Integrand size = 32, antiderivative size = 176 \[ \int \frac {B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\frac {6 C E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^4 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {10 B \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {b \sec (c+d x)}}{21 b^5 d}+\frac {2 B \sin (c+d x)}{7 b^2 d (b \sec (c+d x))^{5/2}}+\frac {2 C \sin (c+d x)}{5 b^3 d (b \sec (c+d x))^{3/2}}+\frac {10 B \sin (c+d x)}{21 b^4 d \sqrt {b \sec (c+d x)}} \]
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Time = 0.15 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {4132, 3854, 3856, 2720, 12, 16, 2719} \[ \int \frac {B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\frac {10 B \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {b \sec (c+d x)}}{21 b^5 d}+\frac {10 B \sin (c+d x)}{21 b^4 d \sqrt {b \sec (c+d x)}}+\frac {6 C E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^4 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 C \sin (c+d x)}{5 b^3 d (b \sec (c+d x))^{3/2}}+\frac {2 B \sin (c+d x)}{7 b^2 d (b \sec (c+d x))^{5/2}} \]
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Rule 12
Rule 16
Rule 2719
Rule 2720
Rule 3854
Rule 3856
Rule 4132
Rubi steps \begin{align*} \text {integral}& = \frac {B \int \frac {1}{(b \sec (c+d x))^{7/2}} \, dx}{b}+\int \frac {C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx \\ & = \frac {2 B \sin (c+d x)}{7 b^2 d (b \sec (c+d x))^{5/2}}+\frac {(5 B) \int \frac {1}{(b \sec (c+d x))^{3/2}} \, dx}{7 b^3}+C \int \frac {\sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx \\ & = \frac {2 B \sin (c+d x)}{7 b^2 d (b \sec (c+d x))^{5/2}}+\frac {10 B \sin (c+d x)}{21 b^4 d \sqrt {b \sec (c+d x)}}+\frac {(5 B) \int \sqrt {b \sec (c+d x)} \, dx}{21 b^5}+\frac {C \int \frac {1}{(b \sec (c+d x))^{5/2}} \, dx}{b^2} \\ & = \frac {2 B \sin (c+d x)}{7 b^2 d (b \sec (c+d x))^{5/2}}+\frac {2 C \sin (c+d x)}{5 b^3 d (b \sec (c+d x))^{3/2}}+\frac {10 B \sin (c+d x)}{21 b^4 d \sqrt {b \sec (c+d x)}}+\frac {(3 C) \int \frac {1}{\sqrt {b \sec (c+d x)}} \, dx}{5 b^4}+\frac {\left (5 B \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 b^5} \\ & = \frac {10 B \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {b \sec (c+d x)}}{21 b^5 d}+\frac {2 B \sin (c+d x)}{7 b^2 d (b \sec (c+d x))^{5/2}}+\frac {2 C \sin (c+d x)}{5 b^3 d (b \sec (c+d x))^{3/2}}+\frac {10 B \sin (c+d x)}{21 b^4 d \sqrt {b \sec (c+d x)}}+\frac {(3 C) \int \sqrt {\cos (c+d x)} \, dx}{5 b^4 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}} \\ & = \frac {6 C E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^4 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {10 B \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {b \sec (c+d x)}}{21 b^5 d}+\frac {2 B \sin (c+d x)}{7 b^2 d (b \sec (c+d x))^{5/2}}+\frac {2 C \sin (c+d x)}{5 b^3 d (b \sec (c+d x))^{3/2}}+\frac {10 B \sin (c+d x)}{21 b^4 d \sqrt {b \sec (c+d x)}} \\ \end{align*}
Time = 1.32 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.59 \[ \int \frac {B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\frac {\sqrt {b \sec (c+d x)} \left (252 C \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+100 B \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+(65 B+42 C \cos (c+d x)+15 B \cos (2 (c+d x))) \sin (2 (c+d x))\right )}{210 b^5 d} \]
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Result contains complex when optimal does not.
Time = 1.72 (sec) , antiderivative size = 582, normalized size of antiderivative = 3.31
method | result | size |
parts | \(-\frac {2 B \left (5 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right )+5 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sec \left (d x +c \right )-3 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )-5 \sin \left (d x +c \right )\right )}{21 d \sqrt {b \sec \left (d x +c \right )}\, b^{4}}+\frac {2 C \left (3 i \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-3 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \cos \left (d x +c \right )+6 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right )-6 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right )+3 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sec \left (d x +c \right )-3 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sec \left (d x +c \right )+\cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+\sin \left (d x +c \right ) \cos \left (d x +c \right )+3 \sin \left (d x +c \right )\right )}{5 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {b \sec \left (d x +c \right )}\, b^{4}}\) | \(582\) |
default | \(-\frac {2 \left (-63 i C \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sec \left (d x +c \right )+50 i B \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right )+63 i C \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-126 i C \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right )-15 B \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}+25 i B \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+126 i C \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right )+63 i C \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sec \left (d x +c \right )-15 B \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}-63 i C \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+25 i B \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sec \left (d x +c \right )-21 C \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )-25 B \cos \left (d x +c \right ) \sin \left (d x +c \right )-21 C \cos \left (d x +c \right ) \sin \left (d x +c \right )-25 B \sin \left (d x +c \right )-63 C \sin \left (d x +c \right )\right )}{105 b^{4} d \left (\cos \left (d x +c \right )+1\right ) \sqrt {b \sec \left (d x +c \right )}}\) | \(669\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.99 \[ \int \frac {B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\frac {-25 i \, \sqrt {2} B \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 25 i \, \sqrt {2} B \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 63 i \, \sqrt {2} C \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 63 i \, \sqrt {2} C \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (15 \, B \cos \left (d x + c\right )^{3} + 21 \, C \cos \left (d x + c\right )^{2} + 25 \, B \cos \left (d x + c\right )\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{105 \, b^{5} d} \]
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Timed out. \[ \int \frac {B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\int \frac {\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]
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