3.57 \(\int \frac{\text{csch}^2(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\)
Optimal. Leaf size=215 \[ -\frac{3 b \left (8 a^2-12 a b+5 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{7/2} d (a-b)^{5/2}}-\frac{(4 a-5 b) (2 a-3 b) \coth (c+d x)}{8 a^3 d (a-b)^2}-\frac{b \coth (c+d x) \left (-(4 a-b) \tanh ^2(c+d x)+4 a-5 b\right )}{8 a^2 d (a-b)^2 \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac{b \text{csch}(c+d x) \text{sech}^3(c+d x)}{4 a d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )^2} \]
[Out]
(-3*b*(8*a^2 - 12*a*b + 5*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(7/2)*(a - b)^(5/2)*d) - ((4
*a - 5*b)*(2*a - 3*b)*Coth[c + d*x])/(8*a^3*(a - b)^2*d) - (b*Csch[c + d*x]*Sech[c + d*x]^3)/(4*a*(a - b)*d*(a
- (a - b)*Tanh[c + d*x]^2)^2) - (b*Coth[c + d*x]*(4*a - 5*b - (4*a - b)*Tanh[c + d*x]^2))/(8*a^2*(a - b)^2*d*
(a - (a - b)*Tanh[c + d*x]^2))
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Rubi [A] time = 0.287169, antiderivative size = 215, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used =
{3187, 468, 577, 453, 208} \[ -\frac{3 b \left (8 a^2-12 a b+5 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{7/2} d (a-b)^{5/2}}-\frac{(4 a-5 b) (2 a-3 b) \coth (c+d x)}{8 a^3 d (a-b)^2}-\frac{b \coth (c+d x) \left (-(4 a-b) \tanh ^2(c+d x)+4 a-5 b\right )}{8 a^2 d (a-b)^2 \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac{b \text{csch}(c+d x) \text{sech}^3(c+d x)}{4 a d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3,x]
[Out]
(-3*b*(8*a^2 - 12*a*b + 5*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(7/2)*(a - b)^(5/2)*d) - ((4
*a - 5*b)*(2*a - 3*b)*Coth[c + d*x])/(8*a^3*(a - b)^2*d) - (b*Csch[c + d*x]*Sech[c + d*x]^3)/(4*a*(a - b)*d*(a
- (a - b)*Tanh[c + d*x]^2)^2) - (b*Coth[c + d*x]*(4*a - 5*b - (4*a - b)*Tanh[c + d*x]^2))/(8*a^2*(a - b)^2*d*
(a - (a - b)*Tanh[c + d*x]^2))
Rule 3187
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + (a + b)*ff^2*x^2)^p)/(1 + ff^2*x^2)^(m/2 + p
+ 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
Rule 468
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[((c*b -
a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*b*e*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), I
nt[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(p
+ 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 577
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)/(a*b*g*n*(p + 1)), x] + Dist[
1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(m
+ 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] &&
IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
Rule 453
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{x^2 \left (a-(a-b) x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{b \text{csch}(c+d x) \text{sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right ) \left (4 a-5 b+(-4 a+b) x^2\right )}{x^2 \left (a+(-a+b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a (a-b) d}\\ &=-\frac{b \text{csch}(c+d x) \text{sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac{b \coth (c+d x) \left (4 a-5 b-(4 a-b) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{(4 a-5 b) (2 a-3 b)-(2 a-b) (4 a-b) x^2}{x^2 \left (a+(-a+b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a-b)^2 d}\\ &=-\frac{(4 a-5 b) (2 a-3 b) \coth (c+d x)}{8 a^3 (a-b)^2 d}-\frac{b \text{csch}(c+d x) \text{sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac{b \coth (c+d x) \left (4 a-5 b-(4 a-b) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac{\left (3 b \left (8 a^2-12 a b+5 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^3 (a-b)^2 d}\\ &=-\frac{3 b \left (8 a^2-12 a b+5 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{7/2} (a-b)^{5/2} d}-\frac{(4 a-5 b) (2 a-3 b) \coth (c+d x)}{8 a^3 (a-b)^2 d}-\frac{b \text{csch}(c+d x) \text{sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac{b \coth (c+d x) \left (4 a-5 b-(4 a-b) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 1.81292, size = 225, normalized size = 1.05 \[ \frac{\text{csch}^6(c+d x) (2 a+b \cosh (2 (c+d x))-b) \left (\frac{4 a^{3/2} b^2 \sinh (2 (c+d x))}{a-b}-\frac{3 b \left (8 a^2-12 a b+5 b^2\right ) (2 a+b \cosh (2 (c+d x))-b)^2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{(a-b)^{5/2}}+\frac{\sqrt{a} b^2 (10 a-7 b) \sinh (2 (c+d x)) (2 a+b \cosh (2 (c+d x))-b)}{(a-b)^2}-8 \sqrt{a} \coth (c+d x) (2 a+b \cosh (2 (c+d x))-b)^2\right )}{64 a^{7/2} d \left (a \text{csch}^2(c+d x)+b\right )^3} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3,x]
[Out]
((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^6*((-3*b*(8*a^2 - 12*a*b + 5*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c +
d*x])/Sqrt[a]]*(2*a - b + b*Cosh[2*(c + d*x)])^2)/(a - b)^(5/2) - 8*Sqrt[a]*(2*a - b + b*Cosh[2*(c + d*x)])^2
*Coth[c + d*x] + (4*a^(3/2)*b^2*Sinh[2*(c + d*x)])/(a - b) + (Sqrt[a]*(10*a - 7*b)*b^2*(2*a - b + b*Cosh[2*(c
+ d*x)])*Sinh[2*(c + d*x)])/(a - b)^2))/(64*a^(7/2)*d*(b + a*Csch[c + d*x]^2)^3)
