Integrand size = 22, antiderivative size = 1977 \[ \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {2 i b^2 x^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{5/2}}{5 a^2}+\frac {8 b^2 x^{3/2} \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 {8 b^2 x^{3/2} \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^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^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^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^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 {24 i b^2 x \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 {24 i b^2 x \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 {8 b^3 x^{3/2} \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 {16 b x^{3/2} \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 {8 b^3 x^{3/2} \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 {16 b x^{3/2} \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 {48 b^2 \sqrt {x} \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 {48 b^2 \sqrt {x} \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 {24 i b^3 x \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 {48 i b x \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 {24 i b^3 x \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 {48 i b x \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 {48 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 )}{a^2 \left (a^2-b^2\right ) d^5}+\frac {48 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 )}{a^2 \left (a^2-b^2\right ) d^5}+\frac {48 b^3 \sqrt {x} \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 {96 b \sqrt {x} \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 {48 b^3 \sqrt {x} \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 {96 b \sqrt {x} \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 {48 i b^3 \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 {96 i b \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 {48 i b^3 \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 {96 i b \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 {2 b^2 x^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]
Time = 3.55 (sec) , antiderivative size = 1977, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4290, 4276, 3405, 3404, 2296, 2221, 2611, 6744, 2320, 6724, 4617} \[ \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {2 i x^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 {2 i x^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 {8 x^{3/2} \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 {8 x^{3/2} \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 {24 i x \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 {24 i x \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 {48 \sqrt {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 {48 \sqrt {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 {48 i \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 {48 i \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 {2 i x^2 b^2}{a^2 \left (a^2-b^2\right ) d}+\frac {8 x^{3/2} \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 {8 x^{3/2} \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 {24 i x \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 {24 i x \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 {48 \sqrt {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 {48 \sqrt {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 {48 i \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 {48 i \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 {2 x^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 {4 i x^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 {4 i x^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 {16 x^{3/2} \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 {16 x^{3/2} \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 {48 i x \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 {48 i x \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 {96 \sqrt {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 {96 \sqrt {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 {96 i \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 {96 i \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 {2 x^{5/2}}{5 a^2} \]
[In]
[Out]
Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3404
Rule 3405
Rule 4276
Rule 4290
Rule 4617
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^4}{(a+b \csc (c+d x))^2} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {x^4}{a^2}+\frac {b^2 x^4}{a^2 (b+a \sin (c+d x))^2}-\frac {2 b x^4}{a^2 (b+a \sin (c+d x))}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 x^{5/2}}{5 a^2}-\frac {(4 b) \text {Subst}\left (\int \frac {x^4}{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^4}{(b+a \sin (c+d x))^2} \, dx,x,\sqrt {x}\right )}{a^2} \\ & = \frac {2 x^{5/2}}{5 a^2}-\frac {2 b^2 x^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^4}{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^4}{b+a \sin (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right )}+\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {x^3 \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^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{5/2}}{5 a^2}-\frac {2 b^2 x^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^4}{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^4}{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^4}{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 (8 i b^2\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^3}{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 (8 i b^2\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^3}{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^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{5/2}}{5 a^2}+\frac {8 b^2 x^{3/2} \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 {8 b^2 x^{3/2} \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^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^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^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^4}{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^4}{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 (24 b^2\right ) \text {Subst}\left (\int x^2 \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 (24 b^2\right ) \text {Subst}\left (\int x^2 \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 {(16 i b) \text {Subst}\left (\int x^3 \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 {(16 i b) \text {Subst}\left (\int x^3 \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^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{5/2}}{5 a^2}+\frac {8 b^2 x^{3/2} \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 {8 b^2 x^{3/2} \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^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^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^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^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 {24 i b^2 x \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 {24 i b^2 x \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 {16 b x^{3/2} \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 {16 b x^{3/2} \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^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 (48 i b^2\right ) \text {Subst}\left (\int x \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 (48 i b^2\right ) \text {Subst}\left (\int x \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 {(48 b) \text {Subst}\left (\int x^2 \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 {(48 b) \text {Subst}\left (\int x^2 \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 (8 i b^3\right ) \text {Subst}\left (\int x^3 \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 (8 i b^3\right ) \text {Subst}\left (\int x^3 \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*}
Time = 14.05 (sec) , antiderivative size = 2316, normalized size of antiderivative = 1.17 \[ \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Result too large to show} \]
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\[\int \frac {x^{\frac {3}{2}}}{\left (a +b \csc \left (c +d \sqrt {x}\right )\right )^{2}}d x\]
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\[ \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{\frac {3}{2}}}{{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x^{\frac {3}{2}}}{\left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{\frac {3}{2}}}{{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x^{3/2}}{{\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]
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