\(\int \frac {x^2}{(a+b \sec (c+d \sqrt {x}))^2} \, dx\) [47]

Optimal result

Integrand size = 20, antiderivative size = 2323 \[ \int \frac {x^2}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \] Output:

-2*I*b^2*x^(5/2)/a^2/(a^2-b^2)/d+240*I*b^2*x^(1/2)*polylog(4,-a*exp(I*(c+d 
*x^(1/2)))/(b-I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^5+240*I*b^3*x^(1/2)*poly 
log(5,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d 
^5+480*I*b*x^(1/2)*polylog(5,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2))) 
/a^2/(-a^2+b^2)^(1/2)/d^5+2*I*b^3*x^(5/2)*ln(1+a*exp(I*(c+d*x^(1/2)))/(b+( 
-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d+80*I*b*x^(3/2)*polylog(3,-a*exp(I 
*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^3+4*I*b*x^(5/ 
2)*ln(1+a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/ 
d+240*I*b^2*x^(1/2)*polylog(4,-a*exp(I*(c+d*x^(1/2)))/(b+I*(a^2-b^2)^(1/2) 
))/a^2/(a^2-b^2)/d^5+2*b^2*x^(5/2)*sin(c+d*x^(1/2))/a/(a^2-b^2)/d/(b+a*cos 
(c+d*x^(1/2)))-40*I*b^2*x^(3/2)*polylog(2,-a*exp(I*(c+d*x^(1/2)))/(b+I*(a^ 
2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3-40*I*b^2*x^(3/2)*polylog(2,-a*exp(I*(c+d* 
x^(1/2)))/(b-I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3-40*I*b^3*x^(3/2)*polylo 
g(3,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3 
-2*I*b^3*x^(5/2)*ln(1+a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a 
^2+b^2)^(3/2)/d-80*I*b*x^(3/2)*polylog(3,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+ 
b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^3-4*I*b*x^(5/2)*ln(1+a*exp(I*(c+d*x^(1 
/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d-240*I*b^3*x^(1/2)*polyl 
og(5,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^ 
5-20*b*x^2*polylog(2,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/...
 

Mathematica [A] (verified)

Time = 11.26 (sec) , antiderivative size = 2777, normalized size of antiderivative = 1.20 \[ \int \frac {x^2}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

((-4*I)*b^2*E^((2*I)*c)*x^(5/2)*(b + a*Cos[c + d*Sqrt[x]])^2*Sec[c + d*Sqr 
t[x]]^2)/(a^2*(a^2 - b^2)*d*(1 + E^((2*I)*c))*(a + b*Sec[c + d*Sqrt[x]])^2 
) + (x^3*(b + a*Cos[c + d*Sqrt[x]])^2*Sec[c + d*Sqrt[x]]^2)/(3*a^2*(a + b* 
Sec[c + d*Sqrt[x]])^2) + (2*b*(b + a*Cos[c + d*Sqrt[x]])^2*(5*b*d^4*Sqrt[( 
-a^2 + b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c 
) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (2*I)*a^2*d^5*E^(I*c)*x^(5/2)*Log[1 
 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)] 
)] - I*b^2*d^5*E^(I*c)*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I 
*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 5*b*d^4*Sqrt[(-a^2 + b^2)*E^((2*I 
)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^ 
2)*E^((2*I)*c)])] - (2*I)*a^2*d^5*E^(I*c)*x^(5/2)*Log[1 + (a*E^(I*(2*c + d 
*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + I*b^2*d^5*E^(I 
*c)*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + 
b^2)*E^((2*I)*c)])] - 5*d^3*((4*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^ 
2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*x^(3/2)*PolyLog[2, -((a*E^(I* 
(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 5*d^3 
*((-4*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2* 
d*E^(I*c)*Sqrt[x])*x^(3/2)*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^( 
I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 60*b*d^2*Sqrt[(-a^2 + b^2)*E^(( 
2*I)*c)]*x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(...
 

Rubi [A] (verified)

Time = 3.47 (sec) , antiderivative size = 2324, 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 = {4692, 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.

