3.367 \(\int \text{sech}^7(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\)

Optimal. Leaf size=205 \[ \frac{a^2 (5 a-6 b) \tan ^{-1}\left (\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{16 f (a-b)^{3/2}}+\frac{\tanh (e+f x) \text{sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 f (a-b)}+\frac{(5 a-6 b) \tanh (e+f x) \text{sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 f (a-b)}+\frac{a (5 a-6 b) \tanh (e+f x) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{16 f (a-b)} \]

[Out]

(a^2*(5*a - 6*b)*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(16*(a - b)^(3/2)*f) + (a*(5
*a - 6*b)*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(16*(a - b)*f) + ((5*a - 6*b)*Sech[e + f*x]
^3*(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x])/(24*(a - b)*f) + (Sech[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(5/2
)*Tanh[e + f*x])/(6*(a - b)*f)

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Rubi [A]  time = 0.172294, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3190, 382, 378, 377, 203} \[ \frac{a^2 (5 a-6 b) \tan ^{-1}\left (\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{16 f (a-b)^{3/2}}+\frac{\tanh (e+f x) \text{sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 f (a-b)}+\frac{(5 a-6 b) \tanh (e+f x) \text{sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 f (a-b)}+\frac{a (5 a-6 b) \tanh (e+f x) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{16 f (a-b)} \]

Antiderivative was successfully verified.

[In]

Int[Sech[e + f*x]^7*(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

(a^2*(5*a - 6*b)*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(16*(a - b)^(3/2)*f) + (a*(5
*a - 6*b)*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(16*(a - b)*f) + ((5*a - 6*b)*Sech[e + f*x]
^3*(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x])/(24*(a - b)*f) + (Sech[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(5/2
)*Tanh[e + f*x])/(6*(a - b)*f)

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 382

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[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 378

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

Rule 377

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

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])

Rubi steps

\begin{align*} \int \text{sech}^7(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{3/2}}{\left (1+x^2\right )^4} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{\text{sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2} \tanh (e+f x)}{6 (a-b) f}+\frac{(5 a-6 b) \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{3/2}}{\left (1+x^2\right )^3} \, dx,x,\sinh (e+f x)\right )}{6 (a-b) f}\\ &=\frac{(5 a-6 b) \text{sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{24 (a-b) f}+\frac{\text{sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2} \tanh (e+f x)}{6 (a-b) f}+\frac{(a (5 a-6 b)) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{\left (1+x^2\right )^2} \, dx,x,\sinh (e+f x)\right )}{8 (a-b) f}\\ &=\frac{a (5 a-6 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{16 (a-b) f}+\frac{(5 a-6 b) \text{sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{24 (a-b) f}+\frac{\text{sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2} \tanh (e+f x)}{6 (a-b) f}+\frac{\left (a^2 (5 a-6 b)\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{16 (a-b) f}\\ &=\frac{a (5 a-6 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{16 (a-b) f}+\frac{(5 a-6 b) \text{sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{24 (a-b) f}+\frac{\text{sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2} \tanh (e+f x)}{6 (a-b) f}+\frac{\left (a^2 (5 a-6 b)\right ) \operatorname{Subst}\left (\int \frac{1}{1-(-a+b) x^2} \, dx,x,\frac{\sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{16 (a-b) f}\\ &=\frac{a^2 (5 a-6 b) \tan ^{-1}\left (\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{16 (a-b)^{3/2} f}+\frac{a (5 a-6 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{16 (a-b) f}+\frac{(5 a-6 b) \text{sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{24 (a-b) f}+\frac{\text{sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2} \tanh (e+f x)}{6 (a-b) f}\\ \end{align*}

