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

Optimal. Leaf size=204 \[ \frac{b^{3/2} \left (35 a^2+56 a b+24 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^4 d (a+b)^{5/2}}+\frac{b (4 a+3 b) (a+4 b) \tanh (c+d x)}{8 a^3 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{b (2 a+3 b) \tanh (c+d x)}{4 a^2 d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}+\frac{x (a-6 b)}{2 a^4}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 a d \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

[Out]

((a - 6*b)*x)/(2*a^4) + (b^(3/2)*(35*a^2 + 56*a*b + 24*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a
^4*(a + b)^(5/2)*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*a*d*(a + b - b*Tanh[c + d*x]^2)^2) + (b*(2*a + 3*b)*Tan
h[c + d*x])/(4*a^2*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) + (b*(4*a + 3*b)*(a + 4*b)*Tanh[c + d*x])/(8*a^3*(
a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))

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Rubi [A]  time = 0.377296, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {4146, 414, 527, 522, 206, 208} \[ \frac{b^{3/2} \left (35 a^2+56 a b+24 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^4 d (a+b)^{5/2}}+\frac{b (4 a+3 b) (a+4 b) \tanh (c+d x)}{8 a^3 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{b (2 a+3 b) \tanh (c+d x)}{4 a^2 d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}+\frac{x (a-6 b)}{2 a^4}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 a d \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a - 6*b)*x)/(2*a^4) + (b^(3/2)*(35*a^2 + 56*a*b + 24*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a
^4*(a + b)^(5/2)*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*a*d*(a + b - b*Tanh[c + d*x]^2)^2) + (b*(2*a + 3*b)*Tan
h[c + d*x])/(4*a^2*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) + (b*(4*a + 3*b)*(a + 4*b)*Tanh[c + d*x])/(8*a^3*(
a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))

Rule 4146

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fre
eFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/
2), x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, 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 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{\cosh ^2(c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2 \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\cosh (c+d x) \sinh (c+d x)}{2 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{a-b-5 b x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{2 a d}\\ &=\frac{\cosh (c+d x) \sinh (c+d x)}{2 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac{b (2 a+3 b) \tanh (c+d x)}{4 a^2 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-2 \left (2 a^2-4 a b-3 b^2\right )+6 b (2 a+3 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a+b) d}\\ &=\frac{\cosh (c+d x) \sinh (c+d x)}{2 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac{b (2 a+3 b) \tanh (c+d x)}{4 a^2 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac{b (4 a+3 b) (a+4 b) \tanh (c+d x)}{8 a^3 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{2 \left (4 a^3-12 a^2 b-25 a b^2-12 b^3\right )-2 b (4 a+3 b) (a+4 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{16 a^3 (a+b)^2 d}\\ &=\frac{\cosh (c+d x) \sinh (c+d x)}{2 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac{b (2 a+3 b) \tanh (c+d x)}{4 a^2 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac{b (4 a+3 b) (a+4 b) \tanh (c+d x)}{8 a^3 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac{(a-6 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 a^4 d}+\frac{\left (b^2 \left (35 a^2+56 a b+24 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^4 (a+b)^2 d}\\ &=\frac{(a-6 b) x}{2 a^4}+\frac{b^{3/2} \left (35 a^2+56 a b+24 