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

\(\Big \downarrow \) 4692

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {i x^{5/2} \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {i x^{5/2} \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} 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^{i \left (c+d \sqrt {x}\right )}}{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^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {20 i x^{3/2} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {20 i x^{3/2} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{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^{i \left (c+d \sqrt {x}\right )}}{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^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}+\frac {120 i \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^5}-\frac {120 i \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{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^{i \left (c+d \sqrt {x}\right )}}{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^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^6}-\frac {i x^{5/2} b^2}{a^2 \left (a^2-b^2\right ) d}+\frac {5 x^2 \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-i \sqrt {a^2-b^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac {5 x^2 \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b+i \sqrt {a^2-b^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac {20 i x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}-\frac {20 i x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}+\frac {60 x \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {60 x \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {120 i \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^5}+\frac {120 i \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^5}-\frac {120 \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^6}-\frac {120 \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^6}+\frac {x^{5/2} \sin \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2-b^2\right ) d \left (b+a \cos \left (c+d \sqrt {x}\right )\right )}+\frac {2 i x^{5/2} \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} d}-\frac {2 i x^{5/2} \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} 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^{i \left (c+d \sqrt {x}\right )}}{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^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}+\frac {40 i x^{3/2} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {40 i x^{3/2} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{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^{i \left (c+d \sqrt {x}\right )}}{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^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^4}-\frac {240 i \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^5}+\frac {240 i \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{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^{i \left (c+d \sqrt {x}\right )}}{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^{i \left (c+d \sqrt {x}\right )}}{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*Sec[c + d*Sqrt[x]])^2,x]
 

Output:

2*(((-I)*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^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^ 
2) + (5*b^2*x^2*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2])] 
)/(a^2*(a^2 - b^2)*d^2) - (I*b^3*x^(5/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x]))) 
/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((2*I)*b*x^(5/2)*Lo 
g[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + 
b^2]*d) + (I*b^3*x^(5/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 
+ b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((2*I)*b*x^(5/2)*Log[1 + (a*E^(I*(c 
 + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((20*I 
)*b^2*x^(3/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2 
]))])/(a^2*(a^2 - b^2)*d^3) - ((20*I)*b^2*x^(3/2)*PolyLog[2, -((a*E^(I*(c 
+ d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (5*b^3*x 
^2*PolyLog[2, -((a*E^(I*(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^(I*(c + d*Sqrt[x])))/ 
(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (5*b^3*x^2*PolyLog[ 
2, -((a*E^(I*(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^(I*(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^(I 
*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + (60* 
b^2*x*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*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 4692

Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sec[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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^2/(a+b*sec(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(4*a^2-4*b^2>0)', see `assume?` f 
or more de
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 96*sqrt(x)*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**7*b*d**5*x**2 
+ 13440*sqrt(x)*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**7*b*d**3*x + 
40320*sqrt(x)*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**7*b*d - 192*sqr 
t(x)*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**6*b**2*d**5*x**2 + 14400 
*sqrt(x)*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**6*b**2*d**3*x - 1728 
0*sqrt(x)*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**6*b**2*d - 24*sqrt( 
x)*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**5*b**3*d**5*x**2 - 7680*sq 
rt(x)*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**5*b**3*d**3*x - 57600*s 
qrt(x)*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**5*b**3*d + 168*sqrt(x) 
*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**4*b**4*d**5*x**2 - 13440*sqr 
t(x)*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**4*b**4*d**3*x + 11520*sq 
rt(x)*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**4*b**4*d + 120*sqrt(x)* 
cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**3*b**5*d**5*x**2 - 5760*sqrt( 
x)*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**3*b**5*d**3*x + 17280*sqrt 
(x)*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**3*b**5*d + 24*sqrt(x)*cos 
(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**2*b**6*d**5*x**2 - 960*sqrt(x)*c 
os(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**2*b**6*d**3*x + 5760*sqrt(x)*c 
os(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)*a**2*b**6*d - 192*cos(sqrt(x)*d + 
 c)*int(x**2/(2*tan((sqrt(x)*d + c)/2)**4*a**6 - 5*tan((sqrt(x)*d + c)/2)* 
*4*a**5*b + tan((sqrt(x)*d + c)/2)**4*a**4*b**2 + 6*tan((sqrt(x)*d + c)...