\(\int \frac {x^3}{(a+b \csc (c+d \sqrt {x}))^2} \, dx\) [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=-\frac {2 i b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac {x^4}{4 a^2}+\frac {14 b^2 x^3 \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 {14 b^2 x^3 \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^{7/2} \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^{7/2} \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^{7/2} \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^{7/2} \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 {84 i b^2 x^{5/2} \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 {84 i b^2 x^{5/2} \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 {14 b^3 x^3 \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 {28 b x^3 \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 {14 b^3 x^3 \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 {28 b x^3 \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 {420 b^2 x^2 \operatorname {PolyLog}\left (3,-\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^4}+\frac {420 b^2 x^2 \operatorname {PolyLog}\left (3,-\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^4}-\frac {84 i b^3 x^{5/2} \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 {168 i b x^{5/2} \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 {84 i b^3 x^{5/2} \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 {168 i b x^{5/2} \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 {1680 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,-\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^5}+\frac {1680 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,-\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^5}+\frac {420 b^3 x^2 \operatorname {PolyLog}\left (4,\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^4}-\frac {840 b x^2 \operatorname {PolyLog}\left (4,\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^4}-\frac {420 b^3 x^2 \operatorname {PolyLog}\left (4,\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^4}+\frac {840 b x^2 \operatorname {PolyLog}\left (4,\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^4}-\frac {5040 b^2 x \operatorname {PolyLog}\left (5,-\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^6}-\frac {5040 b^2 x \operatorname {PolyLog}\left (5,-\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^6}+\frac {1680 i b^3 x^{3/2} \operatorname {PolyLog}\left (5,\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^5}-\frac {3360 i b x^{3/2} \operatorname {PolyLog}\left (5,\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^5}-\frac {1680 i b^3 x^{3/2} \operatorname {PolyLog}\left (5,\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^5}+\frac {3360 i b x^{3/2} \operatorname {PolyLog}\left (5,\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^5}-\frac {10080 i b^2 \sqrt {x} \operatorname {PolyLog}\left (6,-\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^7}-\frac {10080 i b^2 \sqrt {x} \operatorname {PolyLog}\left (6,-\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^7}-\frac {5040 b^3 x \operatorname {PolyLog}\left (6,\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^6}+\frac {10080 b x \operatorname {PolyLog}\left (6,\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^6}+\frac {5040 b^3 x \operatorname {PolyLog}\left (6,\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^6}-\frac {10080 b x \operatorname {PolyLog}\left (6,\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^6}+\frac {10080 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^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 i b^3 \sqrt {x} \operatorname {PolyLog}\left (7,\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^7}+\frac {20160 i b \sqrt {x} \operatorname {PolyLog}\left (7,\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^7}+\frac {10080 i b^3 \sqrt {x} \operatorname {PolyLog}\left (7,\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^7}-\frac {20160 i b \sqrt {x} \operatorname {PolyLog}\left (7,\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^7}+\frac {10080 b^3 \operatorname {PolyLog}\left (8,\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^8}-\frac {20160 b \operatorname {PolyLog}\left (8,\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^8}-\frac {10080 b^3 \operatorname {PolyLog}\left (8,\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^8}+\frac {20160 b \operatorname {PolyLog}\left (8,\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^8}-\frac {2 b^2 x^{7/2} \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 )} \]

[Out]