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^4 (a+b)^{5/2} d}+\frac{\cosh (c+d x) \sinh (c+d x)}{2 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac{b (2 a+3 b) \tanh (c+d x)}{4 a^2 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac{b (4 a+3 b) (a+4 b) \tanh (c+d x)}{8 a^3 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 4.0039, size = 156, normalized size = 0.76 \[ \frac{\frac{b^{3/2} \left (35 a^2+56 a b+24 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{(a+b)^{5/2}}+a \sinh (2 (c+d x)) \left (\frac{2 b^3 (5 a \cosh (2 (c+d x))+3 a+8 b)}{(a+b)^2 (a \cosh (2 (c+d x))+a+2 b)^2}+\frac{13 a b^2}{(a+b)^2 (a \cosh (2 (c+d x))+a+2 b)}+2\right )+4 (a-6 b) (c+d x)}{8 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(a - 6*b)*(c + d*x) + (b^(3/2)*(35*a^2 + 56*a*b + 24*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a
+ b)^(5/2) + a*(2 + (13*a*b^2)/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])) + (2*b^3*(3*a + 8*b + 5*a*Cosh[2*(c
+ d*x)]))/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2))*Sinh[2*(c + d*x)])/(8*a^4*d)

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Maple [B]  time = 0.128, size = 1435, normalized size = 7. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/d/a^3/(tanh(1/2*d*x+1/2*c)+1)^2+1/2/d/a^3/(tanh(1/2*d*x+1/2*c)+1)+1/2/d/a^3*ln(tanh(1/2*d*x+1/2*c)+1)-3/d
/a^4*ln(tanh(1/2*d*x+1/2*c)+1)*b+13/4/d*b^2/a^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*
x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)*tanh(1/2*d*x+1/2*c)^7+2/d*b^3/a^3/(tanh(1/2*d*x+1/2*c)^4*a
+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)*tanh(1/2*d*x+1/2*c)^
7+39/4/d*b^2/a/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c
)^2*b+a+b)^2/(a+b)^2*tanh(1/2*d*x+1/2*c)^5+19/4/d*b^3/a^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*t
anh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)^2*tanh(1/2*d*x+1/2*c)^5-2/d*b^4/a^3/(tanh(1/2*d*
x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)^2*tanh(1
/2*d*x+1/2*c)^5+39/4/d*b^2/a/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh
(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)^2*tanh(1/2*d*x+1/2*c)^3+19/4/d*b^3/a^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*
x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)^2*tanh(1/2*d*x+1/2*c)^3-2/d*b^4/a^
3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/
(a+b)^2*tanh(1/2*d*x+1/2*c)^3+13/4/d*b^2/a^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1
/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)*tanh(1/2*d*x+1/2*c)+2/d*b^3/a^3/(tanh(1/2*d*x+1/2*c)^4*a+b*ta
nh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)*tanh(1/2*d*x+1/2*c)+35/16
/d*b^(3/2)/a^2/(a^2+2*a*b+b^2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+
(a+b)^(1/2))+7/2/d*b^(5/2)/a^3/(a^2+2*a*b+b^2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x
+1/2*c)*b^(1/2)+(a+b)^(1/2))+3/2/d*b^(7/2)/a^4/(a^2+2*a*b+b^2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^
2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))-35/16/d*b^(3/2)/a^2/(a^2+2*a*b+b^2)/(a+b)^(1/2)*ln(-(a+b)^(1/2)*t
anh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)-(a+b)^(1/2))-7/2/d*b^(5/2)/a^3/(a^2+2*a*b+b^2)/(a+b)^(1/2)*
ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)-(a+b)^(1/2))-3/2/d*b^(7/2)/a^4/(a^2+2*a*b+
b^2)/(a+b)^(1/2)*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)-(a+b)^(1/2))+1/2/d/a^3/(t
anh(1/2*d*x+1/2*c)-1)^2+1/2/d/a^3/(tanh(1/2*d*x+1/2*c)-1)-1/2/d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+3/d/a^4*ln(tanh(
1/2*d*x+1/2*c)-1)*b

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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(cosh(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.76583, size = 27738, normalized size = 135.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(2*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^12 + 24*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)*sinh(d*x + c)
