\(\int \frac {x^3}{(a+b \csc (c+d \sqrt {x}))^2} \, dx\) [46]

Optimal result

Integrand size = 20, antiderivative size = 3205 \[ \int \frac {x^3}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \] Output:

-2*I*b^2*x^(7/2)/a^2/(a^2-b^2)/d+14*b^2*x^3*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+14*b^2*x^3*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^2-5040*b^3*x*polylog(6,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^6+420*b^2*x^ 
2*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 
^4-5040*b^2*x*polylog(5,-a*exp(I*(c+d*x^(1/2)))/(I*b+(a^2-b^2)^(1/2)))/a^2 
/(a^2-b^2)/d^6-14*b^3*x^3*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+5040*b^3*x*polylog(6,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^6+420*b^2*x^2*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^4-5040*b^2 
*x*polylog(5,-a*exp(I*(c+d*x^(1/2)))/(I*b-(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/ 
d^6+10080*b*x*polylog(6,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^6+28*b*x^3*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^2-10080*b*x*polylog(6,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^6-28*b*x^3*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+ 
840*b*x^2*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^4-840*b*x^2*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+14*b^3*x^3*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^2-420*b^3*x^2*poly...
 

Mathematica [A] (warning: unable to verify)

Time = 13.31 (sec) , antiderivative size = 3831, normalized size of antiderivative = 1.20 \[ \int \frac {x^3}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

(x^4*Csc[c + d*Sqrt[x]]^2*(b + a*Sin[c + d*Sqrt[x]])^2)/(4*a^2*(a + b*Csc[ 
c + d*Sqrt[x]])^2) - ((2*I)*b*E^(I*c)*Csc[c + d*Sqrt[x]]^2*(2*b*E^(I*c)*x^ 
(7/2) - ((-1 + E^((2*I)*c))*((-7*I)*b*d^6*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^ 
3*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^7*E^(I*c)*x^(7/2)*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^7*E^(I*c)*x^ 
(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)* 
E^((2*I)*c)])] - (7*I)*b*d^6*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^3*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^7*E^(I*c)*x^(7/2)*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^7*E^(I*c)*x^(7/2)*Log[1 + 
 (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]) 
] - 7*d^5*(6*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^(5/2)*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)])] + 7*d^5*(-6*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^(5/2)*P 
olyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^ 
((2*I)*c)]))] - (210*I)*b*d^4*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*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) 
])] + (84*I)*a^2*d^5*E^(I*c)*x^(5/2)*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt...
 

Rubi [A] (verified)

Time = 4.64 (sec) , antiderivative size = 3207, 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.

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

\(\Big \downarrow \) 4693

\(\displaystyle 2 \int \frac {x^{7/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \int \frac {x^{7/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 4679