Mathematica [C]  time = 15.325, size = 959, normalized size = 4.68 \[ \frac{a^2 \text{sech}^3(e+f x) \left (\frac{b \sinh ^2(e+f x)}{a}+1\right )^2 \tanh (e+f x) \left (256 b \, _2F_1\left (2,5;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^2(e+f x) \sqrt{\frac{\text{sech}^2(e+f x) \left (b \sinh ^2(e+f x)+a\right )}{a}} \left (\frac{(a-b) \tanh ^2(e+f x)}{a}\right )^{9/2}+256 a \, _2F_1\left (2,5;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right ) \sqrt{\frac{\text{sech}^2(e+f x) \left (b \sinh ^2(e+f x)+a\right )}{a}} \left (\frac{(a-b) \tanh ^2(e+f x)}{a}\right )^{9/2}-512 b \, _2F_1\left (2,5;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^2(e+f x) \sqrt{\frac{\text{sech}^2(e+f x) \left (b \sinh ^2(e+f x)+a\right )}{a}} \left (\frac{(a-b) \tanh ^2(e+f x)}{a}\right )^{7/2}-512 a \, _2F_1\left (2,5;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right ) \sqrt{\frac{\text{sech}^2(e+f x) \left (b \sinh ^2(e+f x)+a\right )}{a}} \left (\frac{(a-b) \tanh ^2(e+f x)}{a}\right )^{7/2}-80 b \sinh ^2(e+f x) \sqrt{\frac{\text{sech}^2(e+f x) \left (b \sinh ^2(e+f x)+a\right )}{a}} \left (\frac{(a-b) \tanh ^2(e+f x)}{a}\right )^{5/2}+256 b \, _2F_1\left (2,5;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^2(e+f x) \sqrt{\frac{\text{sech}^2(e+f x) \left (b \sinh ^2(e+f x)+a\right )}{a}} \left (\frac{(a-b) \tanh ^2(e+f x)}{a}\right )^{5/2}-120 a \sqrt{\frac{\text{sech}^2(e+f x) \left (b \sinh ^2(e+f x)+a\right )}{a}} \left (\frac{(a-b) \tanh ^2(e+f x)}{a}\right )^{5/2}+256 a \, _2F_1\left (2,5;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right ) \sqrt{\frac{\text{sech}^2(e+f x) \left (b \sinh ^2(e+f x)+a\right )}{a}} \left (\frac{(a-b) \tanh ^2(e+f x)}{a}\right )^{5/2}+140 b \sinh ^2(e+f x) \sqrt{\frac{\text{sech}^2(e+f x) \left (b \sinh ^2(e+f x)+a\right )}{a}} \left (\frac{(a-b) \tanh ^2(e+f x)}{a}\right )^{3/2}+210 a \sqrt{\frac{\text{sech}^2(e+f x) \left (b \sinh ^2(e+f x)+a\right )}{a}} \left (\frac{(a-b) \tanh ^2(e+f x)}{a}\right )^{3/2}+30 b \sin ^{-1}\left (\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right ) \sinh ^2(e+f x)+45 a \sin ^{-1}\left (\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right )-30 b \sinh ^2(e+f x) \sqrt{\frac{(a-b) \text{sech}^2(e+f x) \left (b \sinh ^2(e+f x)+a\right ) \tanh ^2(e+f x)}{a^2}}-45 a \sqrt{\frac{(a-b) \text{sech}^2(e+f x) \left (b \sinh ^2(e+f x)+a\right ) \tanh ^2(e+f x)}{a^2}}\right )}{240 f \left (b \sinh ^2(e+f x)+a\right )^{3/2} \sqrt{\frac{\text{sech}^2(e+f x) \left (b \sinh ^2(e+f x)+a\right )}{a}} \left (\frac{(a-b) \tanh ^2(e+f x)}{a}\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sech[e + f*x]^7*(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

