 + 7*(3*a^5 + 17*a^4*b + 33*a
^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^5 + 5*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh
(x)^3 - (a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x))*sinh(x)^3 - (3*a^5 + 17*a^4*b +
33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^2 + (45*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^8 + 28*(3*a^5
+ 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^6 - 3*a^5 - 17*a^4*b - 33*a^3*b^2 - 27*a^2*b^3 - 8*a*
b^4 + 30*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^4 - 12*(a^5 + 7*a^4*b + 23*a^3*b
^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^2 + 2*(5*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^
9 + 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x)^7 + 6*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^
2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^5 - 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*cosh(x)^3
- (3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*cosh(x))*sinh(x))*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*s
qrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(a*cosh(x)^2 + 2*a*cosh
(x)*sinh(x) + a*sinh(x)^2 + a)) + 2*sqrt(2)*((3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^8 + 8*(3*a^5 + 9*a^3*b^2
+ 4*a^2*b^3)*cosh(x)*sinh(x)^7 + (3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*sinh(x)^8 + 4*(3*a^5 + 6*a^4*b + 8*a^2*b^3 +
3*a*b^4)*cosh(x)^6 + 4*(3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4 + 7*(3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^2)*s
inh(x)^6 + 8*(7*(3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^3 + 3*(3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4)*cosh(x))
*sinh(x)^5 + 3*a^5 + 9*a^3*b^2 + 4*a^2*b^3 + 6*(3*a^5 + 8*a^4*b + 5*a^3*b^2 - 12*a^2*b^3 - 4*a*b^4)*cosh(x)^4
+ 2*(9*a^5 + 24*a^4*b + 15*a^3*b^2 - 36*a^2*b^3 - 12*a*b^4 + 35*(3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^4 + 30
*(3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4)*cosh(x)^2)*sinh(x)^4 + 8*(7*(3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^5
+ 10*(3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4)*cosh(x)^3 + 3*(3*a^5 + 8*a^4*b + 5*a^3*b^2 - 12*a^2*b^3 - 4*a*b^
4)*cosh(x))*sinh(x)^3 + 4*(3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4)*cosh(x)^2 + 4*(7*(3*a^5 + 9*a^3*b^2 + 4*a^2*
b^3)*cosh(x)^6 + 3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4 + 15*(3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4)*cosh(x)^4
+ 9*(3*a^5 + 8*a^4*b + 5*a^3*b^2 - 12*a^2*b^3 - 4*a*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((3*a^5 + 9*a^3*b^2 + 4*a^2*
b^3)*cosh(x)^7 + 3*(3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4)*cosh(x)^5 + 3*(3*a^5 + 8*a^4*b + 5*a^3*b^2 - 12*a^2
*b^3 - 4*a*b^4)*cosh(x)^3 + (3*a^5 + 6*a^4*b + 8*a^2*b^3 + 3*a*b^4)*cosh(x))*sinh(x))*sqrt((a*cosh(x)^2 + a*si
nh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x
)^10 + 10*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)*sinh(x)^9 + (a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*sinh
(x)^10 + (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(x)^8 + (3*a^8 + 17*a^7*b + 33*a^6*b^2 +
27*a^5*b^3 + 8*a^4*b^4 + 45*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)^2)*sinh(x)^8 - a^8 - 3*a^7*b - 3*a^
6*b^2 - a^5*b^3 + 8*(15*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)^3 + (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*
a^5*b^3 + 8*a^4*b^4)*cosh(x))*sinh(x)^7 + 2*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)
*cosh(x)^6 + 2*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5 + 105*(a^8 + 3*a^7*b + 3*a^6*
b^2 + a^5*b^3)*cosh(x)^4 + 14*(3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(x)^2)*sinh(x)^6 +
4*(63*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)^5 + 14*(3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4
*b^4)*cosh(x)^3 + 3*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh(x))*sinh(x)^5 - 2*
(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh(x)^4 - 2*(a^8 + 7*a^7*b + 23*a^6*b^2 +
37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5 - 105*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)^6 - 35*(3*a^8 + 17*a^
7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(x)^4 - 15*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^
4 + 8*a^3*b^5)*cosh(x)^2)*sinh(x)^4 + 8*(15*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)^7 + 7*(3*a^8 + 17*a^
7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(x)^5 + 5*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4
+ 8*a^3*b^5)*cosh(x)^3 - (a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh(x))*sinh(x)^
3 - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(x)^2 + (45*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*
b^3)*cosh(x)^8 - 3*a^8 - 17*a^7*b - 33*a^6*b^2 - 27*a^5*b^3 - 8*a^4*b^4 + 28*(3*a^8 + 17*a^7*b + 33*a^6*b^2 +
27*a^5*b^3 + 8*a^4*b^4)*cosh(x)^6 + 30*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh
(x)^4 - 12*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh(x)^2)*sinh(x)^2 + 2*(5*(a^8
+ 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cosh(x)^9 + 4*(3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*cosh(
x)^7 + 6*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh(x)^5 - 4*(a^8 + 7*a^7*b + 23*
a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*cosh(x)^3 - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4
*b^4)*cosh(x))*sinh(x))]
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^2/(a+b*sech(x)^2)^(5/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.51Error: Bad Argument Type
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maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{2}\relax (x )}{\left (a +b \mathrm {sech}\relax (x )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)^2/(a+b*sech(x)^2)^(5/2),x)
[Out]
int(coth(x)^2/(a+b*sech(x)^2)^(5/2),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth \relax (x)^{2}}{{\left (b \operatorname {sech}\relax (x)^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^2/(a+b*sech(x)^2)^(5/2),x, algorithm="maxima")
[Out]
integrate(coth(x)^2/(b*sech(x)^2 + a)^(5/2), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {coth}\relax (x)}^2}{{\left (a+\frac {b}{{\mathrm {cosh}\relax (x)}^2}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)^2/(a + b/cosh(x)^2)^(5/2),x)
[Out]
int(coth(x)^2/(a + b/cosh(x)^2)^(5/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{2}{\relax (x )}}{\left (a + b \operatorname {sech}^{2}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)**2/(a+b*sech(x)**2)**(5/2),x)
[Out]
Integral(coth(x)**2/(a + b*sech(x)**2)**(5/2), x)
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