Optimal. Leaf size=23 \[ b \text{Unintegrable}\left (\frac{\csc \left (c+d x^2\right )}{x^2},x\right )-\frac{a}{x} \]
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Rubi [A] time = 0.0166206, 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 = {} \[ \int \frac{a+b \csc \left (c+d x^2\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{a+b \csc \left (c+d x^2\right )}{x^2} \, dx &=\int \left (\frac{a}{x^2}+\frac{b \csc \left (c+d x^2\right )}{x^2}\right ) \, dx\\ &=-\frac{a}{x}+b \int \frac{\csc \left (c+d x^2\right )}{x^2} \, dx\\ \end{align*}
Mathematica [A] time = 2.38859, size = 0, normalized size = 0. \[ \int \frac{a+b \csc \left (c+d x^2\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.106, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\csc \left ( d{x}^{2}+c \right ) }{{x}^{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*} b{\left (\int \frac{\sin \left (d x^{2} + c\right )}{x^{2} \cos \left (d x^{2} + c\right )^{2} + x^{2} \sin \left (d x^{2} + c\right )^{2} + 2 \, x^{2} \cos \left (d x^{2} + c\right ) + x^{2}}\,{d x} + \int \frac{\sin \left (d x^{2} + c\right )}{x^{2} \cos \left (d x^{2} + c\right )^{2} + x^{2} \sin \left (d x^{2} + c\right )^{2} - 2 \, x^{2} \cos \left (d x^{2} + c\right ) + x^{2}}\,{d x}\right )} - \frac{a}{x} \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 (\frac{b \csc \left (d x^{2} + c\right ) + a}{x^{2}}, 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 \frac{a + b \csc{\left (c + d x^{2} \right )}}{x^{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 \frac{b \csc \left (d x^{2} + c\right ) + a}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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