^11 + 2*(a^5 + 2*a^4*b + a^3*b^2)*sinh(d*x + c)^12 + 8*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b
- 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^10 + 4*(2*a^5 + 8*a^4*b + 10*a^3*b^2 + 4*a^2*b^3 + 2*(a^5 - 4*a^
4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x + 33*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 40*(11*(a
^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^3 + 2*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b
^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^9 + 2*(5*a^5 + 26*a^4*b + 27*a^3*b^2 - 32*a^2*b^3 - 32*a*b^4
+ 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c)^8 + 2*(5*a^5 + 26*a^4*b + 27*a^3
*b^2 - 32*a^2*b^3 - 32*a*b^4 + 495*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2
- 28*a^2*b^3 - 12*a*b^4)*d*x + 180*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^
2*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(99*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^5 + 60*(a^5 + 4*
a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^3 + (5*a^5 + 26*a^
4*b + 27*a^3*b^2 - 32*a^2*b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d
*x + c))*sinh(d*x + c)^7 - 4*(39*a^3*b^2 + 134*a^2*b^3 + 184*a*b^4 + 80*b^5 - 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2
- 138*a^2*b^3 - 136*a*b^4 - 48*b^5)*d*x)*cosh(d*x + c)^6 + 4*(462*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^6 -
39*a^3*b^2 - 134*a^2*b^3 - 184*a*b^4 - 80*b^5 + 420*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b -
11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^4 + 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2 - 138*a^2*b^3 - 136*a*b^4 - 48*
b^5)*d*x + 14*(5*a^5 + 26*a^4*b + 27*a^3*b^2 - 32*a^2*b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2
*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(198*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^7 + 25
2*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^5 + 14*
(5*a^5 + 26*a^4*b + 27*a^3*b^2 - 32*a^2*b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^
4)*d*x)*cosh(d*x + c)^3 - 3*(39*a^3*b^2 + 134*a^2*b^3 + 184*a*b^4 + 80*b^5 - 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2 -
138*a^2*b^3 - 136*a*b^4 - 48*b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*a^5 - 4*a^4*b - 2*a^3*b^2 - 2*(5*a^
5 + 26*a^4*b + 131*a^3*b^2 + 256*a^2*b^3 + 128*a*b^4 - 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)
*d*x)*cosh(d*x + c)^4 + 2*(495*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^8 + 840*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*
a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^6 - 5*a^5 - 26*a^4*b - 131*a^3*b^2 - 256
*a^2*b^3 - 128*a*b^4 + 70*(5*a^5 + 26*a^4*b + 27*a^3*b^2 - 32*a^2*b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b - 19*a^3*
b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c)^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*
x - 30*(39*a^3*b^2 + 134*a^2*b^3 + 184*a*b^4 + 80*b^5 - 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2 - 138*a^2*b^3 - 136*a*
b^4 - 48*b^5)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(55*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^9 + 120*(a
^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^7 + 14*(5*a
^5 + 26*a^4*b + 27*a^3*b^2 - 32*a^2*b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d
*x)*cosh(d*x + c)^5 - 10*(39*a^3*b^2 + 134*a^2*b^3 + 184*a*b^4 + 80*b^5 - 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2 - 13
8*a^2*b^3 - 136*a*b^4 - 48*b^5)*d*x)*cosh(d*x + c)^3 - (5*a^5 + 26*a^4*b + 131*a^3*b^2 + 256*a^2*b^3 + 128*a*b
