\(\int \frac {x^{3/2}}{(a+b \text {sech}(c+d \sqrt {x}))^2} \, dx\) [74]

Optimal result

Integrand size = 22, antiderivative size = 1755 \[ \int \frac {x^{3/2}}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \] Output:

2/5*x^(5/2)/a^2+2*b^2*x^2*sinh(c+d*x^(1/2))/a/(a^2-b^2)/d/(b+a*cosh(c+d*x^ 
(1/2)))-8*b^2*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)/d^2+2*b^3*x^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+4*b*x^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-4*b*x^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-48*b*x*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+48*b*x*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+16*b*x^(3/2)*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^2-16*b*x^(3/2)*po 
lylog(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 
+96*b*x^(1/2)*polylog(4,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^ 
2+b^2)^(1/2)/d^4-96*b*x^(1/2)*polylog(4,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^ 
(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^4-48*b^3*x^(1/2)*polylog(4,-a*exp(c+d*x^(1/ 
2))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^4+48*b^3*x^(1/2)*polylog( 
4,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^4+48*b^ 
2*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 
)/d^4+48*b^2*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)/d^4+24*b^3*x*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^3-24*b^3*x*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^3-24*b^2*x*polylog(2,-a*exp(c+d*...
 

Mathematica [A] (verified)

Time = 5.09 (sec) , antiderivative size = 1769, normalized size of antiderivative = 1.01 \[ \int \frac {x^{3/2}}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*(b + a*Cosh[c + d*Sqrt[x]])*Sech[c + d*Sqrt[x]]^2*(x^(5/2)*(b + a*Cosh[ 
c + d*Sqrt[x]]) + (5*b*E^c*(b + a*Cosh[c + d*Sqrt[x]])*(2*b*E^c*x^2 - ((1 
+ E^(2*c))*(4*b*d^3*Sqrt[(-a^2 + b^2)*E^(2*c)]*x^(3/2)*Log[1 + (a*E^(2*c + 
 d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])] + 2*a^2*d^4*E^c*x^2*Log 
[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])] - b^2*d 
^4*E^c*x^2*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2 
*c)])] + 4*b*d^3*Sqrt[(-a^2 + b^2)*E^(2*c)]*x^(3/2)*Log[1 + (a*E^(2*c + d* 
Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])] - 2*a^2*d^4*E^c*x^2*Log[1 
+ (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])] + b^2*d^4* 
E^c*x^2*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c) 
])] + 4*d^2*(3*b*Sqrt[(-a^2 + b^2)*E^(2*c)] + 2*a^2*d*E^c*Sqrt[x] - b^2*d* 
E^c*Sqrt[x])*x*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + 
b^2)*E^(2*c)]))] + 4*d^2*(3*b*Sqrt[(-a^2 + b^2)*E^(2*c)] - 2*a^2*d*E^c*Sqr 
t[x] + b^2*d*E^c*Sqrt[x])*x*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + 
Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 24*b*d*Sqrt[(-a^2 + b^2)*E^(2*c)]*Sqrt[x]* 
PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])) 
] - 24*a^2*d^2*E^c*x*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(- 
a^2 + b^2)*E^(2*c)]))] + 12*b^2*d^2*E^c*x*PolyLog[3, -((a*E^(2*c + d*Sqrt[ 
x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 24*b*d*Sqrt[(-a^2 + b^2)*E^( 
2*c)]*Sqrt[x]*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 ...
 

Rubi [A] (verified)

Time = 3.99 (sec) , antiderivative size = 1754, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, 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.

