Integrand size = 23, antiderivative size = 23 \[ \int (a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\text {Int}\left ((a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x),x\right ) \]
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Not integrable
Time = 0.05 (sec) , antiderivative size = 23, 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 = {} \[ \int (a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\int (a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int (a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx \\ \end{align*}
Not integrable
Time = 8.89 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\int (a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx \]
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Not integrable
Time = 0.82 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int \left (a +b \sec \left (d x +c \right )\right )^{n} \sin \left (d x +c \right )^{\frac {3}{2}}d x\]
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Not integrable
Time = 0.36 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\text {Timed out} \]
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Not integrable
Time = 1.59 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Not integrable
Time = 22.55 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Not integrable
Time = 16.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\int {\sin \left (c+d\,x\right )}^{3/2}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
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