^4 - 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(2*a^5 +
8*a^4*b + 23*a^3*b^2 + 14*a^2*b^3 - 2*(a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^2 + 4*(33*(a
^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^10 + 90*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3
*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^8 + 14*(5*a^5 + 26*a^4*b + 27*a^3*b^2 - 32*a^2*b^3 - 32*a*b^4 + 16*(a^5 -
2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c)^6 - 2*a^5 - 8*a^4*b - 23*a^3*b^2 - 14*a^2*b^
3 - 15*(39*a^3*b^2 + 134*a^2*b^3 + 184*a*b^4 + 80*b^5 - 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2 - 138*a^2*b^3 - 136*a*
b^4 - 48*b^5)*d*x)*cosh(d*x + c)^4 + 2*(a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x - 3*(5*a^5 + 26*a^4*b + 13
1*a^3*b^2 + 256*a^2*b^3 + 128*a*b^4 - 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + ((35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^10 + 10*(35*a^4*b + 56*a^3*b^2 + 2
4*a^2*b^3)*cosh(d*x + c)*sinh(d*x + c)^9 + (35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*sinh(d*x + c)^10 + 4*(35*a^4*b
+ 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c)^8 + (140*a^4*b + 504*a^3*b^2 + 544*a^2*b^3 + 192*a*b^4
+ 45*(35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(35*a^4*b + 56*a^3*b^2 + 24
*a^2*b^3)*cosh(d*x + c)^3 + 4*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^7
+ 2*(105*a^4*b + 448*a^3*b^2 + 800*a^2*b^3 + 640*a*b^4 + 192*b^5)*cosh(d*x + c)^6 + 2*(105*a^4*b + 448*a^3*b^
2 + 800*a^2*b^3 + 640*a*b^4 + 192*b^5 + 105*(35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^4 + 56*(35*a^4*
b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(35*a^4*b + 56*a^3*b^2 + 24
*a^2*b^3)*cosh(d*x + c)^5 + 56*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c)^3 + 3*(105*a^4*
b + 448*a^3*b^2 + 800*a^2*b^3 + 640*a*b^4 + 192*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 + 4*(35*a^4*b + 126*a^3*b^
2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c)^4 + 2*(105*(35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^6 + 70
*a^4*b + 252*a^3*b^2 + 272*a^2*b^3 + 96*a*b^4 + 140*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x
+ c)^4 + 15*(105*a^4*b + 448*a^3*b^2 + 800*a^2*b^3 + 640*a*b^4 + 192*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 +
8*(15*(35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^7 + 28*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b
^4)*cosh(d*x + c)^5 + 5*(105*a^4*b + 448*a^3*b^2 + 800*a^2*b^3 + 640*a*b^4 + 192*b^5)*cosh(d*x + c)^3 + 2*(35*
a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + (35*a^4*b + 56*a^3*b^2 + 24*a^2
*b^3)*cosh(d*x + c)^2 + (45*(35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^8 + 112*(35*a^4*b + 126*a^3*b^2
+ 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c)^6 + 35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3 + 30*(105*a^4*b + 448*a^3*b^2
+ 800*a^2*b^3 + 640*a*b^4 + 192*b^5)*cosh(d*x + c)^4 + 24*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*co
sh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^9 + 16*(35*a^4*b + 12
6*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c)^7 + 6*(105*a^4*b + 448*a^3*b^2 + 800*a^2*b^3 + 640*a*b^4 + 1
92*b^5)*cosh(d*x + c)^5 + 8*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c)^3 + (35*a^4*b + 56
*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/(a + b))*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x
+ c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2
+ 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b + 8*b^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d
*x + c) - 4*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x +
c)^2 + a^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b)))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d