(a^2*Sech[e + f*x]^3*(1 + (b*Sinh[e + f*x]^2)/a)^2*Tanh[e + f*x]*(45*a*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a
]] + 30*b*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^2 + 210*a*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[
e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2) + 140*b*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[
e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2) - 120*a*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*
(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) + 256*a*Hypergeometric2F1[2, 5, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Se
ch[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) - 80*b*Sinh[e + f*x]^2*Sqrt[(Sec
h[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) + 256*b*Hypergeometric2F1[2, 5, 7
/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*T
anh[e + f*x]^2)/a)^(5/2) - 512*a*Hypergeometric2F1[2, 5, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Sech[e + f*x]
^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2) - 512*b*Hypergeometric2F1[2, 5, 7/2, ((a -
b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*
x]^2)/a)^(7/2) + 256*a*Hypergeometric2F1[2, 5, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Sech[e + f*x]^2*(a + b*
Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(9/2) + 256*b*Hypergeometric2F1[2, 5, 7/2, ((a - b)*Tanh[e
+ f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(
9/2) - 45*a*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2] - 30*b*Sinh[e + f*x]^2
*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2]))/(240*f*(a + b*Sinh[e + f*x]^2)^
(3/2)*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2))

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Maple [C]  time = 0.154, size = 63, normalized size = 0.3 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ({\frac{{b}^{2} \left ( \sinh \left ( fx+e \right ) \right ) ^{4}+2\,ab \left ( \sinh \left ( fx+e \right ) \right ) ^{2}+{a}^{2}}{ \left ( \cosh \left ( fx+e \right ) \right ) ^{8}}{\frac{1}{\sqrt{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}}}},\sinh \left ( fx+e \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(f*x+e)^7*(a+b*sinh(f*x+e)^2)^(3/2),x)

[Out]

`int/indef0`((b^2*sinh(f*x+e)^4+2*a*b*sinh(f*x+e)^2+a^2)/cosh(f*x+e)^8/(a+b*sinh(f*x+e)^2)^(1/2),sinh(f*x+e))/
f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \operatorname{sech}\left (f x + e\right )^{7}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(f*x+e)^7*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*sech(f*x + e)^7, x)

