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

Optimal. Leaf size=159 \[ -\frac{\sqrt{b} \left (15 a^2-10 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} d (a-b)^3}-\frac{b (7 a-3 b) \sinh (c+d x)}{8 a^2 d (a-b)^2 \left (a+b \sinh ^2(c+d x)\right )}-\frac{b \sinh (c+d x)}{4 a d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{\tan ^{-1}(\sinh (c+d x))}{d (a-b)^3} \]

[Out]

ArcTan[Sinh[c + d*x]]/((a - b)^3*d) - (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a
]])/(8*a^(5/2)*(a - b)^3*d) - (b*Sinh[c + d*x])/(4*a*(a - b)*d*(a + b*Sinh[c + d*x]^2)^2) - ((7*a - 3*b)*b*Sin
h[c + d*x])/(8*a^2*(a - b)^2*d*(a + b*Sinh[c + d*x]^2))

________________________________________________________________________________________

Rubi [A]  time = 0.180384, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3190, 414, 527, 522, 203, 205} \[ -\frac{\sqrt{b} \left (15 a^2-10 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} d (a-b)^3}-\frac{b (7 a-3 b) \sinh (c+d x)}{8 a^2 d (a-b)^2 \left (a+b \sinh ^2(c+d x)\right )}-\frac{b \sinh (c+d x)}{4 a d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{\tan ^{-1}(\sinh (c+d x))}{d (a-b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Sinh[c + d*x]]/((a - b)^3*d) - (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a
]])/(8*a^(5/2)*(a - b)^3*d) - (b*Sinh[c + d*x])/(4*a*(a - b)*d*(a + b*Sinh[c + d*x]^2)^2) - ((7*a - 3*b)*b*Sin
h[c + d*x])/(8*a^2*(a - b)^2*d*(a + b*Sinh[c + d*x]^2))

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\text{sech}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac{b \sinh (c+d x)}{4 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{4 a-3 b-3 b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 a (a-b) d}\\ &=-\frac{b \sinh (c+d x)}{4 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}-\frac{(7 a-3 b) b \sinh (c+d x)}{8 a^2 (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{8 a^2-7 a b+3 b^2-(7 a-3 b) b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\sinh (c+d x)\right )}{8 a^2 (a-b)^2 d}\\ &=-\frac{b \sinh (c+d x)}{4 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}-\frac{(7 a-3 b) b \sinh (c+d x)}{8 a^2 (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{(a-b)^3 d}-\frac{\left (b \left (15 a^2-10 a b+3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{8 a^2 (a-b)^3 d}\\ &=\frac{\tan ^{-1}(\sinh (c+d x))}{(a-b)^3 d}-\frac{\sqrt{b} \left (15 a^2-10 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} (a-b)^3 d}-\frac{b \sinh (c+d x)}{4 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}-\frac{(7 a-3 b) b \sinh (c+d x)}{8 a^2 (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 0.730706, size = 321, normalized size = 2.02 \[ \frac{-2 \sqrt{a} b \left (-35 a^2 b+18 a^3+20 a b^2-3 b^3\right ) \sinh (c+d x)+(b-2 a)^2 \left (\sqrt{b} \left (15 a^2-10 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right )+16 a^{5/2} \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )\right )+\cosh ^2(2 (c+d x)) \left (16 a^{5/2} b^2 \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+b^{5/2} \left (15 a^2-10 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right )\right )-2 b \cosh (2 (c+d x)) \left (\sqrt{a} b \left (7 a^2-10 a b+3 b^2\right ) \sinh (c+d x)-(2 a-b) \left (\sqrt{b} \left (15 a^2-10 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right )+16 a^{5/2} \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{8 a^{5/2} d (a-b)^3 (2 a+b \cosh (2 (c+d x))-b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a + b)^2*(Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 16*a^(5/2)*ArcTan[T
anh[(c + d*x)/2]]) + (b^(5/2)*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 16*a^(5/2)*b
^2*ArcTan[Tanh[(c + d*x)/2]])*Cosh[2*(c + d*x)]^2 - 2*Sqrt[a]*b*(18*a^3 - 35*a^2*b + 20*a*b^2 - 3*b^3)*Sinh[c
+ d*x] - 2*b*Cosh[2*(c + d*x)]*(-((2*a - b)*(Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/
Sqrt[b]] + 16*a^(5/2)*ArcTan[Tanh[(c + d*x)/2]])) + Sqrt[a]*b*(7*a^2 - 10*a*b + 3*b^2)*Sinh[c + d*x]))/(8*a^(5
/2)*(a - b)^3*d*(2*a - b + b*Cosh[2*(c + d*x)])^2)

________________________________________________________________________________________

Maple [B]  time = 0.092, size = 2118, normalized size = 13.3 \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)/(a+b*sinh(d*x+c)^2)^3,x)

