3.346 \(\int \frac{\text{sech}^2(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\)

Optimal. Leaf size=172 \[ \frac{3 b^2 (4 a-b) \tanh (c+d x)}{8 a^2 d (a-b)^3 \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac{3 b \left (8 a^2-4 a b+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} d (a-b)^{7/2}}-\frac{b^3 \tanh (c+d x)}{4 a d (a-b)^3 \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac{\tanh (c+d x)}{d (a-b)^3} \]

[Out]

(-3*b*(8*a^2 - 4*a*b + b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^(7/2)*d) + Tanh[c
+ d*x]/((a - b)^3*d) - (b^3*Tanh[c + d*x])/(4*a*(a - b)^3*d*(a - (a - b)*Tanh[c + d*x]^2)^2) + (3*(4*a - b)*b
^2*Tanh[c + d*x])/(8*a^2*(a - b)^3*d*(a - (a - b)*Tanh[c + d*x]^2))

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Rubi [A]  time = 0.264594, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3191, 390, 1157, 385, 208} \[ \frac{3 b^2 (4 a-b) \tanh (c+d x)}{8 a^2 d (a-b)^3 \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac{3 b \left (8 a^2-4 a b+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} d (a-b)^{7/2}}-\frac{b^3 \tanh (c+d x)}{4 a d (a-b)^3 \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac{\tanh (c+d x)}{d (a-b)^3} \]

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(-3*b*(8*a^2 - 4*a*b + b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^(7/2)*d) + Tanh[c
+ d*x]/((a - b)^3*d) - (b^3*Tanh[c + d*x])/(4*a*(a - b)^3*d*(a - (a - b)*Tanh[c + d*x]^2)^2) + (3*(4*a - b)*b
^2*Tanh[c + d*x])/(8*a^2*(a - b)^3*d*(a - (a - b)*Tanh[c + d*x]^2))

Rule 3191

Int[cos[(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/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

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{sech}^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}{\left (a-(a-b) x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{(a-b)^3}-\frac{b \left (3 a^2-3 a b+b^2\right )-3 (a-b) (2 a-b) b x^2+3 (a-b)^2 b x^4}{(a-b)^3 \left (a+(-a+b) x^2\right )^3}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\tanh (c+d x)}{(a-b)^3 d}-\frac{\operatorname{Subst}\left (\int \frac{b \left (3 a^2-3 a b+b^2\right )-3 (a-b) (2 a-b) b x^2+3 (a-b)^2 b x^4}{\left (a+(-a+b) x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{(a-b)^3 d}\\ &=\frac{\tanh (c+d x)}{(a-b)^3 d}-\frac{b^3 \tanh (c+d x)}{4 a (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{-3 (2 a-b)^2 b+12 a (a-b) b x^2}{\left (a+(-a+b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a (a-b)^3 d}\\ &=\frac{\tanh (c+d x)}{(a-b)^3 d}-\frac{b^3 \tanh (c+d x)}{4 a (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac{3 (4 a-b) b^2 \tanh (c+d x)}{8 a^2 (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac{\left (3 b \left (8 a^2-4 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a-b)^3 d}\\ &=-\frac{3 b \left (8 a^2-4 a b+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} (a-b)^{7/2} d}+\frac{\tanh (c+d x)}{(a-b)^3 d}-\frac{b^3 \tanh (c+d x)}{4 a (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac{3 (4 a-b) b^2 \tanh (c+d x)}{8 a^2 (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.07899, size = 165, normalized size = 0.96 \[ \frac{-\frac{3 b \left (8 a^2-4 a b+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{5/2} (a-b)^{7/2}}+\frac{b^2 (10 a-3 b) \sinh (2 (c+d x))}{a^2 (a-b)^3 (2 a+b \cosh (2 (c+d x))-b)}+\frac{4 b^2 \sinh (2 (c+d x))}{a (a-b)^2 (2 a+b \cosh (2 (c+d x))-b)^2}+\frac{8 \tanh (c+d x)}{(a-b)^3}}{8 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

