Integrand size = 23, antiderivative size = 80 \[ \int \cos ^{\frac {9}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (7 A+9 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (7 A+9 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 A \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d} \]
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Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4151, 3093, 2715, 2719} \[ \int \cos ^{\frac {9}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (7 A+9 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (7 A+9 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d} \]
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Rule 2715
Rule 2719
Rule 3093
Rule 4151
Rubi steps \begin{align*} \text {integral}& = \int \cos ^{\frac {5}{2}}(c+d x) \left (C+A \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 A \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{9} (7 A+9 C) \int \cos ^{\frac {5}{2}}(c+d x) \, dx \\ & = \frac {2 (7 A+9 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 A \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{15} (7 A+9 C) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {2 (7 A+9 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (7 A+9 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 A \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d} \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.81 \[ \int \cos ^{\frac {9}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {12 (7 A+9 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sqrt {\cos (c+d x)} (19 A+18 C+5 A \cos (2 (c+d x))) \sin (2 (c+d x))}{90 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(312\) vs. \(2(96)=192\).
Time = 13.52 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.91
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-160 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+320 A \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-296 A -72 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (136 A +72 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-24 A -18 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-21 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-27 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{45 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(313\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.42 \[ \int \cos ^{\frac {9}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (5 \, A \cos \left (d x + c\right )^{3} + {\left (7 \, A + 9 \, C\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 3 \, \sqrt {2} {\left (-7 i \, A - 9 i \, C\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 \, \sqrt {2} {\left (7 i \, A + 9 i \, C\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{45 \, d} \]
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Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^{\frac {9}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
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\[ \int \cos ^{\frac {9}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
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Time = 17.97 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.09 \[ \int \cos ^{\frac {9}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {2\,A\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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