[Out]

-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))-55/4/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)^5+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)^7-25/8/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))-5/4/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))+5/
4/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))+9/4/d*b/(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)^7-27/4/d*b/(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)^5+27/4/d*b/(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)^3-9/4/d*b/(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^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+55/4/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)-25/8/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))+15/8/d*b/(a-b)^3*a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*a
rctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+15/8/d*b/(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))+13/8/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)*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)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+13/8/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))+2/d/(a-b)^3*arctan(tanh(1/2*d*x+1/2*c))-7/2/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+35/2/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)^5-35/2/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+7/2/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)+15/8/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))-1
5/8/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))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \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)/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-1/4*((7*a*b^2*e^(7*c) - 3*b^3*e^(7*c))*e^(7*d*x) + (36*a^2*b*e^(5*c) - 41*a*b^2*e^(5*c) + 9*b^3*e^(5*c))*e^(5
*d*x) - (36*a^2*b*e^(3*c) - 41*a*b^2*e^(3*c) + 9*b^3*e^(3*c))*e^(3*d*x) - (7*a*b^2*e^c - 3*b^3*e^c)*e^(d*x))/(
a^4*b^2*d - 2*a^3*b^3*d + a^2*b^4*d + (a^4*b^2*d*e^(8*c) - 2*a^3*b^3*d*e^(8*c) + a^2*b^4*d*e^(8*c))*e^(8*d*x)
+ 4*(2*a^5*b*d*e^(6*c) - 5*a^4*b^2*d*e^(6*c) + 4*a^3*b^3*d*e^(6*c) - a^2*b^4*d*e^(6*c))*e^(6*d*x) + 2*(8*a^6*d
*e^(4*c) - 24*a^5*b*d*e^(4*c) + 27*a^4*b^2*d*e^(4*c) - 14*a^3*b^3*d*e^(4*c) + 3*a^2*b^4*d*e^(4*c))*e^(4*d*x) +
4*(2*a^5*b*d*e^(2*c) - 5*a^4*b^2*d*e^(2*c) + 4*a^3*b^3*d*e^(2*c) - a^2*b^4*d*e^(2*c))*e^(2*d*x)) + 2*arctan(e
^(d*x + c))/(a^3*d - 3*a^2*b*d + 3*a*b^2*d - b^3*d) - 2*integrate(1/8*((15*a^2*b*e^(3*c) - 10*a*b^2*e^(3*c) +
3*b^3*e^(3*c))*e^(3*d*x) + (15*a^2*b*e^c - 10*a*b^2*e^c + 3*b^3*e^c)*e^(d*x))/(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 -
a^2*b^4 + (a^5*b*e^(4*c) - 3*a^4*b^2*e^(4*c) + 3*a^3*b^3*e^(4*c) - a^2*b^4*e^(4*c))*e^(4*d*x) + 2*(2*a^6*e^(2
*c) - 7*a^5*b*e^(2*c) + 9*a^4*b^2*e^(2*c) - 5*a^3*b^3*e^(2*c) + a^2*b^4*e^(2*c))*e^(2*d*x)), x)

