\(\int \frac {x^2}{(a+b \text {sech}(c+d \sqrt {x}))^2} \, dx\) [55]

Optimal result

Integrand size = 20, antiderivative size = 2123 \[ \int \frac {x^2}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \] Output:

1/3*x^3/a^2+2*b^2*x^(5/2)*sinh(c+d*x^(1/2))/a/(a^2-b^2)/d/(b+a*cosh(c+d*x^ 
(1/2)))-120*b^3*x*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+120*b^3*x*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+120*b^2*x*polylog(3,-a*exp(c+d*x^(1/2))/ 
(b+(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^4+40*b^3*x^(3/2)*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+120*b^2*x*polylo 
g(3,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^4-40*b^3*x^( 
3/2)*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^3-40*b^2*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)/d^3-10*b^3*x^2*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^2-40*b^2*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)/d^3+10*b^3*x^2*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^2-10*b^2*x^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^(5/2)*l 
n(1+a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d-10*b^2 
*x^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^(5/2)*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+240*b*x*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^4-240*b*x*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^4-80*b*x^(3/2)*polylog(3,-a*exp(c+d*x^(1/2))/(b...
 

Mathematica [A] (verified)

Time = 6.22 (sec) , antiderivative size = 2173, normalized size of antiderivative = 1.02 \[ \int \frac {x^2}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

((b + a*Cosh[c + d*Sqrt[x]])*Sech[c + d*Sqrt[x]]^2*(x^3*(b + a*Cosh[c + d* 
Sqrt[x]]) + (6*b*E^c*(b + a*Cosh[c + d*Sqrt[x]])*(2*b*E^c*x^(5/2) + ((1 + 
E^(2*c))*(-5*b*d^4*Sqrt[(-a^2 + b^2)*E^(2*c)]*x^2*Log[1 + (a*E^(2*c + d*Sq 
rt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])] - 2*a^2*d^5*E^c*x^(5/2)*Log[ 
1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])] + b^2*d^ 
5*E^c*x^(5/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E 
^(2*c)])] - 5*b*d^4*Sqrt[(-a^2 + b^2)*E^(2*c)]*x^2*Log[1 + (a*E^(2*c + d*S 
qrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])] + 2*a^2*d^5*E^c*x^(5/2)*Log 
[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])] - b^2*d 
^5*E^c*x^(5/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)* 
E^(2*c)])] + 5*d^3*(-4*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^(3/2)*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - 
 Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 5*d^3*(4*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^(3/2)*PolyLog[2, -((a*E^(2*c + d 
*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 60*b*d^2*Sqrt[(-a^2 + 
b^2)*E^(2*c)]*x*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + 
 b^2)*E^(2*c)]))] + 40*a^2*d^3*E^c*x^(3/2)*PolyLog[3, -((a*E^(2*c + d*Sqrt 
[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 20*b^2*d^3*E^c*x^(3/2)*Poly 
Log[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 
60*b*d^2*Sqrt[(-a^2 + b^2)*E^(2*c)]*x*PolyLog[3, -((a*E^(2*c + d*Sqrt[x...
 

Rubi [A] (verified)

Time = 4.33 (sec) , antiderivative size = 2122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.

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

\(\Big \downarrow \) 5959

\(\displaystyle 2 \int \frac {x^{5/2}}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \int \frac {x^{5/2}}{\left (a+b \csc \left (i c+i d \sqrt {x}+\frac {\pi }{2}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 4679

\(\displaystyle 2 \int \left (-\frac {2 b x^{5/2}}{a^2 \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}+\frac {x^{5/2}}{a^2}+\frac {b^2 x^{5/2}}{a^2 \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )^2}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {x^{5/2} \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^{5/2} \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 {5 x^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 {5 x^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 {20 x^{3/2} \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 {20 x^{3/2} \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 {60 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 {60 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 {120 \sqrt {x} \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 {120 \sqrt {x} \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 {120 \operatorname {PolyLog}\left (6,-\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^6}-\frac {120 \operatorname {PolyLog}\left (6,-\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^6}+\frac {x^{5/2} b^2}{a^2 \left (a^2-b^2\right ) d}-\frac {5 x^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 {5 x^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 {20 x^{3/2} \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 {20 x^{3/2} \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 {60 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 {60 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 {120 \sqrt {x} \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 {120 \sqrt {x} \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 {120 \operatorname {PolyLog}\left (5,-\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^6}+\frac {120 \operatorname {PolyLog}\left (5,-\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^6}+\frac {x^{5/2} \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^{5/2} \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^{5/2} \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 {10 x^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 {10 x^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 {40 x^{3/2} \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 {40 x^{3/2} \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 {120 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 {120 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 {240 \sqrt {x} \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 {240 \sqrt {x} \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 {240 \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^6}+\frac {240 \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^6}+\frac {x^3}{6 a^2}\right )\)

Input:

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

Output:

2*((b^2*x^(5/2))/(a^2*(a^2 - b^2)*d) + x^3/(6*a^2) - (5*b^2*x^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^(5/2)*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^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^ 
2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (5*b^2*x^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^(5/2)*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^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a 
^2*Sqrt[-a^2 + b^2]*d) - (20*b^2*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]) 
)/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (5*b^3*x^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) - (10*b*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]) 
)])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (20*b^2*x^(3/2)*PolyLog[2, -((a*E^(c + d* 
Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) - (5*b^3*x^2*Pol 
yLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2 
)^(3/2)*d^2) + (10*b*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 
 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (60*b^2*x*PolyLog[3, -((a*E^(c + 
d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) - (20*b^3*x^(3 
/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^ 
2 + b^2)^(3/2)*d^3) + (40*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/...
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2880*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 + 5490*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 - 5760*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 + 10980*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 - 2880*sqrt(a**2 - b**2) 
*atan((e**(sqrt(x)*d + c)*a + b)/sqrt(a**2 - b**2))*a**3*b + 5490*sqrt(a** 
2 - b**2)*atan((e**(sqrt(x)*d + c)*a + b)/sqrt(a**2 - b**2))*a*b**3 - 2880 
*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*sqr 
t(x)*d + 2*c)*b**2 + 4*e**(sqrt(x)*d + c)*a*b + a**2),x)*a**6*b*d**2 + 315 
0*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*sq 
rt(x)*d + 2*c)*b**2 + 4*e**(sqrt(x)*d + c)*a*b + a**2),x)*a**4*b**3*d**2 - 
 270*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 - 24*e**(2*sqrt(x)*d + 3*c)*int((e**(sqrt(x)*d)*x**2)/(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...