Integrand size = 16, antiderivative size = 16 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\frac {a x^3}{3}+b \text {Int}\left (x^2 \csc \left (c+d x^2\right ),x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int x^2 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (a x^2+b x^2 \csc \left (c+d x^2\right )\right ) \, dx \\ & = \frac {a x^3}{3}+b \int x^2 \csc \left (c+d x^2\right ) \, dx \\ \end{align*}
Not integrable
Time = 3.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int x^2 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx \]
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Not integrable
Time = 0.11 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00
\[\int x^{2} \left (a +b \csc \left (d \,x^{2}+c \right )\right )d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )} x^{2} \,d x } \]
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Not integrable
Time = 1.83 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int x^{2} \left (a + b \csc {\left (c + d x^{2} \right )}\right )\, dx \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 6.62 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )} x^{2} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )} x^{2} \,d x } \]
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Not integrable
Time = 17.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int x^2\,\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right ) \,d x \]
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