Integrand size = 18, antiderivative size = 18 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\text {Int}\left (x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2,x\right ) \]
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Not integrable
Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx \\ \end{align*}
Not integrable
Time = 35.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx \]
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Not integrable
Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00
\[\int x^{2} {\left (a +b \csc \left (d \,x^{2}+c \right )\right )}^{2}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.33 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2} x^{2} \,d x } \]
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Not integrable
Time = 2.14 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int x^{2} \left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}\, dx \]
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Not integrable
Time = 0.37 (sec) , antiderivative size = 300, normalized size of antiderivative = 16.67 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2} x^{2} \,d x } \]
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Not integrable
Time = 0.94 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2} x^{2} \,d x } \]
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Not integrable
Time = 18.88 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int x^2\,{\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right )}^2 \,d x \]
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