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

Optimal. Leaf size=154 \[ -\frac{b^3 \sinh (c+d x)}{4 a^3 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2}+\frac{3 b^2 (4 a+3 b) \sinh (c+d x)}{8 a^3 d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}-\frac{3 b \left (4 (a+b)^2+(2 a+b)^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{8 a^{7/2} d (a+b)^{5/2}}+\frac{\sinh (c+d x)}{a^3 d} \]

[Out]

(-3*b*(4*(a + b)^2 + (2*a + b)^2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(8*a^(7/2)*(a + b)^(5/2)*d) + S
inh[c + d*x]/(a^3*d) - (b^3*Sinh[c + d*x])/(4*a^3*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)^2) + (3*b^2*(4*a + 3*b
)*Sinh[c + d*x])/(8*a^3*(a + b)^2*d*(a + b + a*Sinh[c + d*x]^2))

________________________________________________________________________________________

Rubi [A]  time = 0.184815, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4147, 390, 1157, 385, 205} \[ -\frac{b^3 \sinh (c+d x)}{4 a^3 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2}+\frac{3 b^2 (4 a+3 b) \sinh (c+d x)}{8 a^3 d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}-\frac{3 b \left (4 (a+b)^2+(2 a+b)^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{8 a^{7/2} d (a+b)^{5/2}}+\frac{\sinh (c+d x)}{a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*(4*(a + b)^2 + (2*a + b)^2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(8*a^(7/2)*(a + b)^(5/2)*d) + S
inh[c + d*x]/(a^3*d) - (b^3*Sinh[c + d*x])/(4*a^3*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)^2) + (3*b^2*(4*a + 3*b
)*Sinh[c + d*x])/(8*a^3*(a + b)^2*d*(a + b + a*Sinh[c + d*x]^2))

Rule 4147

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^
((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n
/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 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{\cosh (c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{\left (a+b+a x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^3}-\frac{b \left (3 a^2+3 a b+b^2\right )+3 a b (2 a+b) x^2+3 a^2 b x^4}{a^3 \left (a+b+a x^2\right )^3}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\sinh (c+d x)}{a^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) x^2+3 a^2 b x^4}{\left (a+b+a x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{a^3 d}\\ &=\frac{\sinh (c+d x)}{a^3 d}-\frac{b^3 \sinh (c+d x)}{4 a^3 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{-3 b (2 a+b)^2-12 a b (a+b) x^2}{\left (a+b+a x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 a^3 (a+b) d}\\ &=\frac{\sinh (c+d x)}{a^3 d}-\frac{b^3 \sinh (c+d x)}{4 a^3 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}+\frac{3 b^2 (4 a+3 b) \sinh (c+d x)}{8 a^3 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )}-\frac{\left (3 b \left (4 (a+b)^2+(2 a+b)^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+a x^2} \, dx,x,\sinh (c+d x)\right )}{8 a^3 (a+b)^2 d}\\ &=-\frac{3 b \left (4 (a+b)^2+(2 a+b)^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{8 a^{7/2} (a+b)^{5/2} d}+\frac{\sinh (c+d x)}{a^3 d}-\frac{b^3 \sinh (c+d x)}{4 a^3 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}+\frac{3 b^2 (4 a+3 b) \sinh (c+d x)}{8 a^3 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 3.65207, size = 292, normalized size = 1.9 \[ \frac{\text{sech}^5(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\frac{3 b \left (8 a^2+12 a b+5 b^2\right ) (\cosh (c)-\sinh (c)) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left (\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right )}{(a+b)^{5/2} \sqrt{(\cosh (c)-\sinh (c))^2}}-\frac{8 \sqrt{a} b^3 \tanh (c+d x)}{a+b}+\frac{6 \sqrt{a} b^2 (4 a+3 b) \tanh (c+d x) (a \cosh (2 (c+d x))+a+2 b)}{(a+b)^2}+8 \sqrt{a} \sinh (c) \cosh (d x) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2+8 \sqrt{a} \cosh (c) \sinh (d x) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2\right )}{64 a^{7/2} d \left (a+b \text{sech}^2(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^5*((3*b*(8*a^2 + 12*a*b + 5*b^2)*ArcTan[(Sqrt[a + b]*Csch[c + d
*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]*
(Cosh[c] - Sinh[c]))/((a + b)^(5/2)*Sqrt[(Cosh[c] - Sinh[c])^2]) + 8*Sqrt[a]*Cosh[d*x]*(a + 2*b + a*Cosh[2*(c
+ d*x)])^2*Sech[c + d*x]*Sinh[c] + 8*Sqrt[a]*Cosh[c]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]*Sinh[d*x]
- (8*Sqrt[a]*b^3*Tanh[c + d*x])/(a + b) + (6*Sqrt[a]*b^2*(4*a + 3*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Tanh[c +
d*x])/(a + b)^2))/(64*a^(7/2)*d*(a + b*Sech[c + d*x]^2)^3)

