\(\int (c+d x)^2 \text {sech}^3(a+b x) \, dx\) [37]

Optimal result

Integrand size = 16, antiderivative size = 175 \[ \int (c+d x)^2 \text {sech}^3(a+b x) \, dx=\frac {(c+d x)^2 \arctan \left (e^{a+b x}\right )}{b}-\frac {d^2 \arctan (\sinh (a+b x))}{b^3}-\frac {i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}+\frac {i d (c+d x) \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {i d^2 \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}-\frac {i d^2 \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \text {sech}(a+b x) \tanh (a+b x)}{2 b} \] Output:

(d*x+c)^2*arctan(exp(b*x+a))/b-d^2*arctan(sinh(b*x+a))/b^3-I*d*(d*x+c)*pol 
ylog(2,-I*exp(b*x+a))/b^2+I*d*(d*x+c)*polylog(2,I*exp(b*x+a))/b^2+I*d^2*po 
lylog(3,-I*exp(b*x+a))/b^3-I*d^2*polylog(3,I*exp(b*x+a))/b^3+d*(d*x+c)*sec 
h(b*x+a)/b^2+1/2*(d*x+c)^2*sech(b*x+a)*tanh(b*x+a)/b
 

Mathematica [A] (verified)

Time = 1.67 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.54 \[ \int (c+d x)^2 \text {sech}^3(a+b x) \, dx=\frac {i \left (-2 i b^2 c^2 \arctan \left (e^{a+b x}\right )+4 i d^2 \arctan \left (e^{a+b x}\right )+2 b^2 c d x \log \left (1-i e^{a+b x}\right )+b^2 d^2 x^2 \log \left (1-i e^{a+b x}\right )-2 b^2 c d x \log \left (1+i e^{a+b x}\right )-b^2 d^2 x^2 \log \left (1+i e^{a+b x}\right )-2 b d (c+d x) \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )+2 b d (c+d x) \operatorname {PolyLog}\left (2,i e^{a+b x}\right )+2 d^2 \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )-2 d^2 \operatorname {PolyLog}\left (3,i e^{a+b x}\right )\right )+b^2 (c+d x)^2 \text {sech}(a) \text {sech}^2(a+b x) \sinh (b x)+b (c+d x) \text {sech}(a+b x) (2 d+b (c+d x) \tanh (a))}{2 b^3} \] Input:

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

Output:

(I*((-2*I)*b^2*c^2*ArcTan[E^(a + b*x)] + (4*I)*d^2*ArcTan[E^(a + b*x)] + 2 
*b^2*c*d*x*Log[1 - I*E^(a + b*x)] + b^2*d^2*x^2*Log[1 - I*E^(a + b*x)] - 2 
*b^2*c*d*x*Log[1 + I*E^(a + b*x)] - b^2*d^2*x^2*Log[1 + I*E^(a + b*x)] - 2 
*b*d*(c + d*x)*PolyLog[2, (-I)*E^(a + b*x)] + 2*b*d*(c + d*x)*PolyLog[2, I 
*E^(a + b*x)] + 2*d^2*PolyLog[3, (-I)*E^(a + b*x)] - 2*d^2*PolyLog[3, I*E^ 
(a + b*x)]) + b^2*(c + d*x)^2*Sech[a]*Sech[a + b*x]^2*Sinh[b*x] + b*(c + d 
*x)*Sech[a + b*x]*(2*d + b*(c + d*x)*Tanh[a]))/(2*b^3)
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4674, 3042, 4257, 4668, 3011, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^2 \csc \left (i a+i b x+\frac {\pi }{2}\right )^3dx\)

\(\Big \downarrow \) 4674

\(\displaystyle -\frac {d^2 \int \text {sech}(a+b x)dx}{b^2}+\frac {1}{2} \int (c+d x)^2 \text {sech}(a+b x)dx+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {d^2 \int \csc \left (i a+i b x+\frac {\pi }{2}\right )dx}{b^2}+\frac {1}{2} \int (c+d x)^2 \csc \left (i a+i b x+\frac {\pi }{2}\right )dx+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {1}{2} \int (c+d x)^2 \csc \left (i a+i b x+\frac {\pi }{2}\right )dx-\frac {d^2 \arctan (\sinh (a+b x))}{b^3}+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b}\)

