Integrand size = 21, antiderivative size = 129 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^m \, dx=\frac {\operatorname {AppellF1}\left (1+m,\frac {5}{2},\frac {5}{2},2+m,\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) \sec ^5(c+d x) (a+b \sin (c+d x))^{1+m} \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{5/2} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{5/2}}{b d (1+m)} \]
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Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2783, 143} \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^m \, dx=\frac {\sec ^5(c+d x) \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{5/2} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{5/2} (a+b \sin (c+d x))^{m+1} \operatorname {AppellF1}\left (m+1,\frac {5}{2},\frac {5}{2},m+2,\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right )}{b d (m+1)} \]
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Rule 143
Rule 2783
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sec ^5(c+d x) \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{5/2} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {(a+b x)^m}{\left (-\frac {b}{a-b}-\frac {b x}{a-b}\right )^{5/2} \left (\frac {b}{a+b}-\frac {b x}{a+b}\right )^{5/2}} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\operatorname {AppellF1}\left (1+m,\frac {5}{2},\frac {5}{2},2+m,\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) \sec ^5(c+d x) (a+b \sin (c+d x))^{1+m} \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{5/2} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{5/2}}{b d (1+m)} \\ \end{align*}
\[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^m \, dx=\int \sec ^4(c+d x) (a+b \sin (c+d x))^m \, dx \]
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\[\int \left (\sec ^{4}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{m}d x\]
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\[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^m \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{4} \,d x } \]
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Timed out. \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^m \, dx=\text {Timed out} \]
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\[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^m \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{4} \,d x } \]
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\[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^m \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{4} \,d x } \]
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Timed out. \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^m \, dx=\int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m}{{\cos \left (c+d\,x\right )}^4} \,d x \]
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