Integrand size = 19, antiderivative size = 72 \[ \int (b \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\frac {2 b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {b \cos (c+d x)}}+\frac {2 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{3 d} \]
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Time = 0.06 (sec) , antiderivative size = 72, 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, 2715, 2721, 2720} \[ \int (b \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\frac {2 b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {b \cos (c+d x)}}+\frac {2 b^2 \sin (c+d x) \sqrt {b \cos (c+d x)}}{3 d} \]
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Rule 16
Rule 2715
Rule 2720
Rule 2721
Rubi steps \begin{align*} \text {integral}& = b \int (b \cos (c+d x))^{3/2} \, dx \\ & = \frac {2 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {1}{3} b^3 \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx \\ & = \frac {2 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {\left (b^3 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 \sqrt {b \cos (c+d x)}} \\ & = \frac {2 b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {b \cos (c+d x)}}+\frac {2 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{3 d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82 \[ \int (b \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\frac {2 b (b \cos (c+d x))^{3/2} \left (\operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} \sin (c+d x)\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(189\) vs. \(2(88)=176\).
Time = 2.77 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.64
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, b^{3} \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b}\, d}\) | \(190\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.10 \[ \int (b \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\frac {-i \, \sqrt {2} b^{\frac {5}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + i \, \sqrt {2} b^{\frac {5}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, \sqrt {b \cos \left (d x + c\right )} b^{2} \sin \left (d x + c\right )}{3 \, d} \]
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Timed out. \[ \int (b \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\text {Timed out} \]
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\[ \int (b \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \]
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\[ \int (b \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int (b \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{\cos \left (c+d\,x\right )} \,d x \]
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