Integrand size = 25, antiderivative size = 25 \[ \int \frac {1}{\sec ^{\frac {7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Int}\left (\frac {1}{\sec ^{\frac {7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}},x\right ) \]
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Not integrable
Time = 0.07 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {1}{\sec ^{\frac {7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {1}{\sec ^{\frac {7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sec ^{\frac {7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx \\ \end{align*}
Not integrable
Time = 147.76 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sec ^{\frac {7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {1}{\sec ^{\frac {7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx \]
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Not integrable
Time = 0.43 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
\[\int \frac {1}{\sec \left (d x +c \right )^{\frac {7}{3}} \left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
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Not integrable
Time = 1.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.48 \[ \int \frac {1}{\sec ^{\frac {7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {7}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sec ^{\frac {7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Not integrable
Time = 1.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sec ^{\frac {7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {7}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sec ^{\frac {7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Not integrable
Time = 18.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sec ^{\frac {7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/3}} \,d x \]
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