*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x +
c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + 8*(3*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^11 + 10*(a^
5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^9 + 2*(5*a^5
+ 26*a^4*b + 27*a^3*b^2 - 32*a^2*b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x
)*cosh(d*x + c)^7 - 3*(39*a^3*b^2 + 134*a^2*b^3 + 184*a*b^4 + 80*b^5 - 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2 - 138*a
^2*b^3 - 136*a*b^4 - 48*b^5)*d*x)*cosh(d*x + c)^5 - (5*a^5 + 26*a^4*b + 131*a^3*b^2 + 256*a^2*b^3 + 128*a*b^4
- 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c)^3 - (2*a^5 + 8*a^4*b + 23*a^3*b^2
+ 14*a^2*b^3 - 2*(a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^8 + 2*a^7*b
+ a^6*b^2)*d*cosh(d*x + c)^10 + 10*(a^8 + 2*a^7*b + a^6*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^8 + 2*a^7*b
+ a^6*b^2)*d*sinh(d*x + c)^10 + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c)^8 + (45*(a^8 + 2*a^7
*b + a^6*b^2)*d*cosh(d*x + c)^2 + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d)*sinh(d*x + c)^8 + 2*(3*a^8 + 14
*a^7*b + 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^6 + 8*(15*(a^8 + 2*a^7*b + a^6*b^2)*d*cosh(d*x +
c)^3 + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^8 + 2*a^7*b + a
^6*b^2)*d*cosh(d*x + c)^4 + 56*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c)^2 + (3*a^8 + 14*a^7*b +
27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*d)*sinh(d*x + c)^6 + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*
x + c)^4 + 4*(63*(a^8 + 2*a^7*b + a^6*b^2)*d*cosh(d*x + c)^5 + 56*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*co
sh(d*x + c)^3 + 3*(3*a^8 + 14*a^7*b + 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 +
2*(105*(a^8 + 2*a^7*b + a^6*b^2)*d*cosh(d*x + c)^6 + 140*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x +
c)^4 + 15*(3*a^8 + 14*a^7*b + 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^2 + 2*(a^8 + 4*a^7*b + 5*a^
6*b^2 + 2*a^5*b^3)*d)*sinh(d*x + c)^4 + (a^8 + 2*a^7*b + a^6*b^2)*d*cosh(d*x + c)^2 + 8*(15*(a^8 + 2*a^7*b + a
^6*b^2)*d*cosh(d*x + c)^7 + 28*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c)^5 + 5*(3*a^8 + 14*a^7*b
+ 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^3 + 2*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d
*x + c))*sinh(d*x + c)^3 + (45*(a^8 + 2*a^7*b + a^6*b^2)*d*cosh(d*x + c)^8 + 112*(a^8 + 4*a^7*b + 5*a^6*b^2 +
2*a^5*b^3)*d*cosh(d*x + c)^6 + 30*(3*a^8 + 14*a^7*b + 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^4 +
24*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c)^2 + (a^8 + 2*a^7*b + a^6*b^2)*d)*sinh(d*x + c)^2 +
2*(5*(a^8 + 2*a^7*b + a^6*b^2)*d*cosh(d*x + c)^9 + 16*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c)
^7 + 6*(3*a^8 + 14*a^7*b + 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^5 + 8*(a^8 + 4*a^7*b + 5*a^6*b
^2 + 2*a^5*b^3)*d*cosh(d*x + c)^3 + (a^8 + 2*a^7*b + a^6*b^2)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*((a^5 + 2*a
^4*b + a^3*b^2)*cosh(d*x + c)^12 + 12*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)*sinh(d*x + c)^11 + (a^5 + 2*a^4*
b + a^3*b^2)*sinh(d*x + c)^12 + 4*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2
*b^3)*d*x)*cosh(d*x + c)^10 + 2*(2*a^5 + 8*a^4*b + 10*a^3*b^2 + 4*a^2*b^3 + 2*(a^5 - 4*a^4*b - 11*a^3*b^2 - 6*
a^2*b^3)*d*x + 33*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 20*(11*(a^5 + 2*a^4*b + a^3*b^
2)*cosh(d*x + c)^3 + 2*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*
cosh(d*x + c))*sinh(d*x + c)^9 + (5*a^5 + 26*a^4*b + 27*a^3*b^2 - 32*a^2*b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b -
19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c)^8 + (5*a^5 + 26*a^4*b + 27*a^3*b^2 - 32*a^2*b^3 - 32*a*
b^4 + 495*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*
