Integrand size = 19, antiderivative size = 65 \[ \int \frac {\sec (c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=-\frac {2 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d \sqrt {\cos (c+d x)}}+\frac {2 \sin (c+d x)}{d \sqrt {b \cos (c+d x)}} \]
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Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {16, 2716, 2721, 2719} \[ \int \frac {\sec (c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {2 \sin (c+d x)}{d \sqrt {b \cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}} \]
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Rule 16
Rule 2716
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = b \int \frac {1}{(b \cos (c+d x))^{3/2}} \, dx \\ & = \frac {2 \sin (c+d x)}{d \sqrt {b \cos (c+d x)}}-\frac {\int \sqrt {b \cos (c+d x)} \, dx}{b} \\ & = \frac {2 \sin (c+d x)}{d \sqrt {b \cos (c+d x)}}-\frac {\sqrt {b \cos (c+d x)} \int \sqrt {\cos (c+d x)} \, dx}{b \sqrt {\cos (c+d x)}} \\ & = -\frac {2 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d \sqrt {\cos (c+d x)}}+\frac {2 \sin (c+d x)}{d \sqrt {b \cos (c+d x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.72 \[ \int \frac {\sec (c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {2 \left (-\sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sin (c+d x)\right )}{d \sqrt {b \cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(194\) vs. \(2(85)=170\).
Time = 1.90 (sec) , antiderivative size = 195, normalized size of antiderivative = 3.00
method | result | size |
default | \(-\frac {2 \left (-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{\sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b}\, d}\) | \(195\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.60 \[ \int \frac {\sec (c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {-i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{b d \cos \left (d x + c\right )} \]
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\[ \int \frac {\sec (c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int \frac {\sec {\left (c + d x \right )}}{\sqrt {b \cos {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {\sec (c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {\sec (c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {\sec (c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int \frac {1}{\cos \left (c+d\,x\right )\,\sqrt {b\,\cos \left (c+d\,x\right )}} \,d x \]
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