Integrand size = 21, antiderivative size = 305 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^m \, dx=-\frac {\left (3 a^2-3 a b (2-m)+b^2 \left (3-4 m+m^2\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \sin (c+d x)}{a-b}\right ) (a+b \sin (c+d x))^{1+m}}{16 (a-b)^3 d (1+m)}+\frac {\left (3 a^2+3 a b (2-m)+b^2 \left (3-4 m+m^2\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \sin (c+d x)}{a+b}\right ) (a+b \sin (c+d x))^{1+m}}{16 (a+b)^3 d (1+m)}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{4 \left (a^2-b^2\right ) d}+\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^{1+m} \left (b \left (b^2 (3-m)-a^2 (1+m)\right )+a \left (3 a^2-b^2 (5-2 m)\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d} \]
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Time = 0.30 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2747, 755, 837, 845, 70} \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^m \, dx=-\frac {\left (3 a^2-3 a b (2-m)+b^2 \left (m^2-4 m+3\right )\right ) (a+b \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \sin (c+d x)}{a-b}\right )}{16 d (m+1) (a-b)^3}+\frac {\left (3 a^2+3 a b (2-m)+b^2 \left (m^2-4 m+3\right )\right ) (a+b \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \sin (c+d x)}{a+b}\right )}{16 d (m+1) (a+b)^3}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{m+1}}{4 d \left (a^2-b^2\right )}+\frac {\sec ^2(c+d x) \left (a \left (3 a^2-b^2 (5-2 m)\right ) \sin (c+d x)+b \left (b^2 (3-m)-a^2 (m+1)\right )\right ) (a+b \sin (c+d x))^{m+1}}{8 d \left (a^2-b^2\right )^2} \]
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Rule 70
Rule 755
Rule 837
Rule 845
Rule 2747
Rubi steps \begin{align*} \text {integral}& = \frac {b^5 \text {Subst}\left (\int \frac {(a+x)^m}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = -\frac {\sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{4 \left (a^2-b^2\right ) d}+\frac {b^3 \text {Subst}\left (\int \frac {(a+x)^m \left (3 a^2-b^2 (3-m)+a (2-m) x\right )}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 \left (a^2-b^2\right ) d} \\ & = -\frac {\sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{4 \left (a^2-b^2\right ) d}+\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^{1+m} \left (b \left (b^2 (3-m)-a^2 (1+m)\right )+a \left (3 a^2-b^2 (5-2 m)\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}-\frac {b \text {Subst}\left (\int \frac {(a+x)^m \left (-3 a^4+a^2 b^2 \left (6-2 m-m^2\right )-b^4 \left (3-4 m+m^2\right )+a \left (3 a^2-b^2 (5-2 m)\right ) m x\right )}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d} \\ & = -\frac {\sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{4 \left (a^2-b^2\right ) d}+\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^{1+m} \left (b \left (b^2 (3-m)-a^2 (1+m)\right )+a \left (3 a^2-b^2 (5-2 m)\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}-\frac {b \text {Subst}\left (\int \left (\frac {\left (a b^2 \left (3 a^2-b^2 (5-2 m)\right ) m+b \left (-3 a^4+a^2 b^2 \left (6-2 m-m^2\right )-b^4 \left (3-4 m+m^2\right )\right )\right ) (a+x)^m}{2 b^2 (b-x)}+\frac {\left (-a b^2 \left (3 a^2-b^2 (5-2 m)\right ) m+b \left (-3 a^4+a^2 b^2 \left (6-2 m-m^2\right )-b^4 \left (3-4 m+m^2\right )\right )\right ) (a+x)^m}{2 b^2 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d} \\ & = -\frac {\sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{4 \left (a^2-b^2\right ) d}+\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^{1+m} \left (b \left (b^2 (3-m)-a^2 (1+m)\right )+a \left (3 a^2-b^2 (5-2 m)\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}+\frac {\left (3 a^2-3 a b (2-m)+b^2 \left (3-4 m+m^2\right )\right ) \text {Subst}\left (\int \frac {(a+x)^m}{b+x} \, dx,x,b \sin (c+d x)\right )}{16 (a-b)^2 d}+\frac {\left (3 a^2+3 a b (2-m)+b^2 \left (3-4 m+m^2\right )\right ) \text {Subst}\left (\int \frac {(a+x)^m}{b-x} \, dx,x,b \sin (c+d x)\right )}{16 (a+b)^2 d} \\ & = -\frac {\left (3 a^2-3 a b (2-m)+b^2 \left (3-4 m+m^2\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \sin (c+d x)}{a-b}\right ) (a+b \sin (c+d x))^{1+m}}{16 (a-b)^3 d (1+m)}+\frac {\left (3 a^2+3 a b (2-m)+b^2 \left (3-4 m+m^2\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \sin (c+d x)}{a+b}\right ) (a+b \sin (c+d x))^{1+m}}{16 (a+b)^3 d (1+m)}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{4 \left (a^2-b^2\right ) d}+\frac {\sec ^2(c+d x) (a+b \sin (c+d x))^{1+m} \left (b \left (b^2 (3-m)-a^2 (1+m)\right )+a \left (3 a^2-b^2 (5-2 m)\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d} \\ \end{align*}
Time = 2.58 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.85 \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^m \, dx=\frac {(a+b \sin (c+d x))^{1+m} \left (\frac {(a+b)^3 \left (3 a^2+3 a b (-2+m)+b^2 \left (3-4 m+m^2\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \sin (c+d x)}{a-b}\right )-(a-b)^3 \left (3 a^2-3 a b (-2+m)+b^2 \left (3-4 m+m^2\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \sin (c+d x)}{a+b}\right )}{(a-b) (a+b) \left (a^2-b^2\right ) (1+m)}+4 \sec ^4(c+d x) (b-a \sin (c+d x))+\frac {2 \sec ^2(c+d x) \left (b^3 (-3+m)+a^2 b (1+m)-a \left (3 a^2+b^2 (-5+2 m)\right ) \sin (c+d x)\right )}{a^2-b^2}\right )}{16 \left (-a^2+b^2\right ) d} \]
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\[\int \left (\sec ^{5}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{m}d x\]
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\[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^m \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{5} \,d x } \]
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Timed out. \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^m \, dx=\text {Timed out} \]
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\[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^m \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{5} \,d x } \]
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\[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^m \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{5} \,d x } \]
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Timed out. \[ \int \sec ^5(c+d x) (a+b \sin (c+d x))^m \, dx=\int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m}{{\cos \left (c+d\,x\right )}^5} \,d x \]
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