Integrand size = 21, antiderivative size = 95 \[ \int \frac {\sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {10 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {10 b \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 2716, 2721, 2720} \[ \int \frac {\sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {2 b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {10 b \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {10 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {b \cos (c+d x)}} \]
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Rule 16
Rule 2716
Rule 2720
Rule 2721
Rubi steps \begin{align*} \text {integral}& = b^4 \int \frac {1}{(b \cos (c+d x))^{9/2}} \, dx \\ & = \frac {2 b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {1}{7} \left (5 b^2\right ) \int \frac {1}{(b \cos (c+d x))^{5/2}} \, dx \\ & = \frac {2 b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {10 b \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {5}{21} \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx \\ & = \frac {2 b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {10 b \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {\left (5 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {b \cos (c+d x)}} \\ & = \frac {10 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {10 b \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int \frac {\sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {10 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (5+3 \sec ^2(c+d x)\right ) \tan (c+d x)}{21 d \sqrt {b \cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(266\) vs. \(2(107)=214\).
Time = 2.52 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.81
method | result | size |
default | \(-\frac {\sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}{28 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )^{4}}-\frac {5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}{21 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )^{2}}+\frac {10 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{21 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b}\, d}\) | \(267\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.21 \[ \int \frac {\sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {-5 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, \sqrt {b \cos \left (d x + c\right )} {\left (5 \, \cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right )}{21 \, b d \cos \left (d x + c\right )^{4}} \]
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\[ \int \frac {\sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{\sqrt {b \cos {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {\sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {\sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^4\,\sqrt {b\,\cos \left (c+d\,x\right )}} \,d x \]
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