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Maple [B] time = 0.089, size = 1850, normalized size = 8.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x)
[Out]
-1/2/d/a^3*tanh(1/2*d*x+1/2*c)-1/2/d/a^3/tanh(1/2*d*x+1/2*c)+3/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c
)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*b^2/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7-9/4/d/a^2*b^3/(tanh(1/2*d*x+1
/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7-3/d/(
tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a/(a^2-2*a*b+b^2)*tanh(1/2*d*
x+1/2*c)^5*b^2+49/4/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a^2*b^
3/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5-7/d/a^3*b^4/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(
1/2*d*x+1/2*c)^2*b+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5-3/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*
c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3*b^2+49/4/d/(tanh(1/2*d*x+1/2*c)^
4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a^2*b^3/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3-7/d
/a^3*b^4/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/(a^2-2*a*b+b^2)*tan
h(1/2*d*x+1/2*c)^3+3/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*b^2/a
/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)-9/4/d/a^2*b^3/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1
/2*d*x+1/2*c)^2*b+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)-3/d/(a^2-2*a*b+b^2)*b/a/((2*(-b*(a-b))^(1/2)+a-2*b)
*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+9/2/d/a^2/(a^2-2*a*b+b^2)/((2*(-
b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))*b^2-15/8/d/
a^3*b^3/(a^2-2*a*b+b^2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2
)+a-2*b)*a)^(1/2))+3/d/(a^2-2*a*b+b^2)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(
1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))*b^2-9/2/d/a^2/(a^2-2*a*b+b^2)/(-b*(a-b))^(1/2)/((2*(-b*(a
-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))*b^3+15/8/d/a^3*
b^4/(a^2-2*a*b+b^2)/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-
b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+3/d/(a^2-2*a*b+b^2)*b/a/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2
*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-9/2/d/a^2/(a^2-2*a*b+b^2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/
2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))*b^2+15/8/d/a^3*b^3/(a^2-2*a*b+b^2)/((2*(
-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+3/d/(a^2-2*
a*b+b^2)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(
1/2)-a+2*b)*a)^(1/2))*b^2-9/2/d/a^2/(a^2-2*a*b+b^2)/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arct
an(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))*b^3+15/8/d/a^3*b^4/(a^2-2*a*b+b^2)/(-b*(a-b))^(
1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 3.58571, size = 21173, normalized size = 98.48 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")
[Out]
[-1/16*(12*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^8 + 96*(8*a^4*b^2 - 20*a^3*b^3 + 17*a
^2*b^4 - 5*a*b^5)*cosh(d*x + c)*sinh(d*x + c)^7 + 12*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*sinh(d*x
+ c)^8 + 24*(24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x + c)^6 + 24*(24*a^5*b - 76*a
^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5 + 14*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)
^2)*sinh(d*x + c)^6 + 32*a^4*b^2 - 136*a^3*b^3 + 164*a^2*b^4 - 60*a*b^5 + 48*(14*(8*a^4*b^2 - 20*a^3*b^3 + 17*
a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^3 + 3*(24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x +
c))*sinh(d*x + c)^5 + 8*(64*a^6 - 296*a^5*b + 548*a^4*b^2 - 509*a^3*b^3 + 238*a^2*b^4 - 45*a*b^5)*cosh(d*x +
c)^4 + 8*(64*a^6 - 296*a^5*b + 548*a^4*b^2 - 509*a^3*b^3 + 238*a^2*b^4 - 45*a*b^5 + 105*(8*a^4*b^2 - 20*a^3*b^
3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^4 + 45*(24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*co
sh(d*x + c)^2)*sinh(d*x + c)^4 + 32*(21*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^5 + 15*(
24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x + c)^3 + (64*a^6 - 296*a^5*b + 548*a^4*b^
2 - 509*a^3*b^3 + 238*a^2*b^4 - 45*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 8*(32*a^5*b - 144*a^4*b^2 + 219*a^3
*b^3 - 137*a^2*b^4 + 30*a*b^5)*cosh(d*x + c)^2 + 8*(42*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*
x + c)^6 + 32*a^5*b - 144*a^4*b^2 + 219*a^3*b^3 - 137*a^2*b^4 + 30*a*b^5 + 45*(24*a^5*b - 76*a^4*b^2 + 91*a^3*
b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x + c)^4 + 6*(64*a^6 - 296*a^5*b + 548*a^4*b^2 - 509*a^3*b^3 + 238*a^2*b^4
- 45*a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*((8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^10 + 10*(8*a^2
*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)*sinh(d*x + c)^9 + (8*a^2*b^3 - 12*a*b^4 + 5*b^5)*sinh(d*x + c)^10 + (64
*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^8 + (64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5