((-3*b*(8*a^2 - 4*a*b + b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*(a - b)^(7/2)) + (4*b^2*Si
nh[2*(c + d*x)])/(a*(a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])^2) + ((10*a - 3*b)*b^2*Sinh[2*(c + d*x)])/(a^2*(
a - b)^3*(2*a - b + b*Cosh[2*(c + d*x)])) + (8*Tanh[c + d*x])/(a - b)^3)/(8*d)

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Maple [B]  time = 0.084, size = 1694, normalized size = 9.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x)

[Out]

-5/4/d*b^3/(a-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*tanh(1/
2*d*x+1/2*c)^7+3/d*b^2/(a-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*tanh(1/2*d*x+1/2*c)^7-3/d*b^2/(a-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*tanh(1/2*d*x+1/2*c)^5+45/4/d*b^3/(a-b)^3/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tan
h(1/2*d*x+1/2*c)^2*b+a)^2/a*tanh(1/2*d*x+1/2*c)^5-3/d*b^4/(a-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*tanh(1/2*d*x+1/2*c)^5-3/d*b^2/(a-b)^3/(tanh(1/2*d*x+1/2*c)^4*a-2*tan
h(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)^3+45/4/d*b^3/(a-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*tanh(1/2*d*x+1/2*c)^3-3/d*b^4/(a-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*tanh(1/2*d*x+1/2*c)^3-5/4/d*b
^3/(a-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*tanh(1/2*d*x+1/
2*c)+3/d*b^2/(a-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*tanh(1/
2*d*x+1/2*c)-3/d*b/(a-b)^3/((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))+3/2/d*b^2/(a-b)^3/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))-3/8/d*b^3/(a-b)^3/a^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*b^2/(a-b)^3/(-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))-3/2/d*b^3/(a-b)^3/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))+3/8/d*b^4/(a-b)^3/a^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*b/(a-b)^3/((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))-3/2/d*b^2/(a-b)^3/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))+3/8/d*b^3/(a-b)^3/a^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*b^2/(a-b)^3/(-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))-3/2/d*b
^3/(a-b)^3/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))+3/8/d*b^4/(a-b)^3/a^2/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*t
anh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+2/d/(a-b)^3*tanh(1/2*d*x+1/2*c)/(tanh(1/2*d*x+1/2*c)^
2+1)

