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

Optimal result

Integrand size = 20, antiderivative size = 2385 \[ \int \frac {x^2}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \] Output:

-2*I*b^2*x^(5/2)/a^2/(a^2-b^2)/d+120*b^3*x*polylog(4,I*a*exp(I*(c+d*x^(1/2 
)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^4+10*b^3*x^2*polylog(2,I* 
a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2-10*b 
^3*x^2*polylog(2,I*a*exp(I*(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(I*(c+d*x^(1/2)))/(I*b+(a^2-b^2)^(1/2) 
))/a^2/(a^2-b^2)/d^2+10*b^2*x^2*ln(1+a*exp(I*(c+d*x^(1/2)))/(I*b-(a^2-b^2) 
^(1/2)))/a^2/(a^2-b^2)/d^2+120*b^2*x*polylog(3,-a*exp(I*(c+d*x^(1/2)))/(I* 
b+(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^4-120*b^3*x*polylog(4,I*a*exp(I*(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(I*(c+d*x^(1/2)))/(I*b-(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^4-240*b*x 
*polylog(4,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^( 
1/2)/d^4-20*b*x^2*polylog(2,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2))) 
/a^2/(-a^2+b^2)^(1/2)/d^2+20*b*x^2*polylog(2,I*a*exp(I*(c+d*x^(1/2)))/(b-( 
-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^2+240*b*x*polylog(4,I*a*exp(I*(c+ 
d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^4-2*b^2*x^(5/2)*c 
os(c+d*x^(1/2))/a/(a^2-b^2)/d/(b+a*sin(c+d*x^(1/2)))-480*I*b*x^(1/2)*polyl 
og(5,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d 
^5-240*I*b^3*x^(1/2)*polylog(5,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2 
)))/a^2/(-a^2+b^2)^(3/2)/d^5-4*I*b*x^(5/2)*ln(1-I*a*exp(I*(c+d*x^(1/2)))/( 
b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d-40*I*b^3*x^(3/2)*polylog(3,...
 

Mathematica [A] (warning: unable to verify)

Time = 12.54 (sec) , antiderivative size = 2806, normalized size of antiderivative = 1.18 \[ \int \frac {x^2}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

(x^3*Csc[c + d*Sqrt[x]]^2*(b + a*Sin[c + d*Sqrt[x]])^2)/(3*a^2*(a + b*Csc[ 
c + d*Sqrt[x]])^2) - ((2*I)*b*Csc[c + d*Sqrt[x]]^2*((2*b*d^5*E^((2*I)*c)*x 
^(5/2))/(-1 + E^((2*I)*c)) + ((5*I)*b*d^4*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^ 
2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*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 
])))/(I*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])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)* 
E^((2*I)*c)])] + (5*I)*b*d^4*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a* 
E^(I*(2*c + d*Sqrt[x])))/(I*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])))/(I*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])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]) 
] + 5*d^3*(4*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, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b* 
E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 5*d^3*(4*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)*Po 
lyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^( 
(2*I)*c)]))] + (60*I)*b*d^2*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x*PolyLog[3, (I* 
a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] 
- (40*I)*a^2*d^3*E^(I*c)*x^(3/2)*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])...
 

Rubi [A] (verified)

Time = 3.54 (sec) , antiderivative size = 2387, 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 = {4693, 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.

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

\(\Big \downarrow \) 4693

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {i x^{5/2} \log \left (1-\frac {i 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}+\frac {i x^{5/2} \log \left (1-\frac {i 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}-\frac {5 x^2 \operatorname {PolyLog}\left (2,\frac {i 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 {i 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 {i 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 {i 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 {i 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 {i 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 {i 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 {i 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 {i 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 {i 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}{i b-\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}{i b+\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 )}}{i b-\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 )}}{i b+\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 )}}{i b-\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 )}}{i b+\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 )}}{i b-\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 )}}{i b+\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 )}}{i b-\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 )}}{i b+\sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^6}-\frac {x^{5/2} \cos \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt {x}\right )\right )}+\frac {2 i x^{5/2} \log \left (1-\frac {i 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}-\frac {2 i x^{5/2} \log \left (1-\frac {i 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}+\frac {10 x^2 \operatorname {PolyLog}\left (2,\frac {i 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 {i 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 {i 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 {i 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 {i 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 {i 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 {i 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 {i 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 {i 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 {i 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*Csc[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])))/(I*b - 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])))/(I*b + Sqrt[a^2 - b^2])] 
)/(a^2*(a^2 - b^2)*d^2) - (I*b^3*x^(5/2)*Log[1 - (I*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 - (I*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 - (I*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 - (I*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])))/(I*b - 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])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - 
(5*b^3*x^2*PolyLog[2, (I*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, (I*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*Po 
lyLog[2, (I*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, (I*a*E^(I*(c + d*Sqrt[x])))/(b + S 
qrt[-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])))/(I*b - 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])))/(I*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 4693

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^2/(a+b*csc(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*(233280*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*b + a)/sqrt( 
- a**2 + b**2))*sin(sqrt(x)*d + c)*a**7 - 233280*sqrt( - a**2 + b**2)*atan 
((tan((sqrt(x)*d + c)/2)*b + a)/sqrt( - a**2 + b**2))*sin(sqrt(x)*d + c)*a 
**5*b**2 + 50400*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*b + a)/ 
sqrt( - a**2 + b**2))*sin(sqrt(x)*d + c)*a**3*b**4 - 1440*sqrt( - a**2 + b 
**2)*atan((tan((sqrt(x)*d + c)/2)*b + a)/sqrt( - a**2 + b**2))*sin(sqrt(x) 
*d + c)*a*b**6 + 233280*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)* 
b + a)/sqrt( - a**2 + b**2))*a**6*b - 233280*sqrt( - a**2 + b**2)*atan((ta 
n((sqrt(x)*d + c)/2)*b + a)/sqrt( - a**2 + b**2))*a**4*b**3 + 50400*sqrt( 
- a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*b + a)/sqrt( - a**2 + b**2))*a 
**2*b**5 - 1440*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*b + a)/s 
qrt( - a**2 + b**2))*b**7 + 116640*sqrt(x)*cos(sqrt(x)*d + c)*a**8*d + 216 
0*sqrt(x)*cos(sqrt(x)*d + c)*a**6*b**2*d**3*x - 174960*sqrt(x)*cos(sqrt(x) 
*d + c)*a**6*b**2*d + 12*sqrt(x)*cos(sqrt(x)*d + c)*a**4*b**4*d**5*x**2 - 
2760*sqrt(x)*cos(sqrt(x)*d + c)*a**4*b**4*d**3*x + 64080*sqrt(x)*cos(sqrt( 
x)*d + c)*a**4*b**4*d - 12*sqrt(x)*cos(sqrt(x)*d + c)*a**2*b**6*d**5*x**2 
+ 600*sqrt(x)*cos(sqrt(x)*d + c)*a**2*b**6*d**3*x - 5760*sqrt(x)*cos(sqrt( 
x)*d + c)*a**2*b**6*d - 19440*cos(sqrt(x)*d + c)*a**7*b*d**2*x - 180*cos(s 
qrt(x)*d + c)*a**5*b**3*d**4*x**2 + 27000*cos(sqrt(x)*d + c)*a**5*b**3*d** 
2*x + 210*cos(sqrt(x)*d + c)*a**3*b**5*d**4*x**2 - 7920*cos(sqrt(x)*d +...