Integrand size = 23, antiderivative size = 77 \[ \int \frac {(c \sin (a+b x))^m}{(d \cos (a+b x))^{3/2}} \, dx=\frac {\sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c d (1+m) \sqrt {d \cos (a+b x)}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2657} \[ \int \frac {(c \sin (a+b x))^m}{(d \cos (a+b x))^{3/2}} \, dx=\frac {\sqrt [4]{\cos ^2(a+b x)} (c \sin (a+b x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(a+b x)\right )}{b c d (m+1) \sqrt {d \cos (a+b x)}} \]
[In]
[Out]
Rule 2657
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c d (1+m) \sqrt {d \cos (a+b x)}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int \frac {(c \sin (a+b x))^m}{(d \cos (a+b x))^{3/2}} \, dx=\frac {\sqrt {d \cos (a+b x)} \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^m \tan (a+b x)}{b d^2 (1+m)} \]
[In]
[Out]
\[\int \frac {\left (c \sin \left (b x +a \right )\right )^{m}}{\left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}}}d x\]
[In]
[Out]
\[ \int \frac {(c \sin (a+b x))^m}{(d \cos (a+b x))^{3/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{m}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(c \sin (a+b x))^m}{(d \cos (a+b x))^{3/2}} \, dx=\int \frac {\left (c \sin {\left (a + b x \right )}\right )^{m}}{\left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {(c \sin (a+b x))^m}{(d \cos (a+b x))^{3/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{m}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(c \sin (a+b x))^m}{(d \cos (a+b x))^{3/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{m}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(c \sin (a+b x))^m}{(d \cos (a+b x))^{3/2}} \, dx=\int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^m}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}} \,d x \]
[In]
[Out]