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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(sech(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 = 2.84346, size = 21442, normalized size = 124.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(12*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*x + c)^8 + 96*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b
^4 - a*b^5)*cosh(d*x + c)*sinh(d*x + c)^7 + 12*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*sinh(d*x + c)^8 +
24*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x + c)^6 + 24*(24*a^5*b - 44*a^4*b^2 + 27*a
^3*b^3 - 8*a^2*b^4 + a*b^5 + 14*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^6
+ 32*a^4*b^2 + 8*a^3*b^3 - 52*a^2*b^4 + 12*a*b^5 + 48*(14*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*
x + c)^3 + 3*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 + 8*(64*a
^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*b^3 - a^2*b^4)*cosh(d*x + c)^4 + 8*(64*a^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*
b^3 - a^2*b^4 + 105*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*x + c)^4 + 45*(24*a^5*b - 44*a^4*b^2 +
27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 32*(21*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4
- a*b^5)*cosh(d*x + c)^5 + 15*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x + c)^3 + (64*
a^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*b^3 - a^2*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 8*(32*a^5*b - 16*a^4*b^2 -
37*a^3*b^3 + 24*a^2*b^4 - 3*a*b^5)*cosh(d*x + c)^2 + 8*(42*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(
d*x + c)^6 + 32*a^5*b - 16*a^4*b^2 - 37*a^3*b^3 + 24*a^2*b^4 - 3*a*b^5 + 45*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^
3 - 8*a^2*b^4 + a*b^5)*cosh(d*x + c)^4 + 6*(64*a^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*b^3 - a^2*b^4)*cosh(d*x + c
)^2)*sinh(d*x + c)^2 + 3*((8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^10 + 10*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d
*x + c)*sinh(d*x + c)^9 + (8*a^2*b^3 - 4*a*b^4 + b^5)*sinh(d*x + c)^10 + (64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 -
3*b^5)*cosh(d*x + c)^8 + (64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5 + 45*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*
x + c)^2)*sinh(d*x + c)^8 + 8*(15*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^3 + (64*a^3*b^2 - 56*a^2*b^3 + 20*
a*b^4 - 3*b^5)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*
x + c)^6 + 2*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5 + 105*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x +
c)^4 + 14*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(8*a^2*b^3 - 4
*a*b^4 + b^5)*cosh(d*x + c)^5 + 14*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c)^3 + 3*(64*a^4*b
- 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c))*sinh(d*x + c)^5 + 8*a^2*b^3 - 4*a*b^4 + b^5 + 2*(64*
a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^4 + 2*(105*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x
+ c)^6 + 64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5 + 35*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)
*cosh(d*x + c)^4 + 15*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 +
8*(15*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^7 + 7*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x +
c)^5 + 5*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^3 + (64*a^4*b - 64*a^3*b^2 + 32*a^
2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + (64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x
+ c)^2 + (45*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^8 + 28*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cos
h(d*x + c)^6 + 64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5 + 30*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 +
b^5)*cosh(d*x + c)^4 + 12*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^2)*sinh(d*x + c)
^2 + 2*(5*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^9 + 4*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*
x + c)^7 + 6*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^5 + 4*(64*a^4*b - 64*a^3*b^2 +
32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^3 + (64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c))*sin
h(d*x + c))*sqrt(a^2 - a*b)*log((b^2*cosh(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 - 12*a^3*b^3
+ 5*a^2*b^4 - a*b^5)*cosh(d*x + c)^7 + 9*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x +
c)^5 + 2*(64*a^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*b^3 - a^2*b^4)*cosh(d*x + c)^3 + (32*a^5*b - 16*a^4*b^2 - 37*
a^3*b^3 + 24*a^2*b^4 - 3*a*b^5)*cosh(d*x + c))*sinh(d*x + c))/((a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 +
a^3*b^6)*d*cosh(d*x + c)^10 + 10*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)*sinh(
d*x + c)^9 + (a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*sinh(d*x + c)^10 + (8*a^8*b - 35*a^7*b^
2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^8 + (45*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4
- 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^2 + (8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a
^3*b^6)*d)*sinh(d*x + c)^8 + 2*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)
*d*cosh(d*x + c)^6 + 8*(15*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^3 + (8*a^8*
b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(
a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^4 + 14*(8*a^8*b - 35*a^7*b^2 + 60*a^6*b
^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^2 + (8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30
*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d)*sinh(d*x + c)^6 + 2*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*a^5*b^