10080*I*b^3*polylog(7,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^7+20160*I*
b*polylog(7,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))*x^(1/2)/a^2/d^7/(-a^2+b^2)^(1/2)+2*I*b^3*x^(7/2)*ln
(1-I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d+84*I*b^3*x^(5/2)*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+1680*I*b^2*x^(3/2)*polylog(4,-a*exp(I*(c+d*x^(1/
2)))/(I*b-(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^5+1680*I*b^2*x^(3/2)*polylog(4,-a*exp(I*(c+d*x^(1/2)))/(I*b+(a^2-b
^2)^(1/2)))/a^2/(a^2-b^2)/d^5+1680*I*b^3*x^(3/2)*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^(7/2)*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)+
168*I*b*x^(5/2)*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)+3360*I*b*x^(
3/2)*polylog(5,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/d^5/(-a^2+b^2)^(1/2)-2*b^2*x^(7/2)*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^(7/2)*ln(1-I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)
))/a^2/(-a^2+b^2)^(3/2)/d-84*I*b^2*x^(5/2)*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^3-84*I*b^2*x^(5/2)*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^3-84*I*b^3*x
^(5/2)*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^3-1680*I*b^3*x^(3/2)*po
lylog(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^(7/2)*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)-168*I*b*x^(5/2)*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)-3360*I*b*x^(3/2)*polylog(5,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^
2)^(1/2)))/a^2/d^5/(-a^2+b^2)^(1/2)-10080*I*b^2*polylog(6,-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^7-10080*I*b^2*polylog(6,-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^7-10080*I*b^3*polylog(7,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^7-20
160*I*b*polylog(7,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))*x^(1/2)/a^2/d^7/(-a^2+b^2)^(1/2)+1/4*x^4/a^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/d^4/(-a^2+b^2)^(1/2)+10080*b*x*polylog(
6,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/d^6/(-a^2+b^2)^(1/2)-10080*b*x*polylog(6,I*a*exp(I*(c+d*x
^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/d^6/(-a^2+b^2)^(1/2)-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^2-14*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+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^
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+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+420*b^3*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)^(3/2)/d^4-420*b^3*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)^(3/2)/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^
6-5040*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^6-5040*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+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+28*b*x^3*polylog(2,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2
)^(1/2)))/a^2/d^2/(-a^2+b^2)^(1/2)-28*b*x^3*polylog(2,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/d^2/(
-a^2+b^2)^(1/2)-840*b*x^2*polylog(4,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/d^4/(-a^2+b^2)^(1/2)+10
080*b^3*polylog(8,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^8-10080*b^3*polylog(8,
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+10080*b^2*polylog(7,-a*exp(I*(c+d*x^(1
/2)))/(I*b-(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^8+10080*b^2*polylog(7,-a*exp(I*(c+d*x^(1/2)))/(I*b+(a^2-b^2)^(1/2
)))/a^2/(a^2-b^2)/d^8-20160*b*polylog(8,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/d^8/(-a^2+b^2)^(1/2
)+20160*b*polylog(8,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/d^8/(-a^2+b^2)^(1/2)

Rubi [A] (verified)

Time = 5.18 (sec) , antiderivative size = 3205, normalized size of antiderivative = 1.00, number of steps used = 61, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {4290, 4276, 3405, 3404, 2296, 2221, 2611, 6744, 2320, 6724, 4617} \[ \int \frac {x^3}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {x^4}{4 a^2}+\frac {4 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 {2 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 {4 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 {2 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^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}-\frac {2 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 {14 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^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 {28 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 {14 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 {28 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 {14 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 {84 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^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 {168 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 {84 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 {168 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 {84 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 {420 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^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 {840 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 {420 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 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 {420 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 {1680 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^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 {3360 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 {1680 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 {3360 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 {1680 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 {5040 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^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 {10080 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 {5040 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 {10080 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 {5040 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 {10080 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^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 {20160 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 {10080 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 {20160 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 {10080 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 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^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 {20160 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 {10080 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 {20160 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 {10080 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} \]

[In]

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

[Out]

((-2*I)*b^2*x^(7/2))/(a^2*(a^2 - b^2)*d) + x^4/(4*a^2) + (14*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) + (14*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) - ((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) + ((4*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) + ((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) - ((4*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) - ((84*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) - ((84*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) - (14*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) + (28*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) + (14*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
) - (28*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) + (42
0*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) + (420*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) - ((84*I)*b^3*x^(5/
2)*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) + ((168*I)*b*x
^(5/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((84*I)*b^
3*x^(5/2)*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) - ((168
*I)*b*x^(5/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((1
680*I)*b^2*x^(3/2)*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) + (
(1680*I)*b^2*x^(3/2)*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) +
(420*b^3*x^2*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (
840*b*x^2*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (420*b^
3*x^2*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (840*b*x^
2*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (5040*b^2*x*Pol
yLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^6) - (5040*b^2*x*PolyLog[5,
-((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^6) + ((1680*I)*b^3*x^(3/2)*PolyLog[5
, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^5) - ((3360*I)*b*x^(3/2)*Poly
Log[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^5) - ((1680*I)*b^3*x^(3/2)
*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^5) + ((3360*I)*b*x^
(3/2)*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^5) - ((10080*I)*
b^2*Sqrt[x]*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^7) - ((10080*
I)*b^2*Sqrt[x]*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^7) - (5040
*b^3*x*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^6) + (10080*b
*x*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^6) + (5040*b^3*x*Po
lyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^6) - (10080*b*x*PolyLo
g[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^6) + (10080*b^2*PolyLog[7, -
((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^8) + (10080*b^2*PolyLog[7, -((a*E^(I*
(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^8) - ((10080*I)*b^3*Sqrt[x]*PolyLog[7, (I*a*E^
(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^7) + ((20160*I)*b*Sqrt[x]*PolyLog[7, (
I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^7) + ((10080*I)*b^3*Sqrt[x]*PolyLo
g[7, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^7) - ((20160*I)*b*Sqrt[x]*
PolyLog[7, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^7) + (10080*b^3*PolyLo
g[8, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^8) - (20160*b*PolyLog[8, (
I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^8) - (10080*b^3*PolyLog[8, (I*a*E^
(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^8) + (20160*b*PolyLog[8, (I*a*E^(I*(c
+ d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^8) - (2*b^2*x^(7/2)*Cos[c + d*Sqrt[x]])/(a*(a^
2 - b^2)*d*(b + a*Sin[c + d*Sqrt[x]]))

