Integrand size = 22, antiderivative size = 241 \[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=-\frac {2 i b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {8 i a b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {8 a b \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 a b \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3} \]
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Time = 0.38 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4290, 4275, 4268, 2611, 2320, 6724, 4269, 3798, 2221, 2317, 2438} \[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {8 a b \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 a b \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 i a b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}-\frac {2 i b^2 x}{d} \]
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Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3798
Rule 4268
Rule 4269
Rule 4275
Rule 4290
Rule 6724
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^2 (a+b \csc (c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^2 x^2+2 a b x^2 \csc (c+d x)+b^2 x^2 \csc ^2(c+d x)\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{3} a^2 x^{3/2}+(4 a b) \text {Subst}\left (\int x^2 \csc (c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^2 \csc ^2(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}-\frac {(8 a b) \text {Subst}\left (\int x \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(8 a b) \text {Subst}\left (\int x \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\left (4 b^2\right ) \text {Subst}\left (\int x \cot (c+d x) \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}+\frac {8 i a b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(8 i a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(8 i a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (8 i b^2\right ) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} x}{1-e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {8 i a b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(8 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {(8 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {\left (4 b^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = -\frac {2 i b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {8 i a b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 a b \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 a b \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3} \\ & = -\frac {2 i b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {8 i a b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {8 a b \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 a b \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(681\) vs. \(2(241)=482\).
Time = 5.22 (sec) , antiderivative size = 681, normalized size of antiderivative = 2.83 \[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {-12 i b^2 d^2 x-2 a^2 d^3 x^{3/2}+2 a^2 d^3 e^{2 i c} x^{3/2}-12 b^2 d \sqrt {x} \log \left (1-e^{-i \left (c+d \sqrt {x}\right )}\right )+12 b^2 d e^{2 i c} \sqrt {x} \log \left (1-e^{-i \left (c+d \sqrt {x}\right )}\right )-12 a b d^2 x \log \left (1-e^{-i \left (c+d \sqrt {x}\right )}\right )+12 a b d^2 e^{2 i c} x \log \left (1-e^{-i \left (c+d \sqrt {x}\right )}\right )-12 b^2 d \sqrt {x} \log \left (1+e^{-i \left (c+d \sqrt {x}\right )}\right )+12 b^2 d e^{2 i c} \sqrt {x} \log \left (1+e^{-i \left (c+d \sqrt {x}\right )}\right )+12 a b d^2 x \log \left (1+e^{-i \left (c+d \sqrt {x}\right )}\right )-12 a b d^2 e^{2 i c} x \log \left (1+e^{-i \left (c+d \sqrt {x}\right )}\right )+12 i b \left (-1+e^{2 i c}\right ) \left (b-2 a d \sqrt {x}\right ) \operatorname {PolyLog}\left (2,-e^{-i \left (c+d \sqrt {x}\right )}\right )+12 i b \left (-1+e^{2 i c}\right ) \left (b+2 a d \sqrt {x}\right ) \operatorname {PolyLog}\left (2,e^{-i \left (c+d \sqrt {x}\right )}\right )+24 a b \operatorname {PolyLog}\left (3,-e^{-i \left (c+d \sqrt {x}\right )}\right )-24 a b e^{2 i c} \operatorname {PolyLog}\left (3,-e^{-i \left (c+d \sqrt {x}\right )}\right )-24 a b \operatorname {PolyLog}\left (3,e^{-i \left (c+d \sqrt {x}\right )}\right )+24 a b e^{2 i c} \operatorname {PolyLog}\left (3,e^{-i \left (c+d \sqrt {x}\right )}\right )-3 b^2 d^2 x \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 )+3 b^2 d^2 e^{2 i c} x \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 )-3 b^2 d^2 x \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 )+3 b^2 d^2 e^{2 i c} x \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 )}{3 d^3 \left (-1+e^{2 i c}\right )} \]
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\[\int \left (a +b \csc \left (c +d \sqrt {x}\right )\right )^{2} \sqrt {x}d x\]
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\[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2} \sqrt {x} \,d x } \]
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\[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int \sqrt {x} \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1217 vs. \(2 (190) = 380\).
Time = 0.30 (sec) , antiderivative size = 1217, normalized size of antiderivative = 5.05 \[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\text {Too large to display} \]
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\[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2} \sqrt {x} \,d x } \]
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Timed out. \[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int \sqrt {x}\,{\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \]
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