Integrand size = 19, antiderivative size = 48 \[ \int \sec ^3(a+b x) (c \sin (a+b x))^m \, dx=\frac {\operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c (1+m)} \]
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Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2644, 371} \[ \int \sec ^3(a+b x) (c \sin (a+b x))^m \, dx=\frac {(c \sin (a+b x))^{m+1} \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{2},\frac {m+3}{2},\sin ^2(a+b x)\right )}{b c (m+1)} \]
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Rule 371
Rule 2644
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^m}{\left (1-\frac {x^2}{c^2}\right )^2} \, dx,x,c \sin (a+b x)\right )}{b c} \\ & = \frac {\operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c (1+m)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.06 \[ \int \sec ^3(a+b x) (c \sin (a+b x))^m \, dx=\frac {\operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},1+\frac {1+m}{2},\sin ^2(a+b x)\right ) \sin (a+b x) (c \sin (a+b x))^m}{b (1+m)} \]
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\[\int \left (\sec ^{3}\left (b x +a \right )\right ) \left (c \sin \left (b x +a \right )\right )^{m}d x\]
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\[ \int \sec ^3(a+b x) (c \sin (a+b x))^m \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right )^{3} \,d x } \]
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\[ \int \sec ^3(a+b x) (c \sin (a+b x))^m \, dx=\int \left (c \sin {\left (a + b x \right )}\right )^{m} \sec ^{3}{\left (a + b x \right )}\, dx \]
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\[ \int \sec ^3(a+b x) (c \sin (a+b x))^m \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right )^{3} \,d x } \]
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\[ \int \sec ^3(a+b x) (c \sin (a+b x))^m \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right )^{3} \,d x } \]
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Timed out. \[ \int \sec ^3(a+b x) (c \sin (a+b x))^m \, dx=\int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^m}{{\cos \left (a+b\,x\right )}^3} \,d x \]
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