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Fricas [B]  time = 10.1145, size = 19302, normalized size = 94.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(f*x+e)^7*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[-1/96*(3*((5*a^3 - 6*a^2*b)*cosh(f*x + e)^12 + 12*(5*a^3 - 6*a^2*b)*cosh(f*x + e)*sinh(f*x + e)^11 + (5*a^3 -
6*a^2*b)*sinh(f*x + e)^12 + 6*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^10 + 6*(5*a^3 - 6*a^2*b + 11*(5*a^3 - 6*a^2*b)*
cosh(f*x + e)^2)*sinh(f*x + e)^10 + 20*(11*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^3 + 3*(5*a^3 - 6*a^2*b)*cosh(f*x +
e))*sinh(f*x + e)^9 + 15*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^8 + 15*(33*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^4 + 5*a^3
- 6*a^2*b + 18*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 24*(33*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^5 +
30*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^3 + 5*(5*a^3 - 6*a^2*b)*cosh(f*x + e))*sinh(f*x + e)^7 + 20*(5*a^3 - 6*a^2
*b)*cosh(f*x + e)^6 + 4*(231*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^6 + 315*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^4 + 25*a^
3 - 30*a^2*b + 105*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 24*(33*(5*a^3 - 6*a^2*b)*cosh(f*x + e)
^7 + 63*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^5 + 35*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^3 + 5*(5*a^3 - 6*a^2*b)*cosh(f*
x + e))*sinh(f*x + e)^5 + 15*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^4 + 15*(33*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^8 + 84
*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^6 + 70*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^4 + 5*a^3 - 6*a^2*b + 20*(5*a^3 - 6*a^
2*b)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 20*(11*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^9 + 36*(5*a^3 - 6*a^2*b)*cosh(f
*x + e)^7 + 42*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^5 + 20*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^3 + 3*(5*a^3 - 6*a^2*b)*
cosh(f*x + e))*sinh(f*x + e)^3 + 5*a^3 - 6*a^2*b + 6*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^2 + 6*(11*(5*a^3 - 6*a^2*
b)*cosh(f*x + e)^10 + 45*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^8 + 70*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^6 + 50*(5*a^3
- 6*a^2*b)*cosh(f*x + e)^4 + 5*a^3 - 6*a^2*b + 15*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 12*((5*
a^3 - 6*a^2*b)*cosh(f*x + e)^11 + 5*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^9 + 10*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^7 +
10*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^5 + 5*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^3 + (5*a^3 - 6*a^2*b)*cosh(f*x + e))
*sinh(f*x + e))*sqrt(-a + b)*log(((a - 2*b)*cosh(f*x + e)^4 + 4*(a - 2*b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a -
2*b)*sinh(f*x + e)^4 - 2*(3*a - 2*b)*cosh(f*x + e)^2 + 2*(3*(a - 2*b)*cosh(f*x + e)^2 - 3*a + 2*b)*sinh(f*x +
e)^2 - 2*sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 - 1)*sqrt(-a + b)*sqrt((b
*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x +
e)^2)) + 4*((a - 2*b)*cosh(f*x + e)^3 - (3*a - 2*b)*cosh(f*x + e))*sinh(f*x + e) + a - 2*b)/(cosh(f*x + e)^4 +
4*cosh(f*x + e)*sinh(f*x + e)^3 + sinh(f*x + e)^4 + 2*(3*cosh(f*x + e)^2 + 1)*sinh(f*x + e)^2 + 2*cosh(f*x +
e)^2 + 4*(cosh(f*x + e)^3 + cosh(f*x + e))*sinh(f*x + e) + 1)) - 2*sqrt(2)*((15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b
^3)*cosh(f*x + e)^10 + 10*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)*sinh(f*x + e)^9 + (15*a^3 - 23*a
^2*b + 4*a*b^2 + 4*b^3)*sinh(f*x + e)^10 + (85*a^3 - 133*a^2*b + 20*a*b^2 + 28*b^3)*cosh(f*x + e)^8 + (85*a^3
- 133*a^2*b + 20*a*b^2 + 28*b^3 + 45*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^8 +
8*(15*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^3 + (85*a^3 - 133*a^2*b + 20*a*b^2 + 28*b^3)*cosh(f*
x + e))*sinh(f*x + e)^7 + 2*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^6 + 2*(105*(15*a^3 - 23*a^
2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^4 + 99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3 + 14*(85*a^3 - 133*a^2*b + 20