d*x + 180*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)
^2)*sinh(d*x + c)^8 + 8*(99*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^5 + 60*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*
b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^3 + (5*a^5 + 26*a^4*b + 27*a^3*b^2 - 32*a^2*
b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^7 -
2*(39*a^3*b^2 + 134*a^2*b^3 + 184*a*b^4 + 80*b^5 - 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2 - 138*a^2*b^3 - 136*a*b^4
- 48*b^5)*d*x)*cosh(d*x + c)^6 + 2*(462*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^6 - 39*a^3*b^2 - 134*a^2*b^3 -
184*a*b^4 - 80*b^5 + 420*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*
x)*cosh(d*x + c)^4 + 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2 - 138*a^2*b^3 - 136*a*b^4 - 48*b^5)*d*x + 14*(5*a^5 + 26*
a^4*b + 27*a^3*b^2 - 32*a^2*b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh
(d*x + c)^2)*sinh(d*x + c)^6 + 4*(198*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^7 + 252*(a^5 + 4*a^4*b + 5*a^3*b
^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^5 + 14*(5*a^5 + 26*a^4*b + 27*a^3
*b^2 - 32*a^2*b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c)^3 -
3*(39*a^3*b^2 + 134*a^2*b^3 + 184*a*b^4 + 80*b^5 - 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2 - 138*a^2*b^3 - 136*a*b^4 -
48*b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - a^5 - 2*a^4*b - a^3*b^2 - (5*a^5 + 26*a^4*b + 131*a^3*b^2 + 256
*a^2*b^3 + 128*a*b^4 - 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c)^4 + (495*(a^
5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^8 + 840*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*
b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^6 - 5*a^5 - 26*a^4*b - 131*a^3*b^2 - 256*a^2*b^3 - 128*a*b^4 + 70*(5*a^5 +
26*a^4*b + 27*a^3*b^2 - 32*a^2*b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*
cosh(d*x + c)^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x - 30*(39*a^3*b^2 + 134*a^2*b^3 +
184*a*b^4 + 80*b^5 - 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2 - 138*a^2*b^3 - 136*a*b^4 - 48*b^5)*d*x)*cosh(d*x + c)^2
)*sinh(d*x + c)^4 + 4*(55*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^9 + 120*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b
^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^7 + 14*(5*a^5 + 26*a^4*b + 27*a^3*b^2 - 32*a^
2*b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c)^5 - 10*(39*a^3*b
^2 + 134*a^2*b^3 + 184*a*b^4 + 80*b^5 - 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2 - 138*a^2*b^3 - 136*a*b^4 - 48*b^5)*d*
x)*cosh(d*x + c)^3 - (5*a^5 + 26*a^4*b + 131*a^3*b^2 + 256*a^2*b^3 + 128*a*b^4 - 16*(a^5 - 2*a^4*b - 19*a^3*b^
2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(2*a^5 + 8*a^4*b + 23*a^3*b^2 + 14*a^2*b^3
- 2*(a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^2 + 2*(33*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x +
c)^10 + 90*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x +
c)^8 + 14*(5*a^5 + 26*a^4*b + 27*a^3*b^2 - 32*a^2*b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3
- 12*a*b^4)*d*x)*cosh(d*x + c)^6 - 2*a^5 - 8*a^4*b - 23*a^3*b^2 - 14*a^2*b^3 - 15*(39*a^3*b^2 + 134*a^2*b^3 +
184*a*b^4 + 80*b^5 - 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2 - 138*a^2*b^3 - 136*a*b^4 - 48*b^5)*d*x)*cosh(d*x + c)^4
+ 2*(a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x - 3*(5*a^5 + 26*a^4*b + 131*a^3*b^2 + 256*a^2*b^3 + 128*a*b^
4 - 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((35*a^4*b
+ 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^10 + 10*(35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)*sinh(d*x
+ c)^9 + (35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*sinh(d*x + c)^10 + 4*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*
