Integrand size = 23, antiderivative size = 80 \[ \int \cos ^{\frac {7}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (5 A+7 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d} \]
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Time = 0.12 (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, 2720} \[ \int \cos ^{\frac {7}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (5 A+7 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 (5 A+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d} \]
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Rule 2715
Rule 2720
Rule 3093
Rule 4151
Rubi steps \begin{align*} \text {integral}& = \int \cos ^{\frac {3}{2}}(c+d x) \left (C+A \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 A \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{7} (5 A+7 C) \int \cos ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{21} (5 A+7 C) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 (5 A+7 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d} \\ \end{align*}
Time = 1.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.79 \[ \int \cos ^{\frac {7}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (5 A+7 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} (13 A+14 C+3 A \cos (2 (c+d x))) \sin (c+d x)}{21 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(284\) vs. \(2(96)=192\).
Time = 12.68 (sec) , antiderivative size = 285, normalized size of antiderivative = 3.56
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (48 A \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-72 A \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (56 A +28 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-16 A -14 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+5 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+7 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{21 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(285\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.22 \[ \int \cos ^{\frac {7}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (3 \, A \cos \left (d x + c\right )^{2} + 5 \, A + 7 \, C\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-5 i \, A - 7 i \, C\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {2} {\left (5 i \, A + 7 i \, C\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )}{21 \, d} \]
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Timed out. \[ \int \cos ^{\frac {7}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^{\frac {7}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
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\[ \int \cos ^{\frac {7}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
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Time = 17.75 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \cos ^{\frac {7}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,C\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3\,d}+\frac {2\,C\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3\,d}-\frac {2\,A\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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