________________________________________________________________________________________

Fricas [B]  time = 3.15793, size = 18737, normalized size = 117.84 \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)/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 28*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)*sin
h(d*x + c)^6 + 4*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*sinh(d*x + c)^7 + 4*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)
*cosh(d*x + c)^5 + 4*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4 + 21*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x +
c)^2)*sinh(d*x + c)^5 + 20*(7*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + (36*a^3*b - 77*a^2*b^2 + 50*a*b
^3 - 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c)^3 + 4*
(35*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^4 - 36*a^3*b + 77*a^2*b^2 - 50*a*b^3 + 9*b^4 + 10*(36*a^3*b -
77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d
*x + c)^5 + 10*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c)^3 - 3*(36*a^3*b - 77*a^2*b^2 + 50*a*b^
3 - 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + ((15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^8 + 8*(15*a^2*b^2 -
10*a*b^3 + 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^2*b^2 - 10*a*b^3 + 3*b^4)*sinh(d*x + c)^8 + 4*(30*a^3
*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 4*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4 + 7*(15*a^2*
b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^
3 + 3*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(120*a^4 - 200*a^3*b + 149
*a^2*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 120*a^4
- 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4 + 30*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^2)
*sinh(d*x + c)^4 + 15*a^2*b^2 - 10*a*b^3 + 3*b^4 + 8*(7*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + 10*(
30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (120*a^4 - 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*
b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(15*
a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 15*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^4 +
30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4 + 3*(120*a^4 - 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + 8*((15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(30*a^3*b - 35*a^2*b^2 + 16*a*
b^3 - 3*b^4)*cosh(d*x + c)^5 + (120*a^4 - 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (30*a^
3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/a)*log((b*cosh(d*x + c)^4 + 4*b*cos
h(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) + 4*(a*cosh(d*x + c)^3 + 3*
a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 - a*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 - a)*sinh(d*x + c
))*sqrt(-b/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)*co
sh(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)) - 32*(a^2*b^2*cosh(d*x + c)^8 + 8*a^2*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + a^2*b^2*si
nh(d*x + c)^8 + 4*(2*a^3*b - a^2*b^2)*cosh(d*x + c)^6 + 4*(7*a^2*b^2*cosh(d*x + c)^2 + 2*a^3*b - a^2*b^2)*sinh
(d*x + c)^6 + 8*(7*a^2*b^2*cosh(d*x + c)^3 + 3*(2*a^3*b - a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^4 -
8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^4 + 2*(35*a^2*b^2*cosh(d*x + c)^4 + 8*a^4 - 8*a^3*b + 3*a^2*b^2 + 30*(2*a^
3*b - a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^2*b^2 + 8*(7*a^2*b^2*cosh(d*x + c)^5 + 10*(2*a^3*b - a^2*b
^2)*cosh(d*x + c)^3 + (8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(2*a^3*b - a^2*b^2)*cos
h(d*x + c)^2 + 4*(7*a^2*b^2*cosh(d*x + c)^6 + 15*(2*a^3*b - a^2*b^2)*cosh(d*x + c)^4 + 2*a^3*b - a^2*b^2 + 3*(
8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(a^2*b^2*cosh(d*x + c)^7 + 3*(2*a^3*b - a^2*
b^2)*cosh(d*x + c)^5 + (8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^3 + (2*a^3*b - a^2*b^2)*cosh(d*x + c))*sinh
(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 4*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c) + 4*(7*(7*a^
2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 5*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c)^4 - 7*a
^2*b^2 + 10*a*b^3 - 3*b^4 - 3*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^5
*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^8 + 8*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*co
sh(d*x + c)*sinh(d*x + c)^7 + (a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*sinh(d*x + c)^8 + 4*(2*a^6*b - 7*a
^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^6 + 4*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)
*d*cosh(d*x + c)^2 + (2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d)*sinh(d*x + c)^6 + 2*(8*a^7 - 3
2*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^5*b^2 - 3*a^4*b^3 + 3*
a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^3 + 3*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x +
c))*sinh(d*x + c)^5 + 2*(35*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^4 + 30*(2*a^6*b - 7*a^
5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^2 + (8*a^7 - 32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*
a^3*b^4 - 3*a^2*b^5)*d)*sinh(d*x + c)^4 + 4*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x
+ c)^2 + 8*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^5 + 10*(2*a^6*b - 7*a^5*b^2 + 9*a^4
*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^3 + (8*a^7 - 32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a
^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^6
+ 15*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^4 + 3*(8*a^7 - 32*a^6*b + 51*a^5*
b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*d*cosh(d*x + c)^2 + (2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 +
a^2*b^5)*d)*sinh(d*x + c)^2 + (a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d + 8*((a^5*b^2 - 3*a^4*b^3 + 3*a^3*
b^4 - a^2*b^5)*d*cosh(d*x + c)^7 + 3*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^5
+ (8*a^7 - 32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*d*cosh(d*x + c)^3 + (2*a^6*b - 7*a^5*
b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)), -1/8*(2*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)
*cosh(d*x + c)^7 + 14*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(7*a^2*b^2 - 10*a*b^3 +
3*b^4)*sinh(d*x + c)^7 + 2*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c)^5 + 2*(36*a^3*b - 77*a^2*
b^2 + 50*a*b^3 - 9*b^4 + 21*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(7*a^2*b^2
- 10*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + (36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c))*sinh(d*x + c)
