Integrand size = 19, antiderivative size = 97 \[ \int \frac {\sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=-\frac {6 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt {\cos (c+d x)}}+\frac {2 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {6 \sin (c+d x)}{5 b^2 d \sqrt {b \cos (c+d x)}} \]
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Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {16, 2716, 2721, 2719} \[ \int \frac {\sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{5 b^3 d \sqrt {\cos (c+d x)}}+\frac {6 \sin (c+d x)}{5 b^2 d \sqrt {b \cos (c+d x)}}+\frac {2 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}} \]
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Rule 16
Rule 2716
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = b \int \frac {1}{(b \cos (c+d x))^{7/2}} \, dx \\ & = \frac {2 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {3 \int \frac {1}{(b \cos (c+d x))^{3/2}} \, dx}{5 b} \\ & = \frac {2 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {6 \sin (c+d x)}{5 b^2 d \sqrt {b \cos (c+d x)}}-\frac {3 \int \sqrt {b \cos (c+d x)} \, dx}{5 b^3} \\ & = \frac {2 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {6 \sin (c+d x)}{5 b^2 d \sqrt {b \cos (c+d x)}}-\frac {\left (3 \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 b^3 \sqrt {\cos (c+d x)}} \\ & = -\frac {6 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt {\cos (c+d x)}}+\frac {2 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {6 \sin (c+d x)}{5 b^2 d \sqrt {b \cos (c+d x)}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70 \[ \int \frac {\sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\frac {-6 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+6 \sin (c+d x)+2 \sec (c+d x) \tan (c+d x)}{5 b^2 d \sqrt {b \cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(366\) vs. \(2(109)=218\).
Time = 2.78 (sec) , antiderivative size = 367, normalized size of antiderivative = 3.78
method | result | size |
default | \(-\frac {2 \sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (24 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{5 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b}\, d}\) | \(367\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.25 \[ \int \frac {\sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\frac {-3 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, \sqrt {b \cos \left (d x + c\right )} {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right )}{5 \, b^{3} d \cos \left (d x + c\right )^{3}} \]
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Timed out. \[ \int \frac {\sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\int \frac {1}{\cos \left (c+d\,x\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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