+ 45*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*co
sh(d*x + c)^3 + (64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(64*a^4*b -
192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^6 + 2*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 1
20*a*b^4 + 25*b^5 + 105*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^4 + 14*(64*a^3*b^2 - 136*a^2*b^3 + 100*a*
b^4 - 25*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^5 + 14*(64
*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^3 + 3*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a
*b^4 + 25*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 - 8*a^2*b^3 + 12*a*b^4 - 5*b^5 - 2*(64*a^4*b - 192*a^3*b^2 + 224
*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^4 + 2*(105*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^6 - 64*a^
4*b + 192*a^3*b^2 - 224*a^2*b^3 + 120*a*b^4 - 25*b^5 + 35*(64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh
(d*x + c)^4 + 15*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4
+ 8*(15*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^7 + 7*(64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cos
h(d*x + c)^5 + 5*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^3 - (64*a^4*b - 192
*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 - (64*a^3*b^2 - 136*a^2*b^3 + 100*
a*b^4 - 25*b^5)*cosh(d*x + c)^2 + (45*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^8 + 28*(64*a^3*b^2 - 136*a^
2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^6 - 64*a^3*b^2 + 136*a^2*b^3 - 100*a*b^4 + 25*b^5 + 30*(64*a^4*b - 1
92*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^4 - 12*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 12
0*a*b^4 + 25*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^9 + 4*(
64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^7 + 6*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120
*a*b^4 + 25*b^5)*cosh(d*x + c)^5 - 4*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)
^3 - (64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 - a*b)*log((b^2*co
sh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 +
2*(3*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x + c)^3 + (2*a*
b - b^2)*cosh(d*x + c))*sinh(d*x + c) + 4*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x +
c)^2 + 2*a - b)*sqrt(a^2 - a*b))/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 +
2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a
- b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 16*(6*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^
7 + 9*(24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x + c)^5 + 2*(64*a^6 - 296*a^5*b + 5
48*a^4*b^2 - 509*a^3*b^3 + 238*a^2*b^4 - 45*a*b^5)*cosh(d*x + c)^3 + (32*a^5*b - 144*a^4*b^2 + 219*a^3*b^3 - 1
37*a^2*b^4 + 30*a*b^5)*cosh(d*x + c))*sinh(d*x + c))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x +
c)^10 + 10*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^7*b^2 - 3*a^6*b^3
+ 3*a^5*b^4 - a^4*b^5)*d*sinh(d*x + c)^10 + (8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d*co
sh(d*x + c)^8 + (45*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)^2 + (8*a^8*b - 29*a^7*b^2 + 39
*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d)*sinh(d*x + c)^8 + 2*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*a^5
*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^6 + 8*(15*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)^3 + (8
*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^7*b^2
- 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)^4 + 14*(8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*
a^4*b^5)*d*cosh(d*x + c)^2 + (8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d)*sinh(d*x
+ c)^6 - 2*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^4 + 4*(63*(a
^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)^5 + 14*(8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b
^4 + 5*a^4*b^5)*d*cosh(d*x + c)^3 + 3*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d*
cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)^6 + 35*(8*
a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c)^4 + 15*(8*a^9 - 36*a^8*b + 65*a^7*b^
2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^2 - (8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*
a^5*b^4 - 5*a^4*b^5)*d)*sinh(d*x + c)^4 - (8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d*cosh(
d*x + c)^2 + 8*(15*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)^7 + 7*(8*a^8*b - 29*a^7*b^2 + 3
9*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c)^5 + 5*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*a^5
*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^3 - (8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d*
cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)^8 + 28*(8*a^8
*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c)^6 + 30*(8*a^9 - 36*a^8*b + 65*a^7*b^2 -
59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^4 - 12*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*
