Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{\left (a+b \sec \left (c+d \sqrt{x}\right )\right )^2}{x},x\right ) \]
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Rubi [A] time = 0.0226762, 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 = {} \[ \int \frac{\left (a+b \sec \left (c+d \sqrt{x}\right )\right )^2}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (a+b \sec \left (c+d \sqrt{x}\right )\right )^2}{x} \, dx &=\int \frac{\left (a+b \sec \left (c+d \sqrt{x}\right )\right )^2}{x} \, dx\\ \end{align*}
Mathematica [A] time = 38.0832, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sec \left (c+d \sqrt{x}\right )\right )^2}{x} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+b\sec \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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{4 \, b^{2} \sqrt{x} \sin \left (2 \, d \sqrt{x} + 2 \, c\right ) + 2 \,{\left (d \cos \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt{x} + 2 \, c\right ) + d\right )} x \int \frac{b^{2} \sqrt{x} \sin \left (2 \, d \sqrt{x} + 2 \, c\right ) + 2 \,{\left (a b d \cos \left (2 \, d \sqrt{x} + 2 \, c\right ) \cos \left (d \sqrt{x} + c\right ) + a b d \sin \left (2 \, d \sqrt{x} + 2 \, c\right ) \sin \left (d \sqrt{x} + c\right ) + a b d \cos \left (d \sqrt{x} + c\right )\right )} x}{{\left (d \cos \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt{x} + 2 \, c\right ) + d\right )} x^{2}}\,{d x} +{\left (a^{2} d \cos \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + a^{2} d \sin \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + 2 \, a^{2} d \cos \left (2 \, d \sqrt{x} + 2 \, c\right ) + a^{2} d\right )} x \log \left (x\right )}{{\left (d \cos \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt{x} + 2 \, c\right ) + d\right )} 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^{2} \sec \left (d \sqrt{x} + c\right )^{2} + 2 \, a b \sec \left (d \sqrt{x} + c\right ) + a^{2}}{x}, 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{\left (a + b \sec{\left (c + d \sqrt{x} \right )}\right )^{2}}{x}\, 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{{\left (b \sec \left (d \sqrt{x} + c\right ) + a\right )}^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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