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3404

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[(c + d*x)^m*(E
^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3405

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[b*(c + d*x)^m*(Cos[
e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f*x]))), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[b*d*(m/(f*(a^2 - b^2))), Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/(a + b*Sin[e + f*x])), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4276

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*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 4290

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(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]

Rule 4617

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Dist[I, Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a - Rt[-a^2 + b
^2, 2] + b*E^(I*(c + d*x)))), x], x] + Dist[I, Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a + Rt[-a^2 + b^2, 2] + b*E
^(I*(c + d*x)))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^7}{(a+b \csc (c+d x))^2} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {x^7}{a^2}+\frac {b^2 x^7}{a^2 (b+a \sin (c+d x))^2}-\frac {2 b x^7}{a^2 (b+a \sin (c+d x))}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {x^4}{4 a^2}-\frac {(4 b) \text {Subst}\left (\int \frac {x^7}{b+a \sin (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {x^7}{(b+a \sin (c+d x))^2} \, dx,x,\sqrt {x}\right )}{a^2} \\ & = \frac {x^4}{4 a^2}-\frac {2 b^2 x^{7/2} \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 )}-\frac {(8 b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^7}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{a^2}-\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {x^7}{b+a \sin (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right )}+\frac {\left (14 b^2\right ) \text {Subst}\left (\int \frac {x^6 \cos (c+d x)}{b+a \sin (c+d x)} \, dx,x,\sqrt {x}\right )}{a \left (a^2-b^2\right ) d} \\ & = -\frac {2 i b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac {x^4}{4 a^2}-\frac {2 b^2 x^{7/2} \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 )}-\frac {\left (4 b^3\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^7}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right )}+\frac {(8 i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^7}{2 b-2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2}}-\frac {(8 i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^7}{2 b+2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2}}+\frac {\left (14 i b^2\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^6}{i b-\sqrt {a^2-b^2}+a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{a \left (a^2-b^2\right ) d}+\frac {\left (14 i b^2\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^6}{i b+\sqrt {a^2-b^2}+a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{a \left (a^2-b^2\right ) d} \\ & = -\frac {2 i b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac {x^4}{4 a^2}+\frac {14 b^2 x^3 \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 {14 b^2 x^3 \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 i b x^{7/2} \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 x^{7/2} \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 b^2 x^{7/2} \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 )}-\frac {\left (4 i b^3\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^7}{2 b-2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{a \left (-a^2+b^2\right )^{3/2}}+\frac {\left (4 i b^3\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^7}{2 b+2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{a \left (-a^2+b^2\right )^{3/2}}-\frac {\left (84 b^2\right ) \text {Subst}\left (\int x^5 \log \left (1+\frac {a e^{i (c+d x)}}{i b-\sqrt {a^2-b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {\left (84 b^2\right ) \text {Subst}\left (\int x^5 \log \left (1+\frac {a e^{i (c+d x)}}{i b+\sqrt {a^2-b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {(28 i b) \text {Subst}\left (\int x^6 \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {(28 i b) \text {Subst}\left (\int x^6 \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^2+b^2} d} \\ & = -\frac {2 i b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac {x^4}{4 a^2}+\frac {14 b^2 x^3 \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 {14 b^2 x^3 \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^{7/2} \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^{7/2} \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^{7/2} \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^{7/2} \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 {84 i b^2 x^{5/2} \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 {84 i b^2 x^{5/2} \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 {28 b x^3 \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 {28 b x^3 \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 {2 b^2 x^{7/2} \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 )}+\frac {\left (420 i