a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^2 - (8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d)*sinh(
d*x + c)^2 - (a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d + 2*(5*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*
d*cosh(d*x + c)^9 + 4*(8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c)^7 + 6*(8*a^
9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^5 - 4*(8*a^9 - 36*a^8*b + 65*
a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^3 - (8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5
*b^4 + 5*a^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)), -1/8*(6*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cos
h(d*x + c)^8 + 48*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)*sinh(d*x + c)^7 + 6*(8*a^4*b^2
- 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*sinh(d*x + c)^8 + 12*(24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 +
10*a*b^5)*cosh(d*x + c)^6 + 12*(24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5 + 14*(8*a^4*b^2 - 2
0*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 16*a^4*b^2 - 68*a^3*b^3 + 82*a^2*b^4 - 30
*a*b^5 + 24*(14*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^3 + 3*(24*a^5*b - 76*a^4*b^2 + 9
1*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 + 4*(64*a^6 - 296*a^5*b + 548*a^4*b^2 - 509*
a^3*b^3 + 238*a^2*b^4 - 45*a*b^5)*cosh(d*x + c)^4 + 4*(64*a^6 - 296*a^5*b + 548*a^4*b^2 - 509*a^3*b^3 + 238*a^
2*b^4 - 45*a*b^5 + 105*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^4 + 45*(24*a^5*b - 76*a^4
*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(21*(8*a^4*b^2 - 20*a^3*b^3 +
17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^5 + 15*(24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(
d*x + c)^3 + (64*a^6 - 296*a^5*b + 548*a^4*b^2 - 509*a^3*b^3 + 238*a^2*b^4 - 45*a*b^5)*cosh(d*x + c))*sinh(d*x
+ c)^3 + 4*(32*a^5*b - 144*a^4*b^2 + 219*a^3*b^3 - 137*a^2*b^4 + 30*a*b^5)*cosh(d*x + c)^2 + 4*(42*(8*a^4*b^2
- 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^6 + 32*a^5*b - 144*a^4*b^2 + 219*a^3*b^3 - 137*a^2*b^4 + 3
0*a*b^5 + 45*(24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x + c)^4 + 6*(64*a^6 - 296*a^
5*b + 548*a^4*b^2 - 509*a^3*b^3 + 238*a^2*b^4 - 45*a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*((8*a^2*b^3 - 1
2*a*b^4 + 5*b^5)*cosh(d*x + c)^10 + 10*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)*sinh(d*x + c)^9 + (8*a^2*b
^3 - 12*a*b^4 + 5*b^5)*sinh(d*x + c)^10 + (64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^8 + (6
4*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5 + 45*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^2)*sinh(d*x + c
)^8 + 8*(15*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^3 + (64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*c
osh(d*x + c))*sinh(d*x + c)^7 + 2*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^6
+ 2*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5 + 105*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x +
c)^4 + 14*(64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(8*a^2*b^3
- 12*a*b^4 + 5*b^5)*cosh(d*x + c)^5 + 14*(64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^3 + 3*
(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 - 8*a^2*b^3 + 12*a*
b^4 - 5*b^5 - 2*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^4 + 2*(105*(8*a^2*b^
3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^6 - 64*a^4*b + 192*a^3*b^2 - 224*a^2*b^3 + 120*a*b^4 - 25*b^5 + 35*(64*a^3
*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^4 + 15*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^
4 + 25*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(15*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^7 + 7*(64*a^
3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^5 + 5*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^
4 + 25*b^5)*cosh(d*x + c)^3 - (64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c))*sinh(
d*x + c)^3 - (64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^2 + (45*(8*a^2*b^3 - 12*a*b^4 + 5*b
^5)*cosh(d*x + c)^8 + 28*(64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^6 - 64*a^3*b^2 + 136*a^
2*b^3 - 100*a*b^4 + 25*b^5 + 30*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^4 -
12*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(8*a^2*
b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^9 + 4*(64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^7 +
6*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^5 - 4*(64*a^4*b - 192*a^3*b^2 + 22
4*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^3 - (64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c
))*sinh(d*x + c))*sqrt(-a^2 + a*b)*arctan(-1/2*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d
*x + c)^2 + 2*a - b)*sqrt(-a^2 + a*b)/(a^2 - a*b)) + 8*(6*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh
(d*x + c)^7 + 9*(24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x + c)^5 + 2*(64*a^6 - 296