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

\(\Big \downarrow \) 5959

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {x^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^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 {4 x^{3/2} \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 {4 x^{3/2} \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 {12 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 {12 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 {24 \sqrt {x} \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 {24 \sqrt {x} \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 {24 \operatorname {PolyLog}\left (5,-\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^5}+\frac {24 \operatorname {PolyLog}\left (5,-\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^5}+\frac {x^2 b^2}{a^2 \left (a^2-b^2\right ) d}-\frac {4 x^{3/2} \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 {4 x^{3/2} \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 {12 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 {12 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 {24 \sqrt {x} \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 {24 \sqrt {x} \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 {24 \operatorname {PolyLog}\left (4,-\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^5}-\frac {24 \operatorname {PolyLog}\left (4,-\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^5}+\frac {x^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^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^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 {8 x^{3/2} \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 {8 x^{3/2} \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 {24 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 {24 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 {48 \sqrt {x} \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 {48 \sqrt {x} \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 {48 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^5}-\frac {48 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^5}+\frac {x^{5/2}}{5 a^2}\right )\)

Input:

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

Output:

2*((b^2*x^2)/(a^2*(a^2 - b^2)*d) + x^(5/2)/(5*a^2) - (4*b^2*x^(3/2)*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^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^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + 
b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (4*b^2*x^(3/2)*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^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^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[ 
-a^2 + b^2]*d) - (12*b^2*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a 
^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (4*b^3*x^(3/2)*PolyLog[2, -((a*E^(c 
+ d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (8* 
b*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^ 
2*Sqrt[-a^2 + b^2]*d^2) - (12*b^2*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b 
+ Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) - (4*b^3*x^(3/2)*PolyLog[2, - 
((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d 
^2) + (8*b*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2 
]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (24*b^2*Sqrt[x]*PolyLog[3, -((a*E^(c + 
d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) - (12*b^3*x*Po 
lyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^ 
2)^(3/2)*d^3) + (24*b*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a...
 

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]

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

Input:

int(x^(3/2)/(a+b*sech(c+d*x^(1/2)))^2,x)
 

Output:

int(x^(3/2)/(a+b*sech(c+d*x^(1/2)))^2,x)
 

Fricas [F]

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

integrate(x^(3/2)/(a+b*sech(c+d*x^(1/2)))^2,x, algorithm="fricas")
 

Output:

integral(x^(3/2)/(b^2*sech(d*sqrt(x) + c)^2 + 2*a*b*sech(d*sqrt(x) + c) + 
a^2), x)
 

Sympy [F]

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

integrate(x**(3/2)/(a+b*sech(c+d*x**(1/2)))**2,x)
 

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^{3/2}}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^(3/2)/(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]

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

integrate(x^(3/2)/(a+b*sech(c+d*x^(1/2)))^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(x^(3/2)/(a + b/cosh(c + d*x^(1/2)))^2,x)
 

Output:

int(x^(3/2)/(a + b/cosh(c + d*x^(1/2)))^2, x)
 

Reduce [F]

\[ \int \frac {x^{3/2}}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {too large to display} \] Input:

int(x^(3/2)/(a+b*sech(c+d*x^(1/2)))^2,x)
 

Output:

(2*( - 480*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 + 870*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 - 960*e 
**(sqrt(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 + 1740*e**(sqrt(x)*d + c)*sqrt(a**2 - b**2)*atan((e 
**(sqrt(x)*d + c)*a + b)/sqrt(a**2 - b**2))*b**4 - 480*sqrt(a**2 - b**2)*a 
tan((e**(sqrt(x)*d + c)*a + b)/sqrt(a**2 - b**2))*a**3*b + 870*sqrt(a**2 - 
 b**2)*atan((e**(sqrt(x)*d + c)*a + b)/sqrt(a**2 - b**2))*a*b**3 - 480*e** 
(2*sqrt(x)*d + 3*c)*int(e**(sqrt(x)*d)/(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**2 + 570*e** 
(2*sqrt(x)*d + 3*c)*int(e**(sqrt(x)*d)/(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**4*b**3*d**2 - 90*e 
**(2*sqrt(x)*d + 3*c)*int(e**(sqrt(x)*d)/(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**2*b**5*d**2 - 80 
*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**...