Integrand size = 18, antiderivative size = 432 \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^4}{4}-\frac {4 b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {5040 i b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {5040 i b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {10080 i b \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {10080 i b \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8} \]
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Time = 0.51 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {14, 4290, 4268, 2611, 6744, 2320, 6724} \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^4}{4}-\frac {4 b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {10080 i b \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {10080 i b \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}-\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {5040 i b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {5040 i b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {420 i b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {14 i b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2} \]
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Rule 14
Rule 2320
Rule 2611
Rule 4268
Rule 4290
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \int \left (a x^3+b x^3 \csc \left (c+d \sqrt {x}\right )\right ) \, dx \\ & = \frac {a x^4}{4}+b \int x^3 \csc \left (c+d \sqrt {x}\right ) \, dx \\ & = \frac {a x^4}{4}+(2 b) \text {Subst}\left (\int x^7 \csc (c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a x^4}{4}-\frac {4 b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {(14 b) \text {Subst}\left (\int x^6 \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(14 b) \text {Subst}\left (\int x^6 \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d} \\ & = \frac {a x^4}{4}-\frac {4 b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(84 i 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 {(84 i 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 {a x^4}{4}-\frac {4 b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {(420 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 {(420 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 {a x^4}{4}-\frac {4 b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {(1680 i 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 {(1680 i 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 {a x^4}{4}-\frac {4 b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {(5040 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 {(5040 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 {a x^4}{4}-\frac {4 b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {5040 i b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {5040 i b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {(10080 i b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^6}+\frac {(10080 i b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^6} \\ & = \frac {a x^4}{4}-\frac {4 b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {5040 i b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {5040 i b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {(10080 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^7}-\frac {(10080 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^7} \\ & = \frac {a x^4}{4}-\frac {4 b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {5040 i b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {5040 i b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {(10080 i 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 {(10080 i 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 {a x^4}{4}-\frac {4 b x^{7/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {5040 i b x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {5040 i b x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {10080 i b \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {10080 i b \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.03 \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^4}{4}+\frac {2 b \left (d^7 x^{7/2} \log \left (1-e^{i \left (c+d \sqrt {x}\right )}\right )-d^7 x^{7/2} \log \left (1+e^{i \left (c+d \sqrt {x}\right )}\right )+7 i d^6 x^3 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )-7 i d^6 x^3 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )-42 d^5 x^{5/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )+42 d^5 x^{5/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )-210 i d^4 x^2 \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )+210 i d^4 x^2 \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )+840 d^3 x^{3/2} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )-840 d^3 x^{3/2} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )+2520 i d^2 x \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )-2520 i d^2 x \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )-5040 d \sqrt {x} \operatorname {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )+5040 d \sqrt {x} \operatorname {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )-5040 i \operatorname {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )+5040 i \operatorname {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )\right )}{d^8} \]
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\[\int x^{3} \left (a +b \csc \left (c +d \sqrt {x}\right )\right )d x\]
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\[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )} x^{3} \,d x } \]
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\[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\int x^{3} \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1498 vs. \(2 (338) = 676\).
Time = 0.33 (sec) , antiderivative size = 1498, normalized size of antiderivative = 3.47 \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\text {Too large to display} \]
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\[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\int x^3\,\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right ) \,d x \]
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