*a^5*b + 548*a^4*b^2 - 509*a^3*b^3 + 238*a^2*b^4 - 45*a*b^5)*cosh(d*x + c)^3 + (32*a^5*b - 144*a^4*b^2 + 219*a
^3*b^3 - 137*a^2*b^4 + 30*a*b^5)*cosh(d*x + c))*sinh(d*x + c))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*
cosh(d*x + c)^10 + 10*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^7*b^2 -
3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*sinh(d*x + c)^10 + (8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4
*b^5)*d*cosh(d*x + c)^8 + (45*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)^2 + (8*a^8*b - 29*a^
7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d)*sinh(d*x + c)^8 + 2*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^
3 + 27*a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^6 + 8*(15*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x +
c)^3 + (8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105
*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)^4 + 14*(8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^
5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c)^2 + (8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d
)*sinh(d*x + c)^6 - 2*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^4
+ 4*(63*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)^5 + 14*(8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3
- 23*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c)^3 + 3*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a
^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)^
6 + 35*(8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c)^4 + 15*(8*a^9 - 36*a^8*b +
65*a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^2 - (8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6
*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d)*sinh(d*x + c)^4 - (8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^
5)*d*cosh(d*x + c)^2 + 8*(15*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)^7 + 7*(8*a^8*b - 29*a
^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c)^5 + 5*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^
3 + 27*a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^3 - (8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a
^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d*cosh(d*x + c)^8 +
28*(8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c)^6 + 30*(8*a^9 - 36*a^8*b + 65
*a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^4 - 12*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6
*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^2 - (8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5
)*d)*sinh(d*x + c)^2 - (a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*d + 2*(5*(a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 -
a^4*b^5)*d*cosh(d*x + c)^9 + 4*(8*a^8*b - 29*a^7*b^2 + 39*a^6*b^3 - 23*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c)^7
+ 6*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^5 - 4*(8*a^9 - 36*a
^8*b + 65*a^7*b^2 - 59*a^6*b^3 + 27*a^5*b^4 - 5*a^4*b^5)*d*cosh(d*x + c)^3 - (8*a^8*b - 29*a^7*b^2 + 39*a^6*b^
3 - 23*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c))]
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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(csch(d*x+c)**2/(a+b*sinh(d*x+c)**2)**3,x)
[Out]
Timed out
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Giac [A] time = 1.5999, size = 455, normalized size = 2.12 \begin{align*} -\frac{3 \,{\left (8 \, a^{2} b - 12 \, a b^{2} + 5 \, b^{3}\right )} \arctan \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt{-a^{2} + a b}}\right )}{8 \,{\left (a^{5} d - 2 \, a^{4} b d + a^{3} b^{2} d\right )} \sqrt{-a^{2} + a b}} - \frac{16 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 20 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 7 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 80 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} - 136 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 86 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 21 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 64 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 76 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 21 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 10 \, a b^{3} - 7 \, b^{4}}{4 \,{\left (a^{5} d - 2 \, a^{4} b d + a^{3} b^{2} d\right )}{\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}^{2}} - \frac{2}{a^{3} d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")
[Out]
-3/8*(8*a^2*b - 12*a*b^2 + 5*b^3)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^5*d - 2*a^4*b
*d + a^3*b^2*d)*sqrt(-a^2 + a*b)) - 1/4*(16*a^2*b^2*e^(6*d*x + 6*c) - 20*a*b^3*e^(6*d*x + 6*c) + 7*b^4*e^(6*d*
x + 6*c) + 80*a^3*b*e^(4*d*x + 4*c) - 136*a^2*b^2*e^(4*d*x + 4*c) + 86*a*b^3*e^(4*d*x + 4*c) - 21*b^4*e^(4*d*x
+ 4*c) + 64*a^2*b^2*e^(2*d*x + 2*c) - 76*a*b^3*e^(2*d*x + 2*c) + 21*b^4*e^(2*d*x + 2*c) + 10*a*b^3 - 7*b^4)/(
(a^5*d - 2*a^4*b*d + a^3*b^2*d)*(b*e^(4*d*x + 4*c) + 4*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + b)^2) - 2/(a^
3*d*(e^(2*d*x + 2*c) - 1))