*a*b^2 + 28*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 4*(63*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^
5 + 14*(85*a^3 - 133*a^2*b + 20*a*b^2 + 28*b^3)*cosh(f*x + e)^3 + 3*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*
cosh(f*x + e))*sinh(f*x + e)^5 - 2*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^4 + 2*(105*(15*a^3
- 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^6 + 35*(85*a^3 - 133*a^2*b + 20*a*b^2 + 28*b^3)*cosh(f*x + e)^4 -
99*a^3 + 247*a^2*b - 200*a*b^2 + 52*b^3 + 15*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^2)*sinh(f
*x + e)^4 + 8*(15*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^7 + 7*(85*a^3 - 133*a^2*b + 20*a*b^2 + 2
8*b^3)*cosh(f*x + e)^5 + 5*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^3 - (99*a^3 - 247*a^2*b + 2
00*a*b^2 - 52*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 - 15*a^3 + 23*a^2*b - 4*a*b^2 - 4*b^3 - (85*a^3 - 133*a^2*b
+ 20*a*b^2 + 28*b^3)*cosh(f*x + e)^2 + (45*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^8 + 28*(85*a^3
- 133*a^2*b + 20*a*b^2 + 28*b^3)*cosh(f*x + e)^6 + 30*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^
4 - 85*a^3 + 133*a^2*b - 20*a*b^2 - 28*b^3 - 12*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^2)*sin
h(f*x + e)^2 + 2*(5*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^9 + 4*(85*a^3 - 133*a^2*b + 20*a*b^2 +
28*b^3)*cosh(f*x + e)^7 + 6*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^5 - 4*(99*a^3 - 247*a^2*b
+ 200*a*b^2 - 52*b^3)*cosh(f*x + e)^3 - (85*a^3 - 133*a^2*b + 20*a*b^2 + 28*b^3)*cosh(f*x + e))*sinh(f*x + e)
)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + si
nh(f*x + e)^2)))/((a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^12 + 12*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)*sinh(f*x + e
)^11 + (a^2 - 2*a*b + b^2)*f*sinh(f*x + e)^12 + 6*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^10 + 6*(11*(a^2 - 2*a*b
+ b^2)*f*cosh(f*x + e)^2 + (a^2 - 2*a*b + b^2)*f)*sinh(f*x + e)^10 + 15*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^8
+ 20*(11*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^3 + 3*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e))*sinh(f*x + e)^9 + 15*(
33*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 18*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^2 + (a^2 - 2*a*b + b^2)*f)*s
inh(f*x + e)^8 + 20*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^6 + 24*(33*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^5 + 30*
(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^3 + 5*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e))*sinh(f*x + e)^7 + 4*(231*(a^2 -
2*a*b + b^2)*f*cosh(f*x + e)^6 + 315*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 105*(a^2 - 2*a*b + b^2)*f*cosh(f
*x + e)^2 + 5*(a^2 - 2*a*b + b^2)*f)*sinh(f*x + e)^6 + 15*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 24*(33*(a^2
- 2*a*b + b^2)*f*cosh(f*x + e)^7 + 63*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^5 + 35*(a^2 - 2*a*b + b^2)*f*cosh(f*
x + e)^3 + 5*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e))*sinh(f*x + e)^5 + 15*(33*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)
^8 + 84*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^6 + 70*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 20*(a^2 - 2*a*b + b
^2)*f*cosh(f*x + e)^2 + (a^2 - 2*a*b + b^2)*f)*sinh(f*x + e)^4 + 6*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^2 + 20*
(11*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^9 + 36*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^7 + 42*(a^2 - 2*a*b + b^2)*
f*cosh(f*x + e)^5 + 20*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^3 + 3*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e))*sinh(f*x
+ e)^3 + 6*(11*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^10 + 45*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^8 + 70*(a^2 -
2*a*b + b^2)*f*cosh(f*x + e)^6 + 50*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 15*(a^2 - 2*a*b + b^2)*f*cosh(f*x
+ e)^2 + (a^2 - 2*a*b + b^2)*f)*sinh(f*x + e)^2 + (a^2 - 2*a*b + b^2)*f + 12*((a^2 - 2*a*b + b^2)*f*cosh(f*x +
e)^11 + 5*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^9 + 10*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^7 + 10*(a^2 - 2*a*b