^4 - 2*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c)^3 + 2*(35*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(
d*x + c)^4 - 36*a^3*b + 77*a^2*b^2 - 50*a*b^3 + 9*b^4 + 10*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x
+ c)^2)*sinh(d*x + c)^3 + 2*(21*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + 10*(36*a^3*b - 77*a^2*b^2 +
50*a*b^3 - 9*b^4)*cosh(d*x + c)^3 - 3*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^
2 + ((15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^8 + 8*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)*sinh(d*
x + c)^7 + (15*a^2*b^2 - 10*a*b^3 + 3*b^4)*sinh(d*x + c)^8 + 4*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh
(d*x + c)^6 + 4*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4 + 7*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^2)
*sinh(d*x + c)^6 + 8*(7*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + 3*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3
- 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(120*a^4 - 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x +
c)^4 + 2*(35*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 120*a^4 - 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 +
9*b^4 + 30*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 15*a^2*b^2 - 10*a*b^3
+ 3*b^4 + 8*(7*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + 10*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4
)*cosh(d*x + c)^3 + (120*a^4 - 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*
(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c
)^6 + 15*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4
+ 3*(120*a^4 - 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((15*a^2*b^2
- 10*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^5 + (120*a^4
- 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cos
h(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*sqrt(b/a)*(cosh(d*x + c) + sinh(d*x + c))) + ((15*a^2*b^2 - 10
*a*b^3 + 3*b^4)*cosh(d*x + c)^8 + 8*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^2*b^
2 - 10*a*b^3 + 3*b^4)*sinh(d*x + c)^8 + 4*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 4*(30*a
^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4 + 7*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*
(7*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + 3*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c
))*sinh(d*x + c)^5 + 2*(120*a^4 - 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^2*
b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 120*a^4 - 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4 + 30*(30*a^3*b
- 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 15*a^2*b^2 - 10*a*b^3 + 3*b^4 + 8*(7*(15*a
^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + 10*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (
120*a^4 - 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(30*a^3*b - 35*a^2*b^
2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 15*(30*a^3*b -
35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4 + 3*(120*a^4 - 200*a
^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((15*a^2*b^2 - 10*a*b^3 + 3*b^4)*c
osh(d*x + c)^7 + 3*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^5 + (120*a^4 - 200*a^3*b + 149*a^2
*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x
+ c))*sqrt(b/a)*arctan(1/2*(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a -
b)*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 + 4*a - b)*sinh(d*x + c))*sqrt(b/a)/b) - 16*(a^2*b^2*cosh(d*x + c)^8
+ 8*a^2*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + a^2*b^2*sinh(d*x + c)^8 + 4*(2*a^3*b - a^2*b^2)*cosh(d*x + c)^6 +
4*(7*a^2*b^2*cosh(d*x + c)^2 + 2*a^3*b - a^2*b^2)*sinh(d*x + c)^6 + 8*(7*a^2*b^2*cosh(d*x + c)^3 + 3*(2*a^3*b
- a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^4 + 2*(35*a^2*b^2*co
sh(d*x + c)^4 + 8*a^4 - 8*a^3*b + 3*a^2*b^2 + 30*(2*a^3*b - a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^2*b^
2 + 8*(7*a^2*b^2*cosh(d*x + c)^5 + 10*(2*a^3*b - a^2*b^2)*cosh(d*x + c)^3 + (8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh
(d*x + c))*sinh(d*x + c)^3 + 4*(2*a^3*b - a^2*b^2)*cosh(d*x + c)^2 + 4*(7*a^2*b^2*cosh(d*x + c)^6 + 15*(2*a^3*
b - a^2*b^2)*cosh(d*x + c)^4 + 2*a^3*b - a^2*b^2 + 3*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x +
c)^2 + 8*(a^2*b^2*cosh(d*x + c)^7 + 3*(2*a^3*b - a^2*b^2)*cosh(d*x + c)^5 + (8*a^4 - 8*a^3*b + 3*a^2*b^2)*cos
h(d*x + c)^3 + (2*a^3*b - a^2*b^2)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 2*(7*
a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c) + 2*(7*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 5*(36*a^3*b
- 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c)^4 - 7*a^2*b^2 + 10*a*b^3 - 3*b^4 - 3*(36*a^3*b - 77*a^2*b^2 + 5
0*a*b^3 - 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^
8 + 8*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^5*b^2 - 3*a^4*b^3 + 3*a
^3*b^4 - a^2*b^5)*d*sinh(d*x + c)^8 + 4*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c
)^6 + 4*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^2 + (2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 -
5*a^3*b^4 + a^2*b^5)*d)*sinh(d*x + c)^6 + 2*(8*a^7 - 32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b
^5)*d*cosh(d*x + c)^4 + 8*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^3 + 3*(2*a^6*b - 7*a^
5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^5*b^2 - 3*a^4*b^3 + 3*a^3
*b^4 - a^2*b^5)*d*cosh(d*x + c)^4 + 30*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)
^2 + (8*a^7 - 32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*d)*sinh(d*x + c)^4 + 4*(2*a^6*b - 7
*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^2 + 8*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^
5)*d*cosh(d*x + c)^5 + 10*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^3 + (8*a^7 -
32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^5*b^2
- 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^6 + 15*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*
b^5)*d*cosh(d*x + c)^4 + 3*(8*a^7 - 32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*d*cosh(d*x +
c)^2 + (2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d)*sinh(d*x + c)^2 + (a^5*b^2 - 3*a^4*b^3 + 3*a
^3*b^4 - a^2*b^5)*d + 8*((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^7 + 3*(2*a^6*b - 7*a^5*b^
2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^5 + (8*a^7 - 32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*
b^4 - 3*a^2*b^5)*d*cosh(d*x + c)^3 + (2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c))*
sinh(d*x + c))]

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError