Integrand size = 31, antiderivative size = 108 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\frac {2 (3 a A+5 b B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 (A b+a B) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 (A b+a B) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2 a A \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d} \]
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Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3033, 3047, 3102, 2827, 2719, 2715, 2720} \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\frac {2 (a B+A b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 (3 a A+5 b B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 (a B+A b) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}+\frac {2 a A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d} \]
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Rule 2715
Rule 2719
Rule 2720
Rule 2827
Rule 3033
Rule 3047
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\cos (c+d x)} (b+a \cos (c+d x)) (B+A \cos (c+d x)) \, dx \\ & = \int \sqrt {\cos (c+d x)} \left (b B+(A b+a B) \cos (c+d x)+a A \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 a A \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2}{5} \int \sqrt {\cos (c+d x)} \left (\frac {1}{2} (3 a A+5 b B)+\frac {5}{2} (A b+a B) \cos (c+d x)\right ) \, dx \\ & = \frac {2 a A \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+(A b+a B) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{5} (3 a A+5 b B) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {2 (3 a A+5 b B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 (A b+a B) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2 a A \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {1}{3} (A b+a B) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 (3 a A+5 b B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 (A b+a B) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 (A b+a B) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2 a A \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d} \\ \end{align*}
Time = 1.72 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\frac {2 \left (3 (3 a A+5 b B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 (A b+a B) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} (5 A b+5 a B+3 a A \cos (c+d x)) \sin (c+d x)\right )}{15 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(370\) vs. \(2(148)=296\).
Time = 27.95 (sec) , antiderivative size = 371, normalized size of antiderivative = 3.44
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-24 a A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (24 a A +20 A b +20 B a \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-6 a A -10 A b -10 B a \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+5 A \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, b -9 A \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, a +5 B \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, a -15 B \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, b \right )}{15 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(371\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.62 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\frac {2 \, {\left (3 \, A a \cos \left (d x + c\right ) + 5 \, B a + 5 \, A b\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 5 \, \sqrt {2} {\left (i \, B a + i \, A b\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 \, \sqrt {2} {\left (-i \, B a - i \, A b\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 3 \, \sqrt {2} {\left (-3 i \, A a - 5 i \, B b\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 \, \sqrt {2} {\left (3 i \, A a + 5 i \, B b\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{15 \, d} \]
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Timed out. \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
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\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
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\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
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Time = 0.65 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.19 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx=\frac {2\,A\,b\,\left (\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )+\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{3\,d}+\frac {2\,B\,a\,\left (\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )+\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{3\,d}+\frac {2\,B\,b\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}-\frac {2\,A\,a\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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