Integrand size = 20, antiderivative size = 695 \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=-\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {315 i b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {315 b^2 \operatorname {PolyLog}\left (7,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8} \]
[Out]
Time = 0.97 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4290, 4275, 4268, 2611, 6744, 2320, 6724, 4269, 3798, 2221} \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {315 b^2 \operatorname {PolyLog}\left (7,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {315 i b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}-\frac {2 i b^2 x^{7/2}}{d} \]
[In]
[Out]
Rule 2221
Rule 2320
Rule 2611
Rule 3798
Rule 4268
Rule 4269
Rule 4275
Rule 4290
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^7 (a+b \csc (c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^2 x^7+2 a b x^7 \csc (c+d x)+b^2 x^7 \csc ^2(c+d x)\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^4}{4}+(4 a b) \text {Subst}\left (\int x^7 \csc (c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^7 \csc ^2(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}-\frac {(28 a b) \text {Subst}\left (\int x^6 \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(28 a b) \text {Subst}\left (\int x^6 \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\left (14 b^2\right ) \text {Subst}\left (\int x^6 \cot (c+d x) \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(168 i a b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(168 i a b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (28 i b^2\right ) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} x^6}{1-e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {(840 a b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {(840 a b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (84 b^2\right ) \text {Subst}\left (\int x^5 \log \left (1-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {(3360 i a b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {(3360 i a b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4}+\frac {\left (210 i b^2\right ) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {(10080 a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (5,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5}+\frac {(10080 a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (5,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5}-\frac {\left (420 b^2\right ) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {(20160 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^6}+\frac {(20160 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^6}-\frac {\left (630 i b^2\right ) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (4,e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {(20160 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^7}-\frac {(20160 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^7}+\frac {\left (630 b^2\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (5,e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^6} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {315 i b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {(20160 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(7,-x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {(20160 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(7,x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {\left (315 i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (6,e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^7} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {315 i b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {\left (315 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(6,x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{2 d^8} \\ & = -\frac {2 i b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{7/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {42 i b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {210 i b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {315 i b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {315 b^2 \operatorname {PolyLog}\left (7,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8} \\ \end{align*}
Time = 21.47 (sec) , antiderivative size = 1323, normalized size of antiderivative = 1.90 \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^4 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \sin ^2\left (c+d \sqrt {x}\right )}{4 \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2}-\frac {b^2 e^{i c} \csc (c) \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \left (2 d^7 e^{-2 i c} x^{7/2}+7 i d^6 \left (1-e^{-2 i c}\right ) x^3 \log \left (1-e^{-i \left (c+d \sqrt {x}\right )}\right )+7 i d^6 \left (1-e^{-2 i c}\right ) x^3 \log \left (1+e^{-i \left (c+d \sqrt {x}\right )}\right )-42 d^5 \left (1-e^{-2 i c}\right ) x^{5/2} \operatorname {PolyLog}\left (2,-e^{-i \left (c+d \sqrt {x}\right )}\right )-42 d^5 \left (1-e^{-2 i c}\right ) x^{5/2} \operatorname {PolyLog}\left (2,e^{-i \left (c+d \sqrt {x}\right )}\right )+210 i d^4 \left (1-e^{-2 i c}\right ) x^2 \operatorname {PolyLog}\left (3,-e^{-i \left (c+d \sqrt {x}\right )}\right )+210 i d^4 \left (1-e^{-2 i c}\right ) x^2 \operatorname {PolyLog}\left (3,e^{-i \left (c+d \sqrt {x}\right )}\right )+840 d^3 \left (1-e^{-2 i c}\right ) x^{3/2} \operatorname {PolyLog}\left (4,-e^{-i \left (c+d \sqrt {x}\right )}\right )+840 d^3 \left (1-e^{-2 i c}\right ) x^{3/2} \operatorname {PolyLog}\left (4,e^{-i \left (c+d \sqrt {x}\right )}\right )-2520 i d^2 \left (1-e^{-2 i c}\right ) x \operatorname {PolyLog}\left (5,-e^{-i \left (c+d \sqrt {x}\right )}\right )-2520 i d^2 \left (1-e^{-2 i c}\right ) x \operatorname {PolyLog}\left (5,e^{-i \left (c+d \sqrt {x}\right )}\right )-5040 d \left (1-e^{-2 i c}\right ) \sqrt {x} \operatorname {PolyLog}\left (6,-e^{-i \left (c+d \sqrt {x}\right )}\right )-5040 d \left (1-e^{-2 i c}\right ) \sqrt {x} \operatorname {PolyLog}\left (6,e^{-i \left (c+d \sqrt {x}\right )}\right )+5040 i \left (1-e^{-2 i c}\right ) \operatorname {PolyLog}\left (7,-e^{-i \left (c+d \sqrt {x}\right )}\right )+5040 i \left (1-e^{-2 i c}\right ) \operatorname {PolyLog}\left (7,e^{-i \left (c+d \sqrt {x}\right )}\right )\right ) \sin ^2\left (c+d \sqrt {x}\right )}{d^8 \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2}+\frac {4 a b \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \left (d^7 x^{7/2} \log \left (1-e^{i \left (c+d \sqrt {x}\right )}\right )-d^7 x^{7/2} \log \left (1+e^{i \left (c+d \sqrt {x}\right )}\right )+7 i d^6 x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )-7 i d^6 x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )-42 d^5 x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )+42 d^5 x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )-210 i d^4 x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )+210 i d^4 x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )+840 d^3 x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )-840 d^3 x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )+2520 i d^2 x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )-2520 i d^2 x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )-5040 d \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )+5040 d \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )-5040 i \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )+5040 i \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )\right ) \sin ^2\left (c+d \sqrt {x}\right )}{d^8 \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2}+\frac {b^2 x^{7/2} \csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right ) \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \sin ^2\left (c+d \sqrt {x}\right ) \sin \left (\frac {d \sqrt {x}}{2}\right )}{d \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2}+\frac {b^2 x^{7/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right ) \sin ^2\left (c+d \sqrt {x}\right ) \sin \left (\frac {d \sqrt {x}}{2}\right )}{d \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2} \]
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\[\int x^{3} \left (a +b \csc \left (c +d \sqrt {x}\right )\right )^{2}d x\]
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\[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{3} \,d x } \]
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\[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x^{3} \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 6462 vs. \(2 (550) = 1100\).
Time = 0.67 (sec) , antiderivative size = 6462, normalized size of antiderivative = 9.30 \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\text {Too large to display} \]
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\[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x^3\,{\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \]
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