+ b^2)*f*cosh(f*x + e)^5 + 5*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^3 + (a^2 - 2*a*b + b^2)*f*cosh(f*x + e))*sinh
(f*x + e)), 1/48*(3*((5*a^3 - 6*a^2*b)*cosh(f*x + e)^12 + 12*(5*a^3 - 6*a^2*b)*cosh(f*x + e)*sinh(f*x + e)^11
+ (5*a^3 - 6*a^2*b)*sinh(f*x + e)^12 + 6*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^10 + 6*(5*a^3 - 6*a^2*b + 11*(5*a^3 -
6*a^2*b)*cosh(f*x + e)^2)*sinh(f*x + e)^10 + 20*(11*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^3 + 3*(5*a^3 - 6*a^2*b)*c
osh(f*x + e))*sinh(f*x + e)^9 + 15*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^8 + 15*(33*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^
4 + 5*a^3 - 6*a^2*b + 18*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 24*(33*(5*a^3 - 6*a^2*b)*cosh(f*
x + e)^5 + 30*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^3 + 5*(5*a^3 - 6*a^2*b)*cosh(f*x + e))*sinh(f*x + e)^7 + 20*(5*a
^3 - 6*a^2*b)*cosh(f*x + e)^6 + 4*(231*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^6 + 315*(5*a^3 - 6*a^2*b)*cosh(f*x + e)
^4 + 25*a^3 - 30*a^2*b + 105*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 24*(33*(5*a^3 - 6*a^2*b)*cos
h(f*x + e)^7 + 63*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^5 + 35*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^3 + 5*(5*a^3 - 6*a^2*
b)*cosh(f*x + e))*sinh(f*x + e)^5 + 15*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^4 + 15*(33*(5*a^3 - 6*a^2*b)*cosh(f*x +
e)^8 + 84*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^6 + 70*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^4 + 5*a^3 - 6*a^2*b + 20*(5*
a^3 - 6*a^2*b)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 20*(11*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^9 + 36*(5*a^3 - 6*a^2
*b)*cosh(f*x + e)^7 + 42*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^5 + 20*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^3 + 3*(5*a^3 -
6*a^2*b)*cosh(f*x + e))*sinh(f*x + e)^3 + 5*a^3 - 6*a^2*b + 6*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^2 + 6*(11*(5*a^
3 - 6*a^2*b)*cosh(f*x + e)^10 + 45*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^8 + 70*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^6 +
50*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^4 + 5*a^3 - 6*a^2*b + 15*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^2)*sinh(f*x + e)^2
+ 12*((5*a^3 - 6*a^2*b)*cosh(f*x + e)^11 + 5*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^9 + 10*(5*a^3 - 6*a^2*b)*cosh(f*
x + e)^7 + 10*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^5 + 5*(5*a^3 - 6*a^2*b)*cosh(f*x + e)^3 + (5*a^3 - 6*a^2*b)*cosh
(f*x + e))*sinh(f*x + e))*sqrt(a - b)*arctan(sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f
*x + e)^2 - 1)*sqrt(a - b)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*
x + e)*sinh(f*x + e) + sinh(f*x + e)^2))/(b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sinh(f*x + e)^3 + b*sinh(f*x +
e)^4 + 2*(2*a - b)*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + 2*a - b)*sinh(f*x + e)^2 + 4*(b*cosh(f*x + e)^3
+ (2*a - b)*cosh(f*x + e))*sinh(f*x + e) + b)) + sqrt(2)*((15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)
^10 + 10*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)*sinh(f*x + e)^9 + (15*a^3 - 23*a^2*b + 4*a*b^2 +
4*b^3)*sinh(f*x + e)^10 + (85*a^3 - 133*a^2*b + 20*a*b^2 + 28*b^3)*cosh(f*x + e)^8 + (85*a^3 - 133*a^2*b + 20*
a*b^2 + 28*b^3 + 45*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 8*(15*(15*a^3 - 2
3*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^3 + (85*a^3 - 133*a^2*b + 20*a*b^2 + 28*b^3)*cosh(f*x + e))*sinh(f*x
+ e)^7 + 2*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^6 + 2*(105*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4
*b^3)*cosh(f*x + e)^4 + 99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3 + 14*(85*a^3 - 133*a^2*b + 20*a*b^2 + 28*b^3)*
cosh(f*x + e)^2)*sinh(f*x + e)^6 + 4*(63*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^5 + 14*(85*a^3 -
133*a^2*b + 20*a*b^2 + 28*b^3)*cosh(f*x + e)^3 + 3*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e))*si
nh(f*x + e)^5 - 2*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^4 + 2*(105*(15*a^3 - 23*a^2*b + 4*a*
b^2 + 4*b^3)*cosh(f*x + e)^6 + 35*(85*a^3 - 133*a^2*b + 20*a*b^2 + 28*b^3)*cosh(f*x + e)^4 - 99*a^3 + 247*a^2*
