Optimal. Leaf size=513 \[ \frac{20 i a b x^2 \text{PolyLog}\left (2,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i a b x^2 \text{PolyLog}\left (2,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{80 a b x^{3/2} \text{PolyLog}\left (3,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{80 a b x^{3/2} \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{240 i a b x \text{PolyLog}\left (4,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{240 i a b x \text{PolyLog}\left (4,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{480 a b \sqrt{x} \text{PolyLog}\left (5,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{480 a b \sqrt{x} \text{PolyLog}\left (5,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{480 i a b \text{PolyLog}\left (6,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{480 i a b \text{PolyLog}\left (6,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{20 i b^2 x^{3/2} \text{PolyLog}\left (2,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{30 b^2 x \text{PolyLog}\left (3,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{30 i b^2 \sqrt{x} \text{PolyLog}\left (4,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{15 b^2 \text{PolyLog}\left (5,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^6}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{2 b^2 x^{5/2} \cot \left (c+d \sqrt{x}\right )}{d}-\frac{2 i b^2 x^{5/2}}{d} \]
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Rubi [A] time = 0.612184, antiderivative size = 513, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4205, 4190, 4183, 2531, 6609, 2282, 6589, 4184, 3717, 2190} \[ \frac{20 i a b x^2 \text{PolyLog}\left (2,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i a b x^2 \text{PolyLog}\left (2,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{80 a b x^{3/2} \text{PolyLog}\left (3,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{80 a b x^{3/2} \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{240 i a b x \text{PolyLog}\left (4,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{240 i a b x \text{PolyLog}\left (4,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{480 a b \sqrt{x} \text{PolyLog}\left (5,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{480 a b \sqrt{x} \text{PolyLog}\left (5,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{480 i a b \text{PolyLog}\left (6,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{480 i a b \text{PolyLog}\left (6,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{20 i b^2 x^{3/2} \text{PolyLog}\left (2,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{30 b^2 x \text{PolyLog}\left (3,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{30 i b^2 \sqrt{x} \text{PolyLog}\left (4,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{15 b^2 \text{PolyLog}\left (5,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^6}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{2 b^2 x^{5/2} \cot \left (c+d \sqrt{x}\right )}{d}-\frac{2 i b^2 x^{5/2}}{d} \]
Antiderivative was successfully verified.
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Rule 4205
Rule 4190
Rule 4183
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 4184
Rule 3717
Rule 2190
Rubi steps
\begin{align*} \int x^2 \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2 \, dx &=2 \operatorname{Subst}\left (\int x^5 (a+b \csc (c+d x))^2 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^2 x^5+2 a b x^5 \csc (c+d x)+b^2 x^5 \csc ^2(c+d x)\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^3}{3}+(4 a b) \operatorname{Subst}\left (\int x^5 \csc (c+d x) \, dx,x,\sqrt{x}\right )+\left (2 b^2\right ) \operatorname{Subst}\left (\int x^5 \csc ^2(c+d x) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{5/2} \cot \left (c+d \sqrt{x}\right )}{d}-\frac{(20 a b) \operatorname{Subst}\left (\int x^4 \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{(20 a b) \operatorname{Subst}\left (\int x^4 \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{\left (10 b^2\right ) \operatorname{Subst}\left (\int x^4 \cot (c+d x) \, dx,x,\sqrt{x}\right )}{d}\\ &=-\frac{2 i b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{5/2} \cot \left (c+d \sqrt{x}\right )}{d}+\frac{20 i a b x^2 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i a b x^2 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{(80 i a b) \operatorname{Subst}\left (\int x^3 \text{Li}_2\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}+\frac{(80 i a b) \operatorname{Subst}\left (\int x^3 \text{Li}_2\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}-\frac{\left (20 i b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 i (c+d x)} x^4}{1-e^{2 i (c+d x)}} \, dx,x,\sqrt{x}\right )}{d}\\ &=-\frac{2 i b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{5/2} \cot \left (c+d \sqrt{x}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}+\frac{20 i a b x^2 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i a b x^2 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{80 a b x^{3/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{80 a b x^{3/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{(240 a b) \operatorname{Subst}\left (\int x^2 \text{Li}_3\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}-\frac{(240 a b) \operatorname{Subst}\left (\int x^2 \text{Li}_3\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}-\frac{\left (40 b^2\right ) \operatorname{Subst}\left (\int x^3 \log \left (1-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}\\ &=-\frac{2 i b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{5/2} \cot \left (c+d \sqrt{x}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}+\frac{20 i a b x^2 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i a b x^2 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i b^2 x^{3/2} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{80 a b x^{3/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{80 a b x^{3/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{240 i a b x \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{240 i a b x \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{(480 i a b) \operatorname{Subst}\left (\int x \text{Li}_4\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^4}-\frac{(480 i a b) \operatorname{Subst}\left (\int x \text{Li}_4\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^4}+\frac{\left (60 i b^2\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_2\left (e^{2 i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}\\ &=-\frac{2 i b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{5/2} \cot \left (c+d \sqrt{x}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}+\frac{20 i a b x^2 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i a b x^2 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i b^2 x^{3/2} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{80 a b x^{3/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{80 a b x^{3/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{30 b^2 x \text{Li}_3\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{240 i a b x \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{240 i a b x \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{480 a b \sqrt{x} \text{Li}_5\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{480 a b \sqrt{x} \text{Li}_5\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{(480 a b) \operatorname{Subst}\left (\int \text{Li}_5\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^5}+\frac{(480 a b) \operatorname{Subst}\left (\int \text{Li}_5\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^5}-\frac{\left (60 b^2\right ) \operatorname{Subst}\left (\int x \text{Li}_3\left (e^{2 i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^4}\\ &=-\frac{2 i b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{5/2} \cot \left (c+d \sqrt{x}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}+\frac{20 i a b x^2 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i a b x^2 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i b^2 x^{3/2} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{80 a b x^{3/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{80 a b x^{3/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{30 b^2 x \text{Li}_3\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{240 i a b x \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{240 i a b x \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{30 i b^2 \sqrt{x} \text{Li}_4\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{480 a b \sqrt{x} \text{Li}_5\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{480 a b \sqrt{x} \text{Li}_5\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{(480 i a b) \operatorname{Subst}\left (\int \frac{\text{Li}_5(-x)}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{(480 i a b) \operatorname{Subst}\left (\int \frac{\text{Li}_5(x)}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{\left (30 i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_4\left (e^{2 i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^5}\\ &=-\frac{2 i b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{5/2} \cot \left (c+d \sqrt{x}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}+\frac{20 i a b x^2 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i a b x^2 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i b^2 x^{3/2} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{80 a b x^{3/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{80 a b x^{3/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{30 b^2 x \text{Li}_3\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{240 i a b x \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{240 i a b x \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{30 i b^2 \sqrt{x} \text{Li}_4\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{480 a b \sqrt{x} \text{Li}_5\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{480 a b \sqrt{x} \text{Li}_5\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{480 i a b \text{Li}_6\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{480 i a b \text{Li}_6\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_4(x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^6}\\ &=-\frac{2 i b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{5/2} \cot \left (c+d \sqrt{x}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}+\frac{20 i a b x^2 \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i a b x^2 \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 i b^2 x^{3/2} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{80 a b x^{3/2} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{80 a b x^{3/2} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{30 b^2 x \text{Li}_3\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{240 i a b x \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{240 i a b x \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{30 i b^2 \sqrt{x} \text{Li}_4\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{480 a b \sqrt{x} \text{Li}_5\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{480 a b \sqrt{x} \text{Li}_5\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{15 b^2 \text{Li}_5\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^6}+\frac{480 i a b \text{Li}_6\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{480 i a b \text{Li}_6\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^6}\\ \end{align*}
Mathematica [A] time = 13.7114, size = 779, normalized size = 1.52 \[ -\frac{i b \sin ^2\left (c+d \sqrt{x}\right ) \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2 \left (i \left (-80 a d^3 x^{3/2} \text{PolyLog}\left (3,-e^{i \left (c+d \sqrt{x}\right )}\right )+80 a d^3 x^{3/2} \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right )-240 i a d^2 x \text{PolyLog}\left (4,-e^{i \left (c+d \sqrt{x}\right )}\right )+240 i a d^2 x \text{PolyLog}\left (4,e^{i \left (c+d \sqrt{x}\right )}\right )+480 a d \sqrt{x} \text{PolyLog}\left (5,-e^{i \left (c+d \sqrt{x}\right )}\right )-480 a d \sqrt{x} \text{PolyLog}\left (5,e^{i \left (c+d \sqrt{x}\right )}\right )+480 i a \text{PolyLog}\left (6,-e^{i \left (c+d \sqrt{x}\right )}\right )-480 i a \text{PolyLog}\left (6,e^{i \left (c+d \sqrt{x}\right )}\right )-20 i b d^3 x^{3/2} \text{PolyLog}\left (2,e^{2 i \left (c+d \sqrt{x}\right )}\right )+30 b d^2 x \text{PolyLog}\left (3,e^{2 i \left (c+d \sqrt{x}\right )}\right )+30 i b d \sqrt{x} \text{PolyLog}\left (4,e^{2 i \left (c+d \sqrt{x}\right )}\right )-15 b \text{PolyLog}\left (5,e^{2 i \left (c+d \sqrt{x}\right )}\right )+4 a d^5 x^{5/2} \log \left (1-e^{i \left (c+d \sqrt{x}\right )}\right )-4 a d^5 x^{5/2} \log \left (1+e^{i \left (c+d \sqrt{x}\right )}\right )+10 b d^4 x^2 \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )\right )-20 a d^4 x^2 \text{PolyLog}\left (2,-e^{i \left (c+d \sqrt{x}\right )}\right )+20 a d^4 x^2 \text{PolyLog}\left (2,e^{i \left (c+d \sqrt{x}\right )}\right )+\frac{4 b e^{2 i c} d^5 x^{5/2}}{-1+e^{2 i c}}\right )}{d^6 \left (a \sin \left (c+d \sqrt{x}\right )+b\right )^2}+\frac{a^2 x^3 \sin ^2\left (c+d \sqrt{x}\right ) \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2}{3 \left (a \sin \left (c+d \sqrt{x}\right )+b\right )^2}+\frac{b^2 x^{5/2} \csc \left (\frac{c}{2}\right ) \sin \left (\frac{d \sqrt{x}}{2}\right ) \sin ^2\left (c+d \sqrt{x}\right ) \csc \left (\frac{c}{2}+\frac{d \sqrt{x}}{2}\right ) \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2}{d \left (a \sin \left (c+d \sqrt{x}\right )+b\right )^2}+\frac{b^2 x^{5/2} \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d \sqrt{x}}{2}\right ) \sin ^2\left (c+d \sqrt{x}\right ) \sec \left (\frac{c}{2}+\frac{d \sqrt{x}}{2}\right ) \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2}{d \left (a \sin \left (c+d \sqrt{x}\right )+b\right )^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.199, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\csc \left ( c+d\sqrt{x} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.27891, size = 5207, normalized size = 10.15 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} x^{2} \csc \left (d \sqrt{x} + c\right )^{2} + 2 \, a b x^{2} \csc \left (d \sqrt{x} + c\right ) + a^{2} x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \csc{\left (c + d \sqrt{x} \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \csc \left (d \sqrt{x} + c\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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