a*b^4)*cosh(d*x + c)^8 + (140*a^4*b + 504*a^3*b^2 + 544*a^2*b^3 + 192*a*b^4 + 45*(35*a^4*b + 56*a^3*b^2 + 24*a
^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^3 + 4*(35*
a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*a^4*b + 448*a^3*b^2 + 80
0*a^2*b^3 + 640*a*b^4 + 192*b^5)*cosh(d*x + c)^6 + 2*(105*a^4*b + 448*a^3*b^2 + 800*a^2*b^3 + 640*a*b^4 + 192*
b^5 + 105*(35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^4 + 56*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48
*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^5 + 56*(35
*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c)^3 + 3*(105*a^4*b + 448*a^3*b^2 + 800*a^2*b^3 + 64
0*a*b^4 + 192*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 + 4*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d
*x + c)^4 + 2*(105*(35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^6 + 70*a^4*b + 252*a^3*b^2 + 272*a^2*b^3
+ 96*a*b^4 + 140*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c)^4 + 15*(105*a^4*b + 448*a^3*
b^2 + 800*a^2*b^3 + 640*a*b^4 + 192*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(15*(35*a^4*b + 56*a^3*b^2 + 24*
a^2*b^3)*cosh(d*x + c)^7 + 28*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c)^5 + 5*(105*a^4*b
+ 448*a^3*b^2 + 800*a^2*b^3 + 640*a*b^4 + 192*b^5)*cosh(d*x + c)^3 + 2*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3
+ 48*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + (35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^2 + (45*(35*a^
4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^8 + 112*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*
x + c)^6 + 35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3 + 30*(105*a^4*b + 448*a^3*b^2 + 800*a^2*b^3 + 640*a*b^4 + 192*b^
5)*cosh(d*x + c)^4 + 24*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2
*(5*(35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x + c)^9 + 16*(35*a^4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4
)*cosh(d*x + c)^7 + 6*(105*a^4*b + 448*a^3*b^2 + 800*a^2*b^3 + 640*a*b^4 + 192*b^5)*cosh(d*x + c)^5 + 8*(35*a^
4*b + 126*a^3*b^2 + 136*a^2*b^3 + 48*a*b^4)*cosh(d*x + c)^3 + (35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cosh(d*x +
c))*sinh(d*x + c))*sqrt(-b/(a + b))*arctan(1/2*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d
*x + c)^2 + a + 2*b)*sqrt(-b/(a + b))/b) + 4*(3*(a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^11 + 10*(a^5 + 4*a^4*b
+ 5*a^3*b^2 + 2*a^2*b^3 + (a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c)^9 + 2*(5*a^5 + 26*a^4*b
+ 27*a^3*b^2 - 32*a^2*b^3 - 32*a*b^4 + 16*(a^5 - 2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x
+ c)^7 - 3*(39*a^3*b^2 + 134*a^2*b^3 + 184*a*b^4 + 80*b^5 - 4*(3*a^5 - 4*a^4*b - 57*a^3*b^2 - 138*a^2*b^3 - 13
6*a*b^4 - 48*b^5)*d*x)*cosh(d*x + c)^5 - (5*a^5 + 26*a^4*b + 131*a^3*b^2 + 256*a^2*b^3 + 128*a*b^4 - 16*(a^5 -
2*a^4*b - 19*a^3*b^2 - 28*a^2*b^3 - 12*a*b^4)*d*x)*cosh(d*x + c)^3 - (2*a^5 + 8*a^4*b + 23*a^3*b^2 + 14*a^2*b
^3 - 2*(a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^8 + 2*a^7*b + a^6*b^2)*
d*cosh(d*x + c)^10 + 10*(a^8 + 2*a^7*b + a^6*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^8 + 2*a^7*b + a^6*b^2)*
d*sinh(d*x + c)^10 + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c)^8 + (45*(a^8 + 2*a^7*b + a^6*b^
2)*d*cosh(d*x + c)^2 + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d)*sinh(d*x + c)^8 + 2*(3*a^8 + 14*a^7*b + 27
*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^6 + 8*(15*(a^8 + 2*a^7*b + a^6*b^2)*d*cosh(d*x + c)^3 + 4*(
a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^8 + 2*a^7*b + a^6*b^2)*d*c
osh(d*x + c)^4 + 56*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c)^2 + (3*a^8 + 14*a^7*b + 27*a^6*b^2
