Integrand size = 25, antiderivative size = 25 \[ \int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx=\text {Int}\left ((a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m,x\right ) \]
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Not integrable
Time = 0.08 (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 = {} \[ \int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx=\int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx \\ \end{align*}
Not integrable
Time = 39.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx=\int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx \]
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Not integrable
Time = 0.91 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int \left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}} \left (e \sin \left (d x +c \right )\right )^{m}d x\]
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Not integrable
Time = 0.40 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \left (e \sin \left (d x + c\right )\right )^{m} \,d x } \]
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Timed out. \[ \int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx=\text {Timed out} \]
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Not integrable
Time = 1.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \left (e \sin \left (d x + c\right )\right )^{m} \,d x } \]
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Not integrable
Time = 1.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \left (e \sin \left (d x + c\right )\right )^{m} \,d x } \]
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Not integrable
Time = 18.52 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx=\int {\left (e\,\sin \left (c+d\,x\right )\right )}^m\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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