Optimal. Leaf size=21 \[ \text {Int}\left (\frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x},x\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx &=\int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 17.86, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \sec \left (d x^{2} + c\right )^{2} + 2 \, a b \sec \left (d x^{2} + c\right ) + a^{2}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.16, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (d \,x^{2}+c \right )\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \log \relax (x) + \frac {b^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) + 2 \, {\left (d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) + d x^{2}\right )} \int \frac {2 \, a b d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) \cos \left (d x^{2} + c\right ) + 2 \, a b d x^{2} \cos \left (d x^{2} + c\right ) + {\left (2 \, a b d x^{2} \sin \left (d x^{2} + c\right ) + b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )}{d x^{3} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d x^{3} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, d x^{3} \cos \left (2 \, d x^{2} + 2 \, c\right ) + d x^{3}}\,{d x}}{d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) + d x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (a+\frac {b}{\cos \left (d\,x^2+c\right )}\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \sec {\left (c + d x^{2} \right )}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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