________________________________________________________________________________________

Maple [B]  time = 0.118, size = 1238, normalized size = 8. \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)/(a+b*sech(d*x+c)^2)^3,x)

[Out]

-1/d/a^3/(tanh(1/2*d*x+1/2*c)+1)-3/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)^7-7/4/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
-3/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+21/4/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)^5+21/4/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+3/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)^3-21/4/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)^3-21/4/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+3/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)+7/4/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)-3/d*b/a
^(3/2)/(a^2+2*a*b+b^2)/(a+b)^(1/2)*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))-9/2/d*b^2
/a^(5/2)/(a^2+2*a*b+b^2)/(a+b)^(1/2)*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))-15/8/d*
b^3/a^(7/2)/(a^2+2*a*b+b^2)/(a+b)^(1/2)*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))-3/d*
b/a^(3/2)/(a^2+2*a*b+b^2)/(a+b)^(1/2)*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)-2*b^(1/2))/a^(1/2))-9/2/d*
b^2/a^(5/2)/(a^2+2*a*b+b^2)/(a+b)^(1/2)*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)-2*b^(1/2))/a^(1/2))-15/8
/d*b^3/a^(7/2)/(a^2+2*a*b+b^2)/(a+b)^(1/2)*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)-2*b^(1/2))/a^(1/2))-1
/d/a^3/(tanh(1/2*d*x+1/2*c)-1)

________________________________________________________________________________________

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

[Out]

-1/4*(2*a^4 + 4*a^3*b + 2*a^2*b^2 - 2*(a^4*e^(10*c) + 2*a^3*b*e^(10*c) + a^2*b^2*e^(10*c))*e^(10*d*x) - (6*a^4
*e^(8*c) + 28*a^3*b*e^(8*c) + 50*a^2*b^2*e^(8*c) + 25*a*b^3*e^(8*c))*e^(8*d*x) - (4*a^4*e^(6*c) + 24*a^3*b*e^(
6*c) + 80*a^2*b^2*e^(6*c) + 129*a*b^3*e^(6*c) + 60*b^4*e^(6*c))*e^(6*d*x) + (4*a^4*e^(4*c) + 24*a^3*b*e^(4*c)
+ 80*a^2*b^2*e^(4*c) + 129*a*b^3*e^(4*c) + 60*b^4*e^(4*c))*e^(4*d*x) + (6*a^4*e^(2*c) + 28*a^3*b*e^(2*c) + 50*
a^2*b^2*e^(2*c) + 25*a*b^3*e^(2*c))*e^(2*d*x))/((a^7*d*e^(9*c) + 2*a^6*b*d*e^(9*c) + a^5*b^2*d*e^(9*c))*e^(9*d
*x) + 4*(a^7*d*e^(7*c) + 4*a^6*b*d*e^(7*c) + 5*a^5*b^2*d*e^(7*c) + 2*a^4*b^3*d*e^(7*c))*e^(7*d*x) + 2*(3*a^7*d
*e^(5*c) + 14*a^6*b*d*e^(5*c) + 27*a^5*b^2*d*e^(5*c) + 24*a^4*b^3*d*e^(5*c) + 8*a^3*b^4*d*e^(5*c))*e^(5*d*x) +
4*(a^7*d*e^(3*c) + 4*a^6*b*d*e^(3*c) + 5*a^5*b^2*d*e^(3*c) + 2*a^4*b^3*d*e^(3*c))*e^(3*d*x) + (a^7*d*e^c + 2*
a^6*b*d*e^c + a^5*b^2*d*e^c)*e^(d*x)) - 1/2*integrate(3/2*((8*a^2*b*e^(3*c) + 12*a*b^2*e^(3*c) + 5*b^3*e^(3*c)
)*e^(3*d*x) + (8*a^2*b*e^c + 12*a*b^2*e^c + 5*b^3*e^c)*e^(d*x))/(a^6 + 2*a^5*b + a^4*b^2 + (a^6*e^(4*c) + 2*a^
5*b*e^(4*c) + a^4*b^2*e^(4*c))*e^(4*d*x) + 2*(a^6*e^(2*c) + 4*a^5*b*e^(2*c) + 5*a^4*b^2*e^(2*c) + 2*a^3*b^3*e^
(2*c))*e^(2*d*x)), x)