4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^4 + 4*(63*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*c
osh(d*x + c)^5 + 14*(8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^
3 + 3*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c))*sinh(d*
x + c)^5 + 2*(105*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^6 + 35*(8*a^8*b - 35
*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^4 + 15*(8*a^9 - 36*a^8*b + 65*a^7
*b^2 - 60*a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^2 + (8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*
a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d)*sinh(d*x + c)^4 + (8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b
^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^2 + 8*(15*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)
*d*cosh(d*x + c)^7 + 7*(8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x +
c)^5 + 5*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^3 + (
8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^
3 + (45*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^8 + 28*(8*a^8*b - 35*a^7*b^2 +
60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^6 + 30*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*
a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^4 + 12*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3
+ 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^2 + (8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*
a^4*b^5 - 3*a^3*b^6)*d)*sinh(d*x + c)^2 + (a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d + 2*(5*(a^
7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^9 + 4*(8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3
- 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^7 + 6*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*
a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^5 + 4*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*a^5*b^4
- 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^3 + (8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^
3*b^6)*d*cosh(d*x + c))*sinh(d*x + c)), -1/8*(6*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*x + c)^8 +
48*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*x + c)*sinh(d*x + c)^7 + 6*(8*a^4*b^2 - 12*a^3*b^3 + 5
*a^2*b^4 - a*b^5)*sinh(d*x + c)^8 + 12*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x + c)^
6 + 12*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5 + 14*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^
5)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 16*a^4*b^2 + 4*a^3*b^3 - 26*a^2*b^4 + 6*a*b^5 + 24*(14*(8*a^4*b^2 - 12*a
^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*x + c)^3 + 3*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(
d*x + c))*sinh(d*x + c)^5 + 4*(64*a^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*b^3 - a^2*b^4)*cosh(d*x + c)^4 + 4*(64*a
^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*b^3 - a^2*b^4 + 105*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*x +
c)^4 + 45*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(21*
(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*x + c)^5 + 15*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*
b^4 + a*b^5)*cosh(d*x + c)^3 + (64*a^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*b^3 - a^2*b^4)*cosh(d*x + c))*sinh(d*x
+ c)^3 + 4*(32*a^5*b - 16*a^4*b^2 - 37*a^3*b^3 + 24*a^2*b^4 - 3*a*b^5)*cosh(d*x + c)^2 + 4*(42*(8*a^4*b^2 - 12
*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*x + c)^6 + 32*a^5*b - 16*a^4*b^2 - 37*a^3*b^3 + 24*a^2*b^4 - 3*a*b^5 + 45
*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x + c)^4 + 6*(64*a^6 - 88*a^5*b + 28*a^4*b^2
- 3*a^3*b^3 - a^2*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*((8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^10 + 10
*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)*sinh(d*x + c)^9 + (8*a^2*b^3 - 4*a*b^4 + b^5)*sinh(d*x + c)^10 + (6
4*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c)^8 + (64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5 + 45*
(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)
^3 + (64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(64*a^4*b - 64*a^3*b^2 +
32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^6 + 2*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5 + 105*(8*a
^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^4 + 14*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c)^2)*sin
h(d*x + c)^6 + 4*(63*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^5 + 14*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*
b^5)*cosh(d*x + c)^3 + 3*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c))*sinh(d*x + c)^5 +
8*a^2*b^3 - 4*a*b^4 + b^5 + 2*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^4 + 2*(105*(
8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^6 + 64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5 + 35*(64*a^3*b
^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c)^4 + 15*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*
cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(15*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^7 + 7*(64*a^3*b^2 - 56*a^2*
b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c)^5 + 5*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)
^3 + (64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + (64*a^3*b^2 - 56*a^
2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c)^2 + (45*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^8 + 28*(64*a^3*b^2 -
56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c)^6 + 64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5 + 30*(64*a^4*b -