\(\displaystyle 2 \int \left (-\frac {2 b x^{7/2}}{a^2 \left (b+a \sin \left (c+d \sqrt {x}\right )\right )}+\frac {x^{7/2}}{a^2}+\frac {b^2 x^{7/2}}{a^2 \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {x^4}{8 a^2}+\frac {2 i b \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{7/2}}{a^2 \sqrt {b^2-a^2} d}-\frac {i b^3 \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{7/2}}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {2 i b \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{7/2}}{a^2 \sqrt {b^2-a^2} d}+\frac {i b^3 \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{7/2}}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {i b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}-\frac {b^2 \cos \left (c+d \sqrt {x}\right ) x^{7/2}}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt {x}\right )\right )}+\frac {7 b^2 \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{i b-\sqrt {a^2-b^2}}+1\right ) x^3}{a^2 \left (a^2-b^2\right ) d^2}+\frac {7 b^2 \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{i b+\sqrt {a^2-b^2}}+1\right ) x^3}{a^2 \left (a^2-b^2\right ) d^2}+\frac {14 b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^3}{a^2 \sqrt {b^2-a^2} d^2}-\frac {7 b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {14 b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^3}{a^2 \sqrt {b^2-a^2} d^2}+\frac {7 b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {42 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 ) x^{5/2}}{a^2 \left (a^2-b^2\right ) d^3}-\frac {42 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 ) x^{5/2}}{a^2 \left (a^2-b^2\right ) d^3}+\frac {84 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{5/2}}{a^2 \sqrt {b^2-a^2} d^3}-\frac {42 i b^3 \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{5/2}}{a^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac {84 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{5/2}}{a^2 \sqrt {b^2-a^2} d^3}+\frac {42 i b^3 \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{5/2}}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {210 b^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b-\sqrt {a^2-b^2}}\right ) x^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {210 b^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right ) x^2}{a^2 \left (a^2-b^2\right ) d^4}-\frac {420 b \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a^2 \sqrt {b^2-a^2} d^4}+\frac {210 b^3 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^4}+\frac {420 b \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a^2 \sqrt {b^2-a^2} d^4}-\frac {210 b^3 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^4}+\frac {840 i b^2 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b-\sqrt {a^2-b^2}}\right ) x^{3/2}}{a^2 \left (a^2-b^2\right ) d^5}+\frac {840 i b^2 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right ) x^{3/2}}{a^2 \left (a^2-b^2\right ) d^5}-\frac {1680 i b \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{3/2}}{a^2 \sqrt {b^2-a^2} d^5}+\frac {840 i b^3 \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{3/2}}{a^2 \left (b^2-a^2\right )^{3/2} d^5}+\frac {1680 i b \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{3/2}}{a^2 \sqrt {b^2-a^2} d^5}-\frac {840 i b^3 \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{3/2}}{a^2 \left (b^2-a^2\right )^{3/2} d^5}-\frac {2520 b^2 \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b-\sqrt {a^2-b^2}}\right ) x}{a^2 \left (a^2-b^2\right ) d^6}-\frac {2520 b^2 \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right ) x}{a^2 \left (a^2-b^2\right ) d^6}+\frac {5040 b \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x}{a^2 \sqrt {b^2-a^2} d^6}-\frac {2520 b^3 \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x}{a^2 \left (b^2-a^2\right )^{3/2} d^6}-\frac {5040 b \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x}{a^2 \sqrt {b^2-a^2} d^6}+\frac {2520 b^3 \operatorname {PolyLog}\left (6,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x}{a^2 \left (b^2-a^2\right )^{3/2} d^6}-\frac {5040 i b^2 \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b-\sqrt {a^2-b^2}}\right ) \sqrt {x}}{a^2 \left (a^2-b^2\right ) d^7}-\frac {5040 i b^2 \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right ) \sqrt {x}}{a^2 \left (a^2-b^2\right ) d^7}+\frac {10080 i b \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a^2 \sqrt {b^2-a^2} d^7}-\frac {5040 i b^3 \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a^2 \left (b^2-a^2\right )^{3/2} d^7}-\frac {10080 i b \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a^2 \sqrt {b^2-a^2} d^7}+\frac {5040 i b^3 \operatorname {PolyLog}\left (7,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a^2 \left (b^2-a^2\right )^{3/2} d^7}+\frac {5040 b^2 \operatorname {PolyLog}\left (7,-\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^8}+\frac {5040 b^2 \operatorname {PolyLog}\left (7,-\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^8}-\frac {10080 b \operatorname {PolyLog}\left (8,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^8}+\frac {5040 b^3 \operatorname {PolyLog}\left (8,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^8}+\frac {10080 b \operatorname {PolyLog}\left (8,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^8}-\frac {5040 b^3 \operatorname {PolyLog}\left (8,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^8}\right )\)

Input:

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

Output:

2*(((-I)*b^2*x^(7/2))/(a^2*(a^2 - b^2)*d) + x^4/(8*a^2) + (7*b^2*x^3*Log[1 
 + (a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^ 
2) + (7*b^2*x^3*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^(7/2)*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^(7/2)* 
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^(7/2)*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^(7/2)*Log[1 - (I*a 
*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) 
- ((42*I)*b^2*x^(5/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a 
^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((42*I)*b^2*x^(5/2)*PolyLog[2, -((a* 
E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - 
(7*b^3*x^3*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) + (14*b*x^3*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) + (7*b^3*x^3*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) - (14*b*x^3*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) + (210*b^2*x^2*PolyLog[3, -( 
(a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) 
 + (210*b^2*x^2*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^3/(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(117573120*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**9 - 143700480*sqrt( - a**2 + b**2)*a 
tan((tan((sqrt(x)*d + c)/2)*b + a)/sqrt( - a**2 + b**2))*sin(sqrt(x)*d + c 
)*a**7*b**2 + 50077440*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**4 - 5241600*sqrt( - 
 a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*b + a)/sqrt( - a**2 + b**2))*si 
n(sqrt(x)*d + c)*a**3*b**6 + 80640*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**8 + 117573 
120*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*b + a)/sqrt( - a**2 
+ b**2))*a**8*b - 143700480*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c) 
/2)*b + a)/sqrt( - a**2 + b**2))*a**6*b**3 + 50077440*sqrt( - a**2 + b**2) 
*atan((tan((sqrt(x)*d + c)/2)*b + a)/sqrt( - a**2 + b**2))*a**4*b**5 - 524 
1600*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*b + a)/sqrt( - a**2 
 + b**2))*a**2*b**7 + 80640*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c) 
/2)*b + a)/sqrt( - a**2 + b**2))*b**9 + 58786560*sqrt(x)*cos(sqrt(x)*d + c 
)*a**10*d + 1088640*sqrt(x)*cos(sqrt(x)*d + c)*a**8*b**2*d**3*x - 10124352 
0*sqrt(x)*cos(sqrt(x)*d + c)*a**8*b**2*d + 6048*sqrt(x)*cos(sqrt(x)*d + c) 
*a**6*b**4*d**5*x**2 - 1632960*sqrt(x)*cos(sqrt(x)*d + c)*a**6*b**4*d**3*x 
 + 51166080*sqrt(x)*cos(sqrt(x)*d + c)*a**6*b**4*d + 16*sqrt(x)*cos(sqrt(x 
)*d + c)*a**4*b**6*d**7*x**3 - 7728*sqrt(x)*cos(sqrt(x)*d + c)*a**4*b**...