________________________________________________________________________________________

Fricas [B]  time = 3.49527, size = 23050, normalized size = 149.68 \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)/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(8*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x + c)^10 + 80*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cos
h(d*x + c)*sinh(d*x + c)^9 + 8*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*sinh(d*x + c)^10 + 4*(6*a^6 + 34*a^5*b +
78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c)^8 + 4*(6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2
*b^4 + 90*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 32*(30*(a^6 + 3*a^5*b + 3*a
^4*b^2 + a^3*b^3)*cosh(d*x + c)^3 + (6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c))*s
inh(d*x + c)^7 + 4*(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c)^6 + 4
*(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5 + 420*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3
*b^3)*cosh(d*x + c)^4 + 28*(6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c)^2)*sinh(d*x
+ c)^6 - 8*a^6 - 24*a^5*b - 24*a^4*b^2 - 8*a^3*b^3 + 8*(252*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x +
c)^5 + 28*(6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c)^3 + 3*(4*a^6 + 28*a^5*b + 10
4*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 - 4*(4*a^6 + 28*a^5*b + 104*a
^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c)^4 + 4*(420*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*
cosh(d*x + c)^6 - 4*a^6 - 28*a^5*b - 104*a^4*b^2 - 209*a^3*b^3 - 189*a^2*b^4 - 60*a*b^5 + 70*(6*a^6 + 34*a^5*b
+ 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c)^4 + 15*(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 +
189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(60*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d
*x + c)^7 + 14*(6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c)^5 + 5*(4*a^6 + 28*a^5*b
+ 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c)^3 - (4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209
*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*
b^3 + 25*a^2*b^4)*cosh(d*x + c)^2 + 4*(90*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x + c)^8 + 28*(6*a^6 +
34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c)^6 - 6*a^6 - 34*a^5*b - 78*a^4*b^2 - 75*a^3*b^3
- 25*a^2*b^4 + 15*(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c)^4 - 6*
(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*(
(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^9 + 9*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)*sinh(d
*x + c)^8 + (8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*sinh(d*x + c)^9 + 4*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b
^4)*cosh(d*x + c)^7 + 4*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4 + 9*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*c
osh(d*x + c)^2)*sinh(d*x + c)^7 + 28*(3*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^3 + (8*a^4*b + 28*a^3
*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^6 + 2*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a
*b^4 + 40*b^5)*cosh(d*x + c)^5 + 2*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^5 + 63*(8*a^4*b +
12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^4 + 42*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^2)*s
inh(d*x + c)^5 + 2*(63*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^5 + 70*(8*a^4*b + 28*a^3*b^2 + 29*a^2*
b^3 + 10*a*b^4)*cosh(d*x + c)^3 + 5*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^5)*cosh(d*x + c))
*sinh(d*x + c)^4 + 4*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^3 + 4*(21*(8*a^4*b + 12*a^3*
b^2 + 5*a^2*b^3)*cosh(d*x + c)^6 + 8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4 + 35*(8*a^4*b + 28*a^3*b^2 + 2
9*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^4 + 5*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^5)*cosh(d*x
+ c)^2)*sinh(d*x + c)^3 + 4*(9*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^7 + 21*(8*a^4*b + 28*a^3*b^2
+ 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^5 + 5*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^5)*cosh(
d*x + c)^3 + 3*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + (8*a^4*b + 12*a
^3*b^2 + 5*a^2*b^3)*cosh(d*x + c) + (9*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^8 + 28*(8*a^4*b + 28*a