\(\Big \downarrow \) 4668

\(\displaystyle \frac {1}{2} \left (-\frac {2 i d \int (c+d x) \log \left (1-i e^{a+b x}\right )dx}{b}+\frac {2 i d \int (c+d x) \log \left (1+i e^{a+b x}\right )dx}{b}+\frac {2 (c+d x)^2 \arctan \left (e^{a+b x}\right )}{b}\right )-\frac {d^2 \arctan (\sinh (a+b x))}{b^3}+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {1}{2} \left (\frac {2 i d \left (\frac {d \int \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )dx}{b}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b}\right )}{b}-\frac {2 i d \left (\frac {d \int \operatorname {PolyLog}\left (2,i e^{a+b x}\right )dx}{b}-\frac {(c+d x) \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b}\right )}{b}+\frac {2 (c+d x)^2 \arctan \left (e^{a+b x}\right )}{b}\right )-\frac {d^2 \arctan (\sinh (a+b x))}{b^3}+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {1}{2} \left (\frac {2 i d \left (\frac {d \int e^{-a-b x} \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b}\right )}{b}-\frac {2 i d \left (\frac {d \int e^{-a-b x} \operatorname {PolyLog}\left (2,i e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b}\right )}{b}+\frac {2 (c+d x)^2 \arctan \left (e^{a+b x}\right )}{b}\right )-\frac {d^2 \arctan (\sinh (a+b x))}{b^3}+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {d^2 \arctan (\sinh (a+b x))}{b^3}+\frac {1}{2} \left (\frac {2 (c+d x)^2 \arctan \left (e^{a+b x}\right )}{b}+\frac {2 i d \left (\frac {d \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b}\right )}{b}-\frac {2 i d \left (\frac {d \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b}\right )}{b}\right )+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b}\)

Input:

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

Output:

-((d^2*ArcTan[Sinh[a + b*x]])/b^3) + ((2*(c + d*x)^2*ArcTan[E^(a + b*x)])/ 
b + ((2*I)*d*(-(((c + d*x)*PolyLog[2, (-I)*E^(a + b*x)])/b) + (d*PolyLog[3 
, (-I)*E^(a + b*x)])/b^2))/b - ((2*I)*d*(-(((c + d*x)*PolyLog[2, I*E^(a + 
b*x)])/b) + (d*PolyLog[3, I*E^(a + b*x)])/b^2))/b)/2 + (d*(c + d*x)*Sech[a 
 + b*x])/b^2 + ((c + d*x)^2*Sech[a + b*x]*Tanh[a + b*x])/(2*b)
 

Defintions of rubi rules used

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

rule 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 4674

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo 
l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n 
- 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 
2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) 
Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ 
(n - 1))   Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c 
, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int \left (d x +c \right )^{2} \operatorname {sech}\left (b x +a \right )^{3}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2651 vs. \(2 (150) = 300\).

Time = 0.16 (sec) , antiderivative size = 2651, normalized size of antiderivative = 15.15 \[ \int (c+d x)^2 \text {sech}^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(2*(b^2*d^2*x^2 + b^2*c^2 + 2*b*c*d + 2*(b^2*c*d + b*d^2)*x)*cosh(b*x 
+ a)^3 + 6*(b^2*d^2*x^2 + b^2*c^2 + 2*b*c*d + 2*(b^2*c*d + b*d^2)*x)*cosh( 
b*x + a)*sinh(b*x + a)^2 + 2*(b^2*d^2*x^2 + b^2*c^2 + 2*b*c*d + 2*(b^2*c*d 
 + b*d^2)*x)*sinh(b*x + a)^3 - 2*(b^2*d^2*x^2 + b^2*c^2 - 2*b*c*d + 2*(b^2 
*c*d - b*d^2)*x)*cosh(b*x + a) - 2*((-I*b*d^2*x - I*b*c*d)*cosh(b*x + a)^4 
 + 4*(-I*b*d^2*x - I*b*c*d)*cosh(b*x + a)*sinh(b*x + a)^3 + (-I*b*d^2*x - 
I*b*c*d)*sinh(b*x + a)^4 - I*b*d^2*x - I*b*c*d + 2*(-I*b*d^2*x - I*b*c*d)* 
cosh(b*x + a)^2 + 2*(-I*b*d^2*x - I*b*c*d + 3*(-I*b*d^2*x - I*b*c*d)*cosh( 
b*x + a)^2)*sinh(b*x + a)^2 + 4*((-I*b*d^2*x - I*b*c*d)*cosh(b*x + a)^3 + 
(-I*b*d^2*x - I*b*c*d)*cosh(b*x + a))*sinh(b*x + a))*dilog(I*cosh(b*x + a) 
 + I*sinh(b*x + a)) - 2*((I*b*d^2*x + I*b*c*d)*cosh(b*x + a)^4 + 4*(I*b*d^ 
2*x + I*b*c*d)*cosh(b*x + a)*sinh(b*x + a)^3 + (I*b*d^2*x + I*b*c*d)*sinh( 
b*x + a)^4 + I*b*d^2*x + I*b*c*d + 2*(I*b*d^2*x + I*b*c*d)*cosh(b*x + a)^2 
 + 2*(I*b*d^2*x + I*b*c*d + 3*(I*b*d^2*x + I*b*c*d)*cosh(b*x + a)^2)*sinh( 
b*x + a)^2 + 4*((I*b*d^2*x + I*b*c*d)*cosh(b*x + a)^3 + (I*b*d^2*x + I*b*c 
*d)*cosh(b*x + a))*sinh(b*x + a))*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a) 
) + ((I*b^2*c^2 - 2*I*a*b*c*d + I*(a^2 - 2)*d^2)*cosh(b*x + a)^4 - 4*(-I*b 
^2*c^2 + 2*I*a*b*c*d - I*(a^2 - 2)*d^2)*cosh(b*x + a)*sinh(b*x + a)^3 + (I 
*b^2*c^2 - 2*I*a*b*c*d + I*(a^2 - 2)*d^2)*sinh(b*x + a)^4 + I*b^2*c^2 - 2* 
I*a*b*c*d + I*(a^2 - 2)*d^2 - 2*(-I*b^2*c^2 + 2*I*a*b*c*d - I*(a^2 - 2)...
 

