3.1.68 \(\int \frac {\sqrt {x}}{(a+b \csc (c+d \sqrt {x}))^2} \, dx\) [68]

3.1.68.1 Optimal result

Integrand size = 22, antiderivative size = 1157 \[ \int \frac {\sqrt {x}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {2 i b^2 x}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{3/2}}{3 a^2}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {2 i b^3 x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {4 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {2 i b^3 x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {4 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {4 i b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {4 i b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {4 b^3 \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {4 b^3 \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {4 i b^3 \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac {8 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {4 i b^3 \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac {8 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {2 b^2 x \cos \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt {x}\right )\right )} \]

output

4*I*b*x*ln(1-I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/d/(-a^2+b^ 
2)^(1/2)+2/3*x^(3/2)/a^2-2*I*b^2*x/a^2/(a^2-b^2)/d-2*I*b^3*x*ln(1-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-8*I*b*polylo 
g(3,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/d^3/(-a^2+b^2)^(1/2 
)-4*I*b*x*ln(1-I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/d/(-a^2+ 
b^2)^(1/2)+4*I*b^3*polylog(3,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^3-4*I*b^3*polylog(3,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^3-2*b^2*x*cos(c+d*x^(1/2))/a/(a^2 
-b^2)/d/(b+a*sin(c+d*x^(1/2)))+2*I*b^3*x*ln(1-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*I*b^2*polylog(2,-a*exp(I*(c+d* 
x^(1/2)))/(I*b+(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3+8*I*b*polylog(3,I*a*exp 
(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/d^3/(-a^2+b^2)^(1/2)-4*I*b^2*p 
olylog(2,-a*exp(I*(c+d*x^(1/2)))/(I*b-(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3+ 
4*b^2*ln(1+a*exp(I*(c+d*x^(1/2)))/(I*b-(a^2-b^2)^(1/2)))*x^(1/2)/a^2/(a^2- 
b^2)/d^2+4*b^2*ln(1+a*exp(I*(c+d*x^(1/2)))/(I*b+(a^2-b^2)^(1/2)))*x^(1/2)/ 
a^2/(a^2-b^2)/d^2-4*b^3*polylog(2,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^( 
1/2)))*x^(1/2)/a^2/(-a^2+b^2)^(3/2)/d^2+4*b^3*polylog(2,I*a*exp(I*(c+d*x^( 
1/2)))/(b+(-a^2+b^2)^(1/2)))*x^(1/2)/a^2/(-a^2+b^2)^(3/2)/d^2+8*b*polylog( 
2,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))*x^(1/2)/a^2/d^2/(-a^2+b^2 
)^(1/2)-8*b*polylog(2,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))*x^...
 

3.1.68.2 Mathematica [A] (verified)

Time = 9.71 (sec) , antiderivative size = 846, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {x}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {\csc ^2\left (c+d \sqrt {x}\right ) \left (b+a \sin \left (c+d \sqrt {x}\right )\right ) \left (2 x^{3/2} \left (b+a \sin \left (c+d \sqrt {x}\right )\right )-\frac {6 i b \left (\frac {2 b d^2 e^{2 i c} x}{-1+e^{2 i c}}+\frac {2 \left (b \sqrt {\left (a^2-b^2\right ) e^{2 i c}}-2 a^2 d e^{i c} \sqrt {x}+b^2 d e^{i c} \sqrt {x}\right ) \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}+i \sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+2 \left (b \sqrt {\left (a^2-b^2\right ) e^{2 i c}}+2 a^2 d e^{i c} \sqrt {x}-b^2 d e^{i c} \sqrt {x}\right ) \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{i b e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+i \left (d \sqrt {x} \left (\left (2 b \sqrt {\left (a^2-b^2\right ) e^{2 i c}}-2 a^2 d e^{i c} \sqrt {x}+b^2 d e^{i c} \sqrt {x}\right ) \log \left (1+\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{i b e^{i c}-\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+\left (2 b \sqrt {\left (a^2-b^2\right ) e^{2 i c}}+2 a^2 d e^{i c} \sqrt {x}-b^2 d e^{i c} \sqrt {x}\right ) \log \left (1+\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{i b e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )\right )-2 \left (2 a^2-b^2\right ) e^{i c} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}+i \sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+2 \left (2 a^2-b^2\right ) e^{i c} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{i b e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )\right )}{\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right ) \left (b+a \sin \left (c+d \sqrt {x}\right )\right )}{\left (a^2-b^2\right ) d^3}+\frac {6 b^2 x \csc (c) \left (b \cos (c)+a \sin \left (d \sqrt {x}\right )\right )}{(a-b) (a+b) d}\right )}{3 a^2 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \]

input

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

output

(Csc[c + d*Sqrt[x]]^2*(b + a*Sin[c + d*Sqrt[x]])*(2*x^(3/2)*(b + a*Sin[c + 
 d*Sqrt[x]]) - ((6*I)*b*((2*b*d^2*E^((2*I)*c)*x)/(-1 + E^((2*I)*c)) + (2*( 
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])*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)])] + 2*(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])*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x] 
)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + I*(d*Sqrt[x]*((2*b*S 
qrt[(a^2 - b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqr 
t[x])*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*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])*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*(2*a^2 - b^2)*E^(I*c)*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 
)])] + 2*(2*a^2 - b^2)*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I 
*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))]))/Sqrt[(a^2 - b^2)*E^((2*I)* 
c)])*(b + a*Sin[c + d*Sqrt[x]]))/((a^2 - b^2)*d^3) + (6*b^2*x*Csc[c]*(b*Co 
s[c] + a*Sin[d*Sqrt[x]]))/((a - b)*(a + b)*d)))/(3*a^2*(a + b*Csc[c + d*Sq 
rt[x]])^2)
 

3.1.68.3 Rubi [A] (verified)

Time = 2.22 (sec) , antiderivative size = 1159, 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 = {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.

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

\(\Big \downarrow \) 4693

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {i x \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 \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 {2 \sqrt {x} \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 {2 \sqrt {x} \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 {2 i \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 {2 i \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 {i x b^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 \sqrt {x} \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 {2 \sqrt {x} \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 {2 i \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 {2 i \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 {x \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 \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 \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 {4 \sqrt {x} \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 {4 \sqrt {x} \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 {4 i \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 {4 i \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 {x^{3/2}}{3 a^2}\right )\)

input

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

output

2*(((-I)*b^2*x)/(a^2*(a^2 - b^2)*d) + x^(3/2)/(3*a^2) + (2*b^2*Sqrt[x]*Log 
[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)* 
d^2) + (2*b^2*Sqrt[x]*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*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*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*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*Log[1 - (I*a*E^(I*(c + d*Sqrt[ 
x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((2*I)*b^2*PolyL 
og[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b 
^2)*d^3) - ((2*I)*b^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) - (2*b^3*Sqrt[x]*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) + 
(4*b*Sqrt[x]*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) + (2*b^3*Sqrt[x]*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) - (4*b*S 
qrt[x]*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) - ((2*I)*b^3*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])) 
)/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((4*I)*b*PolyLog 
[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2...
 

3.1.68.3.1 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]
 

3.1.68.4 Maple [F]

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

input

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

output

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

3.1.68.5 Fricas [F]

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

input

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

output

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

3.1.68.6 Sympy [F]

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

input

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

output

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

3.1.68.7 Maxima [F(-2)]

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

input

integrate(x^(1/2)/(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
 

3.1.68.8 Giac [F]

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

input

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

output

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

3.1.68.9 Mupad [F(-1)]

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

input

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

output

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