^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^6 + 8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3 + 10*(24*a^4*b + 100*a^3*b^
2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^5)*cosh(d*x + c)^4 + 12*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh
(d*x + c)^2)*sinh(d*x + c))*sqrt(-a^2 - a*b)*log((a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*si
nh(d*x + c)^4 - 2*(3*a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - 3*a - 2*b)*sinh(d*x + c)^2 + 4*(a*cos
h(d*x + c)^3 - (3*a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2
+ sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a^2 - a*b) + a)/(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*(
10*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x + c)^9 + 4*(6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*
a^2*b^4)*cosh(d*x + c)^7 + 3*(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x
+ c)^5 - 2*(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c)^3 - (6*a^6 +
34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^9 + 3*a^8*b + 3*a^7*b^2 + a
^6*b^3)*d*cosh(d*x + c)^9 + 9*(a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^8 + (a^9 + 3
*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*sinh(d*x + c)^9 + 4*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh
(d*x + c)^7 + 4*(9*(a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x + c)^2 + (a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^
6*b^3 + 2*a^5*b^4)*d)*sinh(d*x + c)^7 + 2*(3*a^9 + 17*a^8*b + 41*a^7*b^2 + 51*a^6*b^3 + 32*a^5*b^4 + 8*a^4*b^5
)*d*cosh(d*x + c)^5 + 28*(3*(a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x + c)^3 + (a^9 + 5*a^8*b + 9*a^7*b
^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^6 + 2*(63*(a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*c
osh(d*x + c)^4 + 42*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c)^2 + (3*a^9 + 17*a^8*b
+ 41*a^7*b^2 + 51*a^6*b^3 + 32*a^5*b^4 + 8*a^4*b^5)*d)*sinh(d*x + c)^5 + 4*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*
b^3 + 2*a^5*b^4)*d*cosh(d*x + c)^3 + 2*(63*(a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x + c)^5 + 70*(a^9 +
5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c)^3 + 5*(3*a^9 + 17*a^8*b + 41*a^7*b^2 + 51*a^6*b^
3 + 32*a^5*b^4 + 8*a^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(21*(a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*c
osh(d*x + c)^6 + 35*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c)^4 + 5*(3*a^9 + 17*a^8*
b + 41*a^7*b^2 + 51*a^6*b^3 + 32*a^5*b^4 + 8*a^4*b^5)*d*cosh(d*x + c)^2 + (a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b
^3 + 2*a^5*b^4)*d)*sinh(d*x + c)^3 + (a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x + c) + 4*(9*(a^9 + 3*a^8
*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x + c)^7 + 21*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*
x + c)^5 + 5*(3*a^9 + 17*a^8*b + 41*a^7*b^2 + 51*a^6*b^3 + 32*a^5*b^4 + 8*a^4*b^5)*d*cosh(d*x + c)^3 + 3*(a^9
+ 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (9*(a^9 + 3*a^8*b + 3*a^7*b^
2 + a^6*b^3)*d*cosh(d*x + c)^8 + 28*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c)^6 + 10
*(3*a^9 + 17*a^8*b + 41*a^7*b^2 + 51*a^6*b^3 + 32*a^5*b^4 + 8*a^4*b^5)*d*cosh(d*x + c)^4 + 12*(a^9 + 5*a^8*b +
9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c)^2 + (a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d)*sinh(d*x + c)
), 1/8*(4*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x + c)^10 + 40*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*co
sh(d*x + c)*sinh(d*x + c)^9 + 4*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*sinh(d*x + c)^10 + 2*(6*a^6 + 34*a^5*b +
78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c)^8 + 2*(6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^
2*b^4 + 90*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(30*(a^6 + 3*a^5*b + 3*
a^4*b^2 + a^3*b^3)*cosh(d*x + c)^3 + (6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c))*
sinh(d*x + c)^7 + 2*(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c)^6 +
2*(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5 + 420*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^
3*b^3)*cosh(d*x + c)^4 + 28*(6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c)^2)*sinh(d*
x + c)^6 - 4*a^6 - 12*a^5*b - 12*a^4*b^2 - 4*a^3*b^3 + 4*(252*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x +
c)^5 + 28*(6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c)^3 + 3*(4*a^6 + 28*a^5*b + 1
