Integrand size = 21, antiderivative size = 123 \[ \int \sqrt {b \cos (c+d x)} \sec ^6(c+d x) \, dx=-\frac {14 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {2 b^5 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b^3 \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 b \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}} \]
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Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 2716, 2721, 2719} \[ \int \sqrt {b \cos (c+d x)} \sec ^6(c+d x) \, dx=\frac {2 b^5 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b^3 \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 b \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}}-\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}} \]
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Rule 16
Rule 2716
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = b^6 \int \frac {1}{(b \cos (c+d x))^{11/2}} \, dx \\ & = \frac {2 b^5 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {1}{9} \left (7 b^4\right ) \int \frac {1}{(b \cos (c+d x))^{7/2}} \, dx \\ & = \frac {2 b^5 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b^3 \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {1}{15} \left (7 b^2\right ) \int \frac {1}{(b \cos (c+d x))^{3/2}} \, dx \\ & = \frac {2 b^5 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b^3 \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 b \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}}-\frac {7}{15} \int \sqrt {b \cos (c+d x)} \, dx \\ & = \frac {2 b^5 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b^3 \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 b \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}}-\frac {\left (7 \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 \sqrt {\cos (c+d x)}} \\ & = -\frac {14 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {2 b^5 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b^3 \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 b \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.64 \[ \int \sqrt {b \cos (c+d x)} \sec ^6(c+d x) \, dx=\frac {\sqrt {b \cos (c+d x)} \sec ^5(c+d x) \left (-336 \cos ^{\frac {9}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+150 \sin (c+d x)+91 \sin (3 (c+d x))+21 \sin (5 (c+d x))\right )}{360 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(413\) vs. \(2(131)=262\).
Time = 3.06 (sec) , antiderivative size = 414, normalized size of antiderivative = 3.37
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 )}\, b \left (-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}{144 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )^{5}}-\frac {7 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}{180 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )^{3}}-\frac {14 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{15 \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 )}}+\frac {7 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}-\frac {7 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \left (F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{15 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b}\, d}\) | \(414\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.04 \[ \int \sqrt {b \cos (c+d x)} \sec ^6(c+d x) \, dx=\frac {-21 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (21 \, \cos \left (d x + c\right )^{4} + 7 \, \cos \left (d x + c\right )^{2} + 5\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{45 \, d \cos \left (d x + c\right )^{5}} \]
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Timed out. \[ \int \sqrt {b \cos (c+d x)} \sec ^6(c+d x) \, dx=\text {Timed out} \]
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\[ \int \sqrt {b \cos (c+d x)} \sec ^6(c+d x) \, dx=\int { \sqrt {b \cos \left (d x + c\right )} \sec \left (d x + c\right )^{6} \,d x } \]
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\[ \int \sqrt {b \cos (c+d x)} \sec ^6(c+d x) \, dx=\int { \sqrt {b \cos \left (d x + c\right )} \sec \left (d x + c\right )^{6} \,d x } \]
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Timed out. \[ \int \sqrt {b \cos (c+d x)} \sec ^6(c+d x) \, dx=\int \frac {\sqrt {b\,\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^6} \,d x \]
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