Integrand size = 18, antiderivative size = 333 \[ \int x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=-\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{3/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {6 b^2 x \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {24 i a b \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {24 i a b \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4} \]
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Time = 0.51 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4290, 4275, 4268, 2611, 6744, 2320, 6724, 4269, 3798, 2221} \[ \int x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {24 i a b \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {24 i a b \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {3 b^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {6 b^2 x \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 b^2 x^{3/2} \cot \left (c+d \sqrt {x}\right )}{d}-\frac {2 i b^2 x^{3/2}}{d} \]
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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^3 (a+b \csc (c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^2 x^3+2 a b x^3 \csc (c+d x)+b^2 x^3 \csc ^2(c+d x)\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^2}{2}+(4 a b) \text {Subst}\left (\int x^3 \csc (c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^3 \csc ^2(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{3/2} \cot \left (c+d \sqrt {x}\right )}{d}-\frac {(12 a b) \text {Subst}\left (\int x^2 \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(12 a b) \text {Subst}\left (\int x^2 \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\left (6 b^2\right ) \text {Subst}\left (\int x^2 \cot (c+d x) \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{3/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(24 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(24 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (12 i b^2\right ) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} x^2}{1-e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{3/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {6 b^2 x \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {(24 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {(24 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int x \log \left (1-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = -\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{3/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {6 b^2 x \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {(24 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {(24 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {\left (6 i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = -\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{3/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {6 b^2 x \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {24 i a b \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {24 i a b \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4} \\ & = -\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 a b x^{3/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^{3/2} \cot \left (c+d \sqrt {x}\right )}{d}+\frac {6 b^2 x \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {24 i a b \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {24 i a b \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4} \\ \end{align*}
Time = 10.31 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.35 \[ \int x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^2}{2}-\frac {2 i b \left (\frac {2 b d^3 e^{2 i c} x^{3/2}}{-1+e^{2 i c}}+\left (6 b d \sqrt {x}-6 a d^2 x\right ) \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )+i \left (3 b d^2 x \log \left (1-e^{i \left (c+d \sqrt {x}\right )}\right )+2 a d^3 x^{3/2} \log \left (1-e^{i \left (c+d \sqrt {x}\right )}\right )+3 b d^2 x \log \left (1+e^{i \left (c+d \sqrt {x}\right )}\right )-2 a d^3 x^{3/2} \log \left (1+e^{i \left (c+d \sqrt {x}\right )}\right )-6 i \left (b d \sqrt {x}+a d^2 x\right ) \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )+6 \left (b-2 a d \sqrt {x}\right ) \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )+6 b \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )+12 a d \sqrt {x} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )-12 i a \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )+12 i a \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )\right )\right )}{d^4}+\frac {b^2 x^{3/2} \csc \left (\frac {c}{2}\right ) \csc \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right ) \sin \left (\frac {d \sqrt {x}}{2}\right )}{d}+\frac {b^2 x^{3/2} \sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right ) \sin \left (\frac {d \sqrt {x}}{2}\right )}{d} \]
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\[\int x \left (a +b \csc \left (c +d \sqrt {x}\right )\right )^{2}d x\]
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\[ \int x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2} x \,d x } \]
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\[ \int x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1950 vs. \(2 (262) = 524\).
Time = 0.32 (sec) , antiderivative size = 1950, normalized size of antiderivative = 5.86 \[ \int x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\text {Too large to display} \]
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\[ \int x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2} x \,d x } \]
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Timed out. \[ \int x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x\,{\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \]
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