04*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(4*a^6 + 28*a^5*b + 104*
a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c)^4 + 2*(420*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)
*cosh(d*x + c)^6 - 4*a^6 - 28*a^5*b - 104*a^4*b^2 - 209*a^3*b^3 - 189*a^2*b^4 - 60*a*b^5 + 70*(6*a^6 + 34*a^5*
b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c)^4 + 15*(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 +
189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(60*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d
*x + c)^7 + 14*(6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c)^5 + 5*(4*a^6 + 28*a^5*b
+ 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c)^3 - (4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209
*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*
b^3 + 25*a^2*b^4)*cosh(d*x + c)^2 + 2*(90*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x + c)^8 + 28*(6*a^6 +
34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c)^6 - 6*a^6 - 34*a^5*b - 78*a^4*b^2 - 75*a^3*b^3
- 25*a^2*b^4 + 15*(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c)^4 - 6*
(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*(
(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^9 + 9*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)*sinh(d
*x + c)^8 + (8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*sinh(d*x + c)^9 + 4*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b
^4)*cosh(d*x + c)^7 + 4*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4 + 9*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*c
osh(d*x + c)^2)*sinh(d*x + c)^7 + 28*(3*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^3 + (8*a^4*b + 28*a^3
*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^6 + 2*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a
*b^4 + 40*b^5)*cosh(d*x + c)^5 + 2*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^5 + 63*(8*a^4*b +
12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^4 + 42*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^2)*s
inh(d*x + c)^5 + 2*(63*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^5 + 70*(8*a^4*b + 28*a^3*b^2 + 29*a^2*
b^3 + 10*a*b^4)*cosh(d*x + c)^3 + 5*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^5)*cosh(d*x + c))
*sinh(d*x + c)^4 + 4*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^3 + 4*(21*(8*a^4*b + 12*a^3*
b^2 + 5*a^2*b^3)*cosh(d*x + c)^6 + 8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4 + 35*(8*a^4*b + 28*a^3*b^2 + 2
9*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^4 + 5*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^5)*cosh(d*x
+ c)^2)*sinh(d*x + c)^3 + 4*(9*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^7 + 21*(8*a^4*b + 28*a^3*b^2
+ 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^5 + 5*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^5)*cosh(
d*x + c)^3 + 3*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + (8*a^4*b + 12*a
^3*b^2 + 5*a^2*b^3)*cosh(d*x + c) + (9*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^8 + 28*(8*a^4*b + 28*a
^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^6 + 8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3 + 10*(24*a^4*b + 100*a^3*b^
2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^5)*cosh(d*x + c)^4 + 12*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh
(d*x + c)^2)*sinh(d*x + c))*sqrt(a^2 + a*b)*arctan(1/2*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2
+ a*sinh(d*x + c)^3 + (3*a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + 3*a + 4*b)*sinh(d*x + c))/sqrt(a^2 +
a*b)) - 3*((8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^9 + 9*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x
+ c)*sinh(d*x + c)^8 + (8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*sinh(d*x + c)^9 + 4*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b
^3 + 10*a*b^4)*cosh(d*x + c)^7 + 4*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4 + 9*(8*a^4*b + 12*a^3*b^2 + 5
*a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 28*(3*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^3 + (8*a^4
*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^6 + 2*(24*a^4*b + 100*a^3*b^2 + 175*a^2*
b^3 + 136*a*b^4 + 40*b^5)*cosh(d*x + c)^5 + 2*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^5 + 63*
(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^4 + 42*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*