Sympy [F]

\[ \int (c+d x)^2 \text {sech}^3(a+b x) \, dx=\int \left (c + d x\right )^{2} \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \] Input:

integrate((d*x+c)**2*sech(b*x+a)**3,x)
 

Output:

Integral((c + d*x)**2*sech(a + b*x)**3, x)
 

Maxima [F]

\[ \int (c+d x)^2 \text {sech}^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \operatorname {sech}\left (b x + a\right )^{3} \,d x } \] Input:

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

Output:

b^2*d^2*integrate(x^2*e^(b*x + a)/(b^2*e^(2*b*x + 2*a) + b^2), x) + 2*b^2* 
c*d*integrate(x*e^(b*x + a)/(b^2*e^(2*b*x + 2*a) + b^2), x) - c^2*(arctan( 
e^(-b*x - a))/b - (e^(-b*x - a) - e^(-3*b*x - 3*a))/(b*(2*e^(-2*b*x - 2*a) 
 + e^(-4*b*x - 4*a) + 1))) + ((b*d^2*x^2*e^(3*a) + 2*c*d*e^(3*a) + 2*(b*c* 
d + d^2)*x*e^(3*a))*e^(3*b*x) - (b*d^2*x^2*e^a - 2*c*d*e^a + 2*(b*c*d - d^ 
2)*x*e^a)*e^(b*x))/(b^2*e^(4*b*x + 4*a) + 2*b^2*e^(2*b*x + 2*a) + b^2) - 2 
*d^2*arctan(e^(b*x + a))/b^3
                                                                                    
                                                                                    
 

Giac [F]

\[ \int (c+d x)^2 \text {sech}^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \operatorname {sech}\left (b x + a\right )^{3} \,d x } \] Input:

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

Output:

integrate((d*x + c)^2*sech(b*x + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^2 \text {sech}^3(a+b x) \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\mathrm {cosh}\left (a+b\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (c+d x)^2 \text {sech}^3(a+b x) \, dx=\frac {48 e^{4 b x +5 a} \left (\int \frac {e^{b x} x}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{3} c d +96 e^{2 b x +3 a} \left (\int \frac {e^{b x} x}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{3} c d +48 e^{a} \left (\int \frac {e^{b x} x}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{3} c d +8 \mathit {atan} \left (e^{b x +a}\right ) d^{2}+8 e^{3 b x +3 a} d^{2}+8 e^{b x +a} d^{2}+24 e^{4 b x +4 a} \mathit {atan} \left (e^{b x +a}\right ) b c d +48 e^{2 b x +2 a} \mathit {atan} \left (e^{b x +a}\right ) b c d -48 e^{b x +a} b^{2} c d x +8 e^{4 b x +4 a} \mathit {atan} \left (e^{b x +a}\right ) d^{2}+16 e^{2 b x +2 a} \mathit {atan} \left (e^{b x +a}\right ) d^{2}+9 \mathit {atan} \left (e^{b x +a}\right ) b^{2} c^{2}+9 e^{3 b x +3 a} b^{2} c^{2}-9 e^{b x +a} b^{2} c^{2}+9 e^{4 b x +4 a} \mathit {atan} \left (e^{b x +a}\right ) b^{2} c^{2}+18 e^{2 b x +2 a} \mathit {atan} \left (e^{b x +a}\right ) b^{2} c^{2}+24 \mathit {atan} \left (e^{b x +a}\right ) b c d +24 e^{3 b x +3 a} b c d -24 e^{b x +a} b^{2} d^{2} x^{2}+24 e^{b x +a} b c d -16 e^{b x +a} b \,d^{2} x +24 e^{4 b x +5 a} \left (\int \frac {e^{b x} x^{2}}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{3} d^{2}+64 e^{4 b x +5 a} \left (\int \frac {e^{b x} x}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{2} d^{2}+48 e^{2 b x +3 a} \left (\int \frac {e^{b x} x^{2}}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{3} d^{2}+128 e^{2 b x +3 a} \left (\int \frac {e^{b x} x}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{2} d^{2}+24 e^{a} \left (\int \frac {e^{b x} x^{2}}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{3} d^{2}+64 e^{a} \left (\int \frac {e^{b x} x}{e^{6 b x +6 a}+3 e^{4 b x +4 a}+3 e^{2 b x +2 a}+1}d x \right ) b^{2} d^{2}}{9 b^{3} \left (e^{4 b x +4 a}+2 e^{2 b x +2 a}+1\right )} \] Input:

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

Output:

(9*e**(4*a + 4*b*x)*atan(e**(a + b*x))*b**2*c**2 + 24*e**(4*a + 4*b*x)*ata 
n(e**(a + b*x))*b*c*d + 8*e**(4*a + 4*b*x)*atan(e**(a + b*x))*d**2 + 18*e* 
*(2*a + 2*b*x)*atan(e**(a + b*x))*b**2*c**2 + 48*e**(2*a + 2*b*x)*atan(e** 
(a + b*x))*b*c*d + 16*e**(2*a + 2*b*x)*atan(e**(a + b*x))*d**2 + 9*atan(e* 
*(a + b*x))*b**2*c**2 + 24*atan(e**(a + b*x))*b*c*d + 8*atan(e**(a + b*x)) 
*d**2 + 24*e**(5*a + 4*b*x)*int((e**(b*x)*x**2)/(e**(6*a + 6*b*x) + 3*e**( 
4*a + 4*b*x) + 3*e**(2*a + 2*b*x) + 1),x)*b**3*d**2 + 48*e**(5*a + 4*b*x)* 
int((e**(b*x)*x)/(e**(6*a + 6*b*x) + 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b* 
x) + 1),x)*b**3*c*d + 64*e**(5*a + 4*b*x)*int((e**(b*x)*x)/(e**(6*a + 6*b* 
x) + 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) + 1),x)*b**2*d**2 + 9*e**(3*a 
 + 3*b*x)*b**2*c**2 + 24*e**(3*a + 3*b*x)*b*c*d + 8*e**(3*a + 3*b*x)*d**2 
+ 48*e**(3*a + 2*b*x)*int((e**(b*x)*x**2)/(e**(6*a + 6*b*x) + 3*e**(4*a + 
4*b*x) + 3*e**(2*a + 2*b*x) + 1),x)*b**3*d**2 + 96*e**(3*a + 2*b*x)*int((e 
**(b*x)*x)/(e**(6*a + 6*b*x) + 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) + 1 
),x)*b**3*c*d + 128*e**(3*a + 2*b*x)*int((e**(b*x)*x)/(e**(6*a + 6*b*x) + 
3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) + 1),x)*b**2*d**2 - 9*e**(a + b*x) 
*b**2*c**2 - 48*e**(a + b*x)*b**2*c*d*x - 24*e**(a + b*x)*b**2*d**2*x**2 + 
 24*e**(a + b*x)*b*c*d - 16*e**(a + b*x)*b*d**2*x + 8*e**(a + b*x)*d**2 + 
24*e**a*int((e**(b*x)*x**2)/(e**(6*a + 6*b*x) + 3*e**(4*a + 4*b*x) + 3*e** 
(2*a + 2*b*x) + 1),x)*b**3*d**2 + 48*e**a*int((e**(b*x)*x)/(e**(6*a + 6...