64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^4 + 12*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 +
b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^9 + 4*(64*a^3*b^2 - 56*
a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c)^7 + 6*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x
+ c)^5 + 4*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^3 + (64*a^3*b^2 - 56*a^2*b^3 + 2
0*a*b^4 - 3*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 - 12*a^3*b^3
+ 5*a^2*b^4 - a*b^5)*cosh(d*x + c)^7 + 9*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x +
c)^5 + 2*(64*a^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*b^3 - a^2*b^4)*cosh(d*x + c)^3 + (32*a^5*b - 16*a^4*b^2 - 37*
a^3*b^3 + 24*a^2*b^4 - 3*a*b^5)*cosh(d*x + c))*sinh(d*x + c))/((a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 +
a^3*b^6)*d*cosh(d*x + c)^10 + 10*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)*sinh(
d*x + c)^9 + (a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*sinh(d*x + c)^10 + (8*a^8*b - 35*a^7*b^
2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^8 + (45*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4
- 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^2 + (8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a
^3*b^6)*d)*sinh(d*x + c)^8 + 2*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)
*d*cosh(d*x + c)^6 + 8*(15*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^3 + (8*a^8*
b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(
a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^4 + 14*(8*a^8*b - 35*a^7*b^2 + 60*a^6*b
^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^2 + (8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30
*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d)*sinh(d*x + c)^6 + 2*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*a^5*b^
4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^4 + 4*(63*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*c
osh(d*x + c)^5 + 14*(8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^
3 + 3*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c))*sinh(d*
x + c)^5 + 2*(105*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^6 + 35*(8*a^8*b - 35
*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^4 + 15*(8*a^9 - 36*a^8*b + 65*a^7
*b^2 - 60*a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^2 + (8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*
a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d)*sinh(d*x + c)^4 + (8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b
^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^2 + 8*(15*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)
*d*cosh(d*x + c)^7 + 7*(8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x +
c)^5 + 5*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^3 + (
8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^
3 + (45*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^8 + 28*(8*a^8*b - 35*a^7*b^2 +
60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^6 + 30*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*
a^6*b^3 + 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^4 + 12*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3
+ 30*a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^2 + (8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*
a^4*b^5 - 3*a^3*b^6)*d)*sinh(d*x + c)^2 + (a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d + 2*(5*(a^
7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^9 + 4*(8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3
- 50*a^5*b^4 + 20*a^4*b^5 - 3*a^3*b^6)*d*cosh(d*x + c)^7 + 6*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*
a^5*b^4 - 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^5 + 4*(8*a^9 - 36*a^8*b + 65*a^7*b^2 - 60*a^6*b^3 + 30*a^5*b^4
- 8*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)^3 + (8*a^8*b - 35*a^7*b^2 + 60*a^6*b^3 - 50*a^5*b^4 + 20*a^4*b^5 - 3*a^
3*b^6)*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(sech(d*x+c)**2/(a+b*sinh(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.398, size = 509, normalized size = 2.96 \begin{align*} -\frac{3 \,{\left (8 \, a^{2} b - 4 \, a b^{2} + 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 - 3 \, a^{4} b d + 3 \, a^{3} b^{2} d - a^{2} b^{3} d\right )} \sqrt{-a^{2} + a b}} - \frac{16 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 12 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 80 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} - 104 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 54 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 9 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 64 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 52 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 10 \, a b^{3} - 3 \, b^{4}}{4 \,{\left (a^{5} d - 3 \, a^{4} b d + 3 \, a^{3} b^{2} d - a^{2} b^{3} 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}{{\left (a^{3} d - 3 \, a^{2} b d + 3 \, a b^{2} d - b^{3} d\right )}{\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(sech(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-3/8*(8*a^2*b - 4*a*b^2 + b^3)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^5*d - 3*a^4*b*d
+ 3*a^3*b^2*d - a^2*b^3*d)*sqrt(-a^2 + a*b)) - 1/4*(16*a^2*b^2*e^(6*d*x + 6*c) - 12*a*b^3*e^(6*d*x + 6*c) + 3*
b^4*e^(6*d*x + 6*c) + 80*a^3*b*e^(4*d*x + 4*c) - 104*a^2*b^2*e^(4*d*x + 4*c) + 54*a*b^3*e^(4*d*x + 4*c) - 9*b^
4*e^(4*d*x + 4*c) + 64*a^2*b^2*e^(2*d*x + 2*c) - 52*a*b^3*e^(2*d*x + 2*c) + 9*b^4*e^(2*d*x + 2*c) + 10*a*b^3 -
3*b^4)/((a^5*d - 3*a^4*b*d + 3*a^3*b^2*d - a^2*b^3*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 - 3*a^2*b*d + 3*a*b^2*d - b^3*d)*(e^(2*d*x + 2*c) + 1))