x + c)^2)*sinh(d*x + c)^5 + 2*(63*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^5 + 70*(8*a^4*b + 28*a^3*b^
2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^3 + 5*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^5)*cos
h(d*x + c))*sinh(d*x + c)^4 + 4*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^3 + 4*(21*(8*a^4*
b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^6 + 8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4 + 35*(8*a^4*b + 28*
a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^4 + 5*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^
5)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(9*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^7 + 21*(8*a^4*b +
28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^5 + 5*(24*a^4*b + 100*a^3*b^2 + 175*a^2*b^3 + 136*a*b^4 + 40
*b^5)*cosh(d*x + c)^3 + 3*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + (8*a
^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c) + (9*(8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^8 + 28*(8*a
^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*a*b^4)*cosh(d*x + c)^6 + 8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3 + 10*(24*a^4*b +
100*a^3*b^2 + 175*a^2*b^3 + 136*a*b^4 + 40*b^5)*cosh(d*x + c)^4 + 12*(8*a^4*b + 28*a^3*b^2 + 29*a^2*b^3 + 10*
a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(a^2 + a*b)*arctan(1/2*sqrt(a^2 + a*b)*(cosh(d*x + c) + sinh(d*x +
c))/(a + b)) + 4*(10*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x + c)^9 + 4*(6*a^6 + 34*a^5*b + 78*a^4*b^2
+ 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c)^7 + 3*(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 6
0*a*b^5)*cosh(d*x + c)^5 - 2*(4*a^6 + 28*a^5*b + 104*a^4*b^2 + 209*a^3*b^3 + 189*a^2*b^4 + 60*a*b^5)*cosh(d*x
+ c)^3 - (6*a^6 + 34*a^5*b + 78*a^4*b^2 + 75*a^3*b^3 + 25*a^2*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^9 + 3*a^8
*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x + c)^9 + 9*(a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x + c)*sinh(d*x
+ c)^8 + (a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*sinh(d*x + c)^9 + 4*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 +
2*a^5*b^4)*d*cosh(d*x + c)^7 + 4*(9*(a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x + c)^2 + (a^9 + 5*a^8*b
+ 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d)*sinh(d*x + c)^7 + 2*(3*a^9 + 17*a^8*b + 41*a^7*b^2 + 51*a^6*b^3 + 32*a
^5*b^4 + 8*a^4*b^5)*d*cosh(d*x + c)^5 + 28*(3*(a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x + c)^3 + (a^9 +
5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^6 + 2*(63*(a^9 + 3*a^8*b + 3*a^7*
b^2 + a^6*b^3)*d*cosh(d*x + c)^4 + 42*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c)^2 +
(3*a^9 + 17*a^8*b + 41*a^7*b^2 + 51*a^6*b^3 + 32*a^5*b^4 + 8*a^4*b^5)*d)*sinh(d*x + c)^5 + 4*(a^9 + 5*a^8*b +
9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c)^3 + 2*(63*(a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x
+ c)^5 + 70*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c)^3 + 5*(3*a^9 + 17*a^8*b + 41*a
^7*b^2 + 51*a^6*b^3 + 32*a^5*b^4 + 8*a^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(21*(a^9 + 3*a^8*b + 3*a^7*
b^2 + a^6*b^3)*d*cosh(d*x + c)^6 + 35*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c)^4 +
5*(3*a^9 + 17*a^8*b + 41*a^7*b^2 + 51*a^6*b^3 + 32*a^5*b^4 + 8*a^4*b^5)*d*cosh(d*x + c)^2 + (a^9 + 5*a^8*b + 9
*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d)*sinh(d*x + c)^3 + (a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x + c) +
4*(9*(a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x + c)^7 + 21*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*
a^5*b^4)*d*cosh(d*x + c)^5 + 5*(3*a^9 + 17*a^8*b + 41*a^7*b^2 + 51*a^6*b^3 + 32*a^5*b^4 + 8*a^4*b^5)*d*cosh(d*
x + c)^3 + 3*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (9*(a^9 +
3*a^8*b + 3*a^7*b^2 + a^6*b^3)*d*cosh(d*x + c)^8 + 28*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*co
sh(d*x + c)^6 + 10*(3*a^9 + 17*a^8*b + 41*a^7*b^2 + 51*a^6*b^3 + 32*a^5*b^4 + 8*a^4*b^5)*d*cosh(d*x + c)^4 + 1
2*(a^9 + 5*a^8*b + 9*a^7*b^2 + 7*a^6*b^3 + 2*a^5*b^4)*d*cosh(d*x + c)^2 + (a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3
)*d)*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(cosh(d*x+c)/(a+b*sech(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(cosh(d*x+c)/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError