Optimal. Leaf size=20 \[ \text{Unintegrable}\left (x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2,x\right ) \]
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Rubi [A] time = 0.0228732, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx &=\int x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx\\ \end{align*}
Mathematica [A] time = 13.661, size = 0, normalized size = 0. \[ \int x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.238, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ( a+b\csc \left ( d{x}^{2}+c \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{5} \, a^{2} x^{5} - \frac{b^{2} x^{3} \sin \left (2 \, d x^{2} + 2 \, c\right ) - \frac{1}{2} \,{\left (d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} - 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d\right )} \int \frac{{\left (4 \, a b d x^{4} - 3 \, b^{2} x^{2}\right )} \sin \left (d x^{2} + c\right )}{d \cos \left (d x^{2} + c\right )^{2} + d \sin \left (d x^{2} + c\right )^{2} + 2 \, d \cos \left (d x^{2} + c\right ) + d}\,{d x} - \frac{1}{2} \,{\left (d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} - 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d\right )} \int \frac{{\left (4 \, a b d x^{4} + 3 \, b^{2} x^{2}\right )} \sin \left (d x^{2} + c\right )}{d \cos \left (d x^{2} + c\right )^{2} + d \sin \left (d x^{2} + c\right )^{2} - 2 \, d \cos \left (d x^{2} + c\right ) + d}\,{d x}}{d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} - 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} x^{4} \csc \left (d x^{2} + c\right )^{2} + 2 \, a b x^{4} \csc \left (d x^{2} + c\right ) + a^{2} x^{4}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \left (a + b \csc{\left (c + d x^{2} \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2} x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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