+ 24*a^5*b^3 + 8*a^4*b^4)*d)*sinh(d*x + c)^6 + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c)^4 +
4*(63*(a^8 + 2*a^7*b + a^6*b^2)*d*cosh(d*x + c)^5 + 56*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c)
^3 + 3*(3*a^8 + 14*a^7*b + 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^8
+ 2*a^7*b + a^6*b^2)*d*cosh(d*x + c)^6 + 140*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c)^4 + 15*(
3*a^8 + 14*a^7*b + 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^2 + 2*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a
^5*b^3)*d)*sinh(d*x + c)^4 + (a^8 + 2*a^7*b + a^6*b^2)*d*cosh(d*x + c)^2 + 8*(15*(a^8 + 2*a^7*b + a^6*b^2)*d*c
osh(d*x + c)^7 + 28*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c)^5 + 5*(3*a^8 + 14*a^7*b + 27*a^6*b
^2 + 24*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^3 + 2*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c))*si
nh(d*x + c)^3 + (45*(a^8 + 2*a^7*b + a^6*b^2)*d*cosh(d*x + c)^8 + 112*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*
d*cosh(d*x + c)^6 + 30*(3*a^8 + 14*a^7*b + 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^4 + 24*(a^8 +
4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c)^2 + (a^8 + 2*a^7*b + a^6*b^2)*d)*sinh(d*x + c)^2 + 2*(5*(a^8
+ 2*a^7*b + a^6*b^2)*d*cosh(d*x + c)^9 + 16*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*d*cosh(d*x + c)^7 + 6*(3*a
^8 + 14*a^7*b + 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^5 + 8*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*
b^3)*d*cosh(d*x + c)^3 + (a^8 + 2*a^7*b + a^6*b^2)*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(cosh(d*x+c)**2/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.2131, size = 552, normalized size = 2.71 \begin{align*} \frac{{\left (35 \, a^{2} b^{2} + 56 \, a b^{3} + 24 \, b^{4}\right )} \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt{-a b - b^{2}}}\right )}{8 \,{\left (a^{6} d + 2 \, a^{5} b d + a^{4} b^{2} d\right )} \sqrt{-a b - b^{2}}} - \frac{13 \, a^{3} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 40 \, a^{2} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 24 \, a b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 39 \, a^{3} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 134 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 184 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 80 \, b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 39 \, a^{3} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 104 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 56 \, a b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 13 \, a^{3} b^{2} + 10 \, a^{2} b^{3}}{4 \,{\left (a^{6} d + 2 \, a^{5} b d + a^{4} b^{2} d\right )}{\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}} + \frac{{\left (d x + c\right )}{\left (a - 6 \, b\right )}}{2 \, a^{4} d} + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a^{3} d} - \frac{{\left (2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 12 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, a^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(35*a^2*b^2 + 56*a*b^3 + 24*b^4)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^6*d + 2*a^
5*b*d + a^4*b^2*d)*sqrt(-a*b - b^2)) - 1/4*(13*a^3*b^2*e^(6*d*x + 6*c) + 40*a^2*b^3*e^(6*d*x + 6*c) + 24*a*b^4
*e^(6*d*x + 6*c) + 39*a^3*b^2*e^(4*d*x + 4*c) + 134*a^2*b^3*e^(4*d*x + 4*c) + 184*a*b^4*e^(4*d*x + 4*c) + 80*b
^5*e^(4*d*x + 4*c) + 39*a^3*b^2*e^(2*d*x + 2*c) + 104*a^2*b^3*e^(2*d*x + 2*c) + 56*a*b^4*e^(2*d*x + 2*c) + 13*
a^3*b^2 + 10*a^2*b^3)/((a^6*d + 2*a^5*b*d + a^4*b^2*d)*(a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x
+ 2*c) + a)^2) + 1/2*(d*x + c)*(a - 6*b)/(a^4*d) + 1/8*e^(2*d*x + 2*c)/(a^3*d) - 1/8*(2*a*e^(2*d*x + 2*c) - 1
2*b*e^(2*d*x + 2*c) + a)*e^(-2*d*x - 2*c)/(a^4*d)