\(\int \frac {x}{(a+b \text {sech}(c+d \sqrt {x}))^2} \, dx\) [56]

Optimal result

Integrand size = 18, antiderivative size = 1395 \[ \int \frac {x}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \] Output:

2*b^2*x^(3/2)*sinh(c+d*x^(1/2))/a/(a^2-b^2)/d/(b+a*cosh(c+d*x^(1/2)))-4*b* 
x^(3/2)*ln(1+a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2) 
/d+12*b*x*polylog(2,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^ 
2)^(1/2)/d^2-12*b*x*polylog(2,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^ 
2/(-a^2+b^2)^(1/2)/d^2+4*b*x^(3/2)*ln(1+a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^( 
1/2)))/a^2/(-a^2+b^2)^(1/2)/d-24*b*x^(1/2)*polylog(3,-a*exp(c+d*x^(1/2))/( 
b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^3+24*b*x^(1/2)*polylog(3,-a*ex 
p(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^3+12*b^3*x^(1/ 
2)*polylog(3,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2 
)/d^3-12*b^3*x^(1/2)*polylog(3,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a 
^2/(-a^2+b^2)^(3/2)/d^3-12*b^2*x^(1/2)*polylog(2,-a*exp(c+d*x^(1/2))/(b+(- 
a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^3-12*b^2*x^(1/2)*polylog(2,-a*exp(c+d*x^( 
1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^3-6*b^3*x*polylog(2,-a*exp(c+d 
*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2+6*b^3*x*polylog(2 
,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2-6*b^2* 
x*ln(1+a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^2-2*b^3*x^ 
(3/2)*ln(1+a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d 
-6*b^2*x*ln(1+a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^2+2 
*b^3*x^(3/2)*ln(1+a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^ 
(3/2)/d-12*b^3*polylog(4,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/...
 

Mathematica [A] (verified)

Time = 8.29 (sec) , antiderivative size = 1393, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((b + a*Cosh[c + d*Sqrt[x]])*Sech[c + d*Sqrt[x]]^2*(x^2*(b + a*Cosh[c + d* 
Sqrt[x]]) + (4*b*E^c*(b + a*Cosh[c + d*Sqrt[x]])*(2*b*E^c*x^(3/2) + ((1 + 
E^(2*c))*(-3*b*d^2*Sqrt[(-a^2 + b^2)*E^(2*c)]*x*Log[1 + (a*E^(2*c + d*Sqrt 
[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])] - 2*a^2*d^3*E^c*x^(3/2)*Log[1 
+ (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])] + b^2*d^3* 
E^c*x^(3/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^( 
2*c)])] - 3*b*d^2*Sqrt[(-a^2 + b^2)*E^(2*c)]*x*Log[1 + (a*E^(2*c + d*Sqrt[ 
x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])] + 2*a^2*d^3*E^c*x^(3/2)*Log[1 + 
 (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])] - b^2*d^3*E 
^c*x^(3/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2 
*c)])] + 3*d*(-2*b*Sqrt[(-a^2 + b^2)*E^(2*c)] - 2*a^2*d*E^c*Sqrt[x] + b^2* 
d*E^c*Sqrt[x])*Sqrt[x]*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[ 
(-a^2 + b^2)*E^(2*c)]))] - 3*d*(2*b*Sqrt[(-a^2 + b^2)*E^(2*c)] - 2*a^2*d*E 
^c*Sqrt[x] + b^2*d*E^c*Sqrt[x])*Sqrt[x]*PolyLog[2, -((a*E^(2*c + d*Sqrt[x] 
))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 6*b*Sqrt[(-a^2 + b^2)*E^(2*c)] 
*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]) 
)] + 12*a^2*d*E^c*Sqrt[x]*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sq 
rt[(-a^2 + b^2)*E^(2*c)]))] - 6*b^2*d*E^c*Sqrt[x]*PolyLog[3, -((a*E^(2*c + 
 d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 6*b*Sqrt[(-a^2 + b^2 
)*E^(2*c)]*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b...
 

Rubi [A] (verified)

Time = 2.90 (sec) , antiderivative size = 1394, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5959, 3042, 4679, 2009}

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.

\(\displaystyle \int \frac {x}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx\)

\(\Big \downarrow \) 5959

\(\displaystyle 2 \int \frac {x^{3/2}}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

\(\displaystyle 2 \int \left (\frac {x^{3/2} b^2}{a^2 \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )^2}-\frac {2 x^{3/2} b}{a^2 \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}+\frac {x^{3/2}}{a^2}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {3 x \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {3 x \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {6 \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {6 \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {6 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}-\frac {6 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}+\frac {x^{3/2} b^2}{a^2 \left (a^2-b^2\right ) d}-\frac {3 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {b^2-a^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac {3 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {b^2-a^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac {6 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}-\frac {6 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}+\frac {6 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {6 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {x^{3/2} \sinh \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}-\frac {2 x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} d}+\frac {2 x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} d}-\frac {6 x \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}+\frac {6 x \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}+\frac {12 \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {12 \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {12 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^4}+\frac {12 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^4}+\frac {x^2}{4 a^2}\right )\)

Input:

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

Output:

