Integrand size = 16, antiderivative size = 16 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x^2} \, dx=-\frac {a}{x}+b \text {Int}\left (\frac {\tan \left (c+d x^2\right )}{x^2},x\right ) \]
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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 = {} \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x^2} \, dx=\int \frac {a+b \tan \left (c+d x^2\right )}{x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{x^2}+\frac {b \tan \left (c+d x^2\right )}{x^2}\right ) \, dx \\ & = -\frac {a}{x}+b \int \frac {\tan \left (c+d x^2\right )}{x^2} \, dx \\ \end{align*}
Not integrable
Time = 1.64 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x^2} \, dx=\int \frac {a+b \tan \left (c+d x^2\right )}{x^2} \, dx \]
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Not integrable
Time = 0.10 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00
\[\int \frac {a +b \tan \left (d \,x^{2}+c \right )}{x^{2}}d x\]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x^2} \, dx=\int { \frac {b \tan \left (d x^{2} + c\right ) + a}{x^{2}} \,d x } \]
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Not integrable
Time = 0.38 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x^2} \, dx=\int \frac {a + b \tan {\left (c + d x^{2} \right )}}{x^{2}}\, dx \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 80, normalized size of antiderivative = 5.00 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x^2} \, dx=\int { \frac {b \tan \left (d x^{2} + c\right ) + a}{x^{2}} \,d x } \]
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Not integrable
Time = 0.37 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x^2} \, dx=\int { \frac {b \tan \left (d x^{2} + c\right ) + a}{x^{2}} \,d x } \]
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Not integrable
Time = 4.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x^2} \, dx=\int \frac {a+b\,\mathrm {tan}\left (d\,x^2+c\right )}{x^2} \,d x \]
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