Integrand size = 25, antiderivative size = 85 \[ \int \frac {(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx=\frac {2^{\frac {1}{4}+m} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {7}{4}-m,\frac {1}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {3}{4}-m} (a+a \sin (c+d x))^m}{3 d e (e \cos (c+d x))^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2768, 72, 71} \[ \int \frac {(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx=\frac {2^{m+\frac {1}{4}} (\sin (c+d x)+1)^{\frac {3}{4}-m} (a \sin (c+d x)+a)^m \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {7}{4}-m,\frac {1}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (e \cos (c+d x))^{3/2}} \]
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Rule 71
Rule 72
Rule 2768
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 (a-a \sin (c+d x))^{3/4} (a+a \sin (c+d x))^{3/4}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {7}{4}+m}}{(a-a x)^{7/4}} \, dx,x,\sin (c+d x)\right )}{d e (e \cos (c+d x))^{3/2}} \\ & = \frac {\left (2^{-\frac {7}{4}+m} a (a-a \sin (c+d x))^{3/4} (a+a \sin (c+d x))^m \left (\frac {a+a \sin (c+d x)}{a}\right )^{\frac {3}{4}-m}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {7}{4}+m}}{(a-a x)^{7/4}} \, dx,x,\sin (c+d x)\right )}{d e (e \cos (c+d x))^{3/2}} \\ & = \frac {2^{\frac {1}{4}+m} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {7}{4}-m,\frac {1}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {3}{4}-m} (a+a \sin (c+d x))^m}{3 d e (e \cos (c+d x))^{3/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx=\frac {2^{\frac {1}{4}+m} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {7}{4}-m,\frac {1}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {3}{4}-m} (a (1+\sin (c+d x)))^m}{3 d e (e \cos (c+d x))^{3/2}} \]
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\[\int \frac {\left (a +a \sin \left (d x +c \right )\right )^{m}}{\left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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