b - 200*a*b^2 + 52*b^3 + 15*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(15
*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^7 + 7*(85*a^3 - 133*a^2*b + 20*a*b^2 + 28*b^3)*cosh(f*x +
e)^5 + 5*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^3 - (99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3
)*cosh(f*x + e))*sinh(f*x + e)^3 - 15*a^3 + 23*a^2*b - 4*a*b^2 - 4*b^3 - (85*a^3 - 133*a^2*b + 20*a*b^2 + 28*b
^3)*cosh(f*x + e)^2 + (45*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^8 + 28*(85*a^3 - 133*a^2*b + 20*
a*b^2 + 28*b^3)*cosh(f*x + e)^6 + 30*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^4 - 85*a^3 + 133*
a^2*b - 20*a*b^2 - 28*b^3 - 12*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*
(5*(15*a^3 - 23*a^2*b + 4*a*b^2 + 4*b^3)*cosh(f*x + e)^9 + 4*(85*a^3 - 133*a^2*b + 20*a*b^2 + 28*b^3)*cosh(f*x
+ e)^7 + 6*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52*b^3)*cosh(f*x + e)^5 - 4*(99*a^3 - 247*a^2*b + 200*a*b^2 - 52
*b^3)*cosh(f*x + e)^3 - (85*a^3 - 133*a^2*b + 20*a*b^2 + 28*b^3)*cosh(f*x + e))*sinh(f*x + e))*sqrt((b*cosh(f*
x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/
((a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^12 + 12*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)*sinh(f*x + e)^11 + (a^2 - 2*a
*b + b^2)*f*sinh(f*x + e)^12 + 6*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^10 + 6*(11*(a^2 - 2*a*b + b^2)*f*cosh(f*x
+ e)^2 + (a^2 - 2*a*b + b^2)*f)*sinh(f*x + e)^10 + 15*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^8 + 20*(11*(a^2 - 2
*a*b + b^2)*f*cosh(f*x + e)^3 + 3*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e))*sinh(f*x + e)^9 + 15*(33*(a^2 - 2*a*b +
b^2)*f*cosh(f*x + e)^4 + 18*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^2 + (a^2 - 2*a*b + b^2)*f)*sinh(f*x + e)^8 +
20*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^6 + 24*(33*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^5 + 30*(a^2 - 2*a*b + b^
2)*f*cosh(f*x + e)^3 + 5*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e))*sinh(f*x + e)^7 + 4*(231*(a^2 - 2*a*b + b^2)*f*c
osh(f*x + e)^6 + 315*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 105*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^2 + 5*(a^
2 - 2*a*b + b^2)*f)*sinh(f*x + e)^6 + 15*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 24*(33*(a^2 - 2*a*b + b^2)*f*
cosh(f*x + e)^7 + 63*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^5 + 35*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^3 + 5*(a^2
- 2*a*b + b^2)*f*cosh(f*x + e))*sinh(f*x + e)^5 + 15*(33*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^8 + 84*(a^2 - 2*
a*b + b^2)*f*cosh(f*x + e)^6 + 70*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 20*(a^2 - 2*a*b + b^2)*f*cosh(f*x +
e)^2 + (a^2 - 2*a*b + b^2)*f)*sinh(f*x + e)^4 + 6*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^2 + 20*(11*(a^2 - 2*a*b
+ b^2)*f*cosh(f*x + e)^9 + 36*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^7 + 42*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^5
+ 20*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^3 + 3*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e))*sinh(f*x + e)^3 + 6*(11*(
a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^10 + 45*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^8 + 70*(a^2 - 2*a*b + b^2)*f*co
sh(f*x + e)^6 + 50*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 15*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^2 + (a^2 - 2
*a*b + b^2)*f)*sinh(f*x + e)^2 + (a^2 - 2*a*b + b^2)*f + 12*((a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^11 + 5*(a^2 -
2*a*b + b^2)*f*cosh(f*x + e)^9 + 10*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^7 + 10*(a^2 - 2*a*b + b^2)*f*cosh(f*x
+ e)^5 + 5*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^3 + (a^2 - 2*a*b + b^2)*f*cosh(f*x + e))*sinh(f*x + e))]

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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(f*x+e)**7*(a+b*sinh(f*x+e)**2)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \operatorname{sech}\left (f x + e\right )^{7}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(f*x+e)^7*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*sech(f*x + e)^7, x)