2*((b^2*x^(3/2))/(a^2*(a^2 - b^2)*d) + x^2/(4*a^2) - (3*b^2*x*Log[1 + (a*E 
^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) + (b^3*x^ 
(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + 
b^2)^(3/2)*d) - (2*b*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 
+ b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (3*b^2*x*Log[1 + (a*E^(c + d*Sqrt[x]) 
)/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) - (b^3*x^(3/2)*Log[1 + (a 
*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + 
(2*b*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*S 
qrt[-a^2 + b^2]*d) - (6*b^2*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b 
- Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (3*b^3*x*PolyLog[2, -((a*E^ 
(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - 
(6*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*S 
qrt[-a^2 + b^2]*d^2) - (6*b^2*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/( 
b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) - (3*b^3*x*PolyLog[2, -((a* 
E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) 
+ (6*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2 
*Sqrt[-a^2 + b^2]*d^2) + (6*b^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sq 
rt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) - (6*b^3*Sqrt[x]*PolyLog[3, -((a* 
E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) 
+ (12*b*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4679

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Si 
n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt 
Q[m, 0]
 

rule 5959

Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbo 
l] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sech[c + d*x] 
)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m 
 + 1)/n], 0] && IntegerQ[p]
 

Maple [F]

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

Input:

int(x/(a+b*sech(c+d*x^(1/2)))^2,x)
 

Output:

int(x/(a+b*sech(c+d*x^(1/2)))^2,x)
 

Fricas [F]

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

integrate(x/(a+b*sech(c+d*x^(1/2)))^2,x, algorithm="fricas")
 

Output:

integral(x/(b^2*sech(d*sqrt(x) + c)^2 + 2*a*b*sech(d*sqrt(x) + c) + a^2), 
x)
 

Sympy [F]

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

integrate(x/(a+b*sech(c+d*x**(1/2)))**2,x)
 

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x/(a+b*sech(c+d*x^(1/2)))^2,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a-b>0)', see `assume?` for more 
details)Is
 

Giac [F]

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

integrate(x/(a+b*sech(c+d*x^(1/2)))^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(x/(a + b/cosh(c + d*x^(1/2)))^2,x)
 

Output:

int(x/(a + b/cosh(c + d*x^(1/2)))^2, x)
 

Reduce [F]

\[ \int \frac {x}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {too large to display} \] Input:

int(x/(a+b*sech(c+d*x^(1/2)))^2,x)
 

Output:

( - 96*e**(2*sqrt(x)*d + 2*c)*sqrt(a**2 - b**2)*atan((e**(sqrt(x)*d + c)*a 
 + b)/sqrt(a**2 - b**2))*a**3*b + 156*e**(2*sqrt(x)*d + 2*c)*sqrt(a**2 - b 
**2)*atan((e**(sqrt(x)*d + c)*a + b)/sqrt(a**2 - b**2))*a*b**3 - 192*e**(s 
qrt(x)*d + c)*sqrt(a**2 - b**2)*atan((e**(sqrt(x)*d + c)*a + b)/sqrt(a**2 
- b**2))*a**2*b**2 + 312*e**(sqrt(x)*d + c)*sqrt(a**2 - b**2)*atan((e**(sq 
rt(x)*d + c)*a + b)/sqrt(a**2 - b**2))*b**4 - 96*sqrt(a**2 - b**2)*atan((e 
**(sqrt(x)*d + c)*a + b)/sqrt(a**2 - b**2))*a**3*b + 156*sqrt(a**2 - b**2) 
*atan((e**(sqrt(x)*d + c)*a + b)/sqrt(a**2 - b**2))*a*b**3 - 96*e**(2*sqrt 
(x)*d + 3*c)*int(e**(sqrt(x)*d)/(e**(4*sqrt(x)*d + 4*c)*a**2 + 4*e**(3*sqr 
t(x)*d + 3*c)*a*b + 2*e**(2*sqrt(x)*d + 2*c)*a**2 + 4*e**(2*sqrt(x)*d + 2* 
c)*b**2 + 4*e**(sqrt(x)*d + c)*a*b + a**2),x)*a**6*b*d**2 + 132*e**(2*sqrt 
(x)*d + 3*c)*int(e**(sqrt(x)*d)/(e**(4*sqrt(x)*d + 4*c)*a**2 + 4*e**(3*sqr 
t(x)*d + 3*c)*a*b + 2*e**(2*sqrt(x)*d + 2*c)*a**2 + 4*e**(2*sqrt(x)*d + 2* 
c)*b**2 + 4*e**(sqrt(x)*d + c)*a*b + a**2),x)*a**4*b**3*d**2 - 36*e**(2*sq 
rt(x)*d + 3*c)*int(e**(sqrt(x)*d)/(e**(4*sqrt(x)*d + 4*c)*a**2 + 4*e**(3*s 
qrt(x)*d + 3*c)*a*b + 2*e**(2*sqrt(x)*d + 2*c)*a**2 + 4*e**(2*sqrt(x)*d + 
2*c)*b**2 + 4*e**(sqrt(x)*d + c)*a*b + a**2),x)*a**2*b**5*d**2 - 16*e**(2* 
sqrt(x)*d + 3*c)*int((e**(sqrt(x)*d)*x)/(e**(4*sqrt(x)*d + 4*c)*a**2 + 4*e 
**(3*sqrt(x)*d + 3*c)*a*b + 2*e**(2*sqrt(x)*d + 2*c)*a**2 + 4*e**(2*sqrt(x 
)*d + 2*c)*b**2 + 4*e**(sqrt(x)*d + c)*a*b + a**2),x)*a**6*b*d**4 + 24*...