b^2\right ) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (2,-\frac {a e^{i (c+d x)}}{i b-\sqrt {a^2-b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {\left (420 i b^2\right ) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (2,-\frac {a e^{i (c+d x)}}{i b+\sqrt {a^2-b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {(168 b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {(168 b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {\left (14 i b^3\right ) \text {Subst}\left (\int x^6 \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {\left (14 i b^3\right ) \text {Subst}\left (\int x^6 \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 15.83 (sec) , antiderivative size = 3831, normalized size of antiderivative = 1.20 \[ \int \frac {x^3}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(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*Sqr
t[(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)*PolyLog[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[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (42*I)*
b^2*d^5*E^(I*c)*x^(5/2)*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)])
] - (210*I)*b*d^4*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sq
rt[(a^2 - b^2)*E^((2*I)*c)]))] - (84*I)*a^2*d^5*E^(I*c)*x^(5/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*
E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (42*I)*b^2*d^5*E^(I*c)*x^(5/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt
[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 840*b*d^3*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^(3/2)*PolyL
og[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 420*a^2*d^4*E^(I*c)*x^2*P
olyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 210*b^2*d^4*E^(I*c)*x
^2*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 840*b*d^3*Sqrt[(a
^2 - b^2)*E^((2*I)*c)]*x^(3/2)*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*
I)*c)]))] + 420*a^2*d^4*E^(I*c)*x^2*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E
^((2*I)*c)]))] - 210*b^2*d^4*E^(I*c)*x^2*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b
^2)*E^((2*I)*c)]))] + (2520*I)*b*d^2*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x*PolyLog[5, (I*a*E^(I*(2*c + d*Sqrt[x])))/
(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (1680*I)*a^2*d^3*E^(I*c)*x^(3/2)*PolyLog[5, (I*a*E^(I*(2*c +
d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (840*I)*b^2*d^3*E^(I*c)*x^(3/2)*PolyLog[5, (I*a*
E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2520*I)*b*d^2*Sqrt[(a^2 - b^2)*E^((
2*I)*c)]*x*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (1680*I)
*a^2*d^3*E^(I*c)*x^(3/2)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]
))] - (840*I)*b^2*d^3*E^(I*c)*x^(3/2)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)
*E^((2*I)*c)]))] - 5040*b*d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[6, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*
E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 5040*a^2*d^2*E^(I*c)*x*PolyLog[6, (I*a*E^(I*(2*c + d*Sqrt[x])))/
(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2520*b^2*d^2*E^(I*c)*x*PolyLog[6, (I*a*E^(I*(2*c + d*Sqrt[x])
))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 5040*b*d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[6,
-((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 5040*a^2*d^2*E^(I*c)*x*PolyLog
[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 2520*b^2*d^2*E^(I*c)*x*Pol
yLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (5040*I)*b*Sqrt[(a^2 -
b^2)*E^((2*I)*c)]*PolyLog[7, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (
10080*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[7, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*
I)*c)])] - (5040*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[7, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 -
b^2)*E^((2*I)*c)])] - (5040*I)*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E
^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (10080*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt
[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (5040*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[7, -((a*E^(I*(
2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 10080*a^2*E^(I*c)*PolyLog[8, (I*a*E^(I*(2
*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 5040*b^2*E^(I*c)*PolyLog[8, (I*a*E^(I*(2*c
+ d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 10080*a^2*E^(I*c)*PolyLog[8, -((a*E^(I*(2*c +
d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 5040*b^2*E^(I*c)*PolyLog[8, -((a*E^(I*(2*c + d*
Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))]))/(d^7*E^(I*c)*Sqrt[(a^2 - b^2)*E^((2*I)*c)]))*(b +
a*Sin[c + d*Sqrt[x]])^2)/(a^2*(a^2 - b^2)*d*(-1 + E^((2*I)*c))*(a + b*Csc[c + d*Sqrt[x]])^2) + (Csc[c/2]*Csc[
c + d*Sqrt[x]]^2*Sec[c/2]*(b + a*Sin[c + d*Sqrt[x]])*(-(b^3*x^(7/2)*Cos[c]) - a*b^2*x^(7/2)*Sin[d*Sqrt[x]]))/(
a^2*(-a + b)*(a + b)*d*(a + b*Csc[c + d*Sqrt[x]])^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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