Integrand size = 21, antiderivative size = 111 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x)) \, dx=\frac {6 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {10 a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {10 a \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d} \]
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Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4310, 2827, 2715, 2719, 2720} \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x)) \, dx=\frac {10 a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {6 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {2 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d} \]
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Rule 2715
Rule 2719
Rule 2720
Rule 2827
Rule 4310
Rubi steps \begin{align*} \text {integral}& = \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \, dx \\ & = a \int \cos ^{\frac {5}{2}}(c+d x) \, dx+a \int \cos ^{\frac {7}{2}}(c+d x) \, dx \\ & = \frac {2 a \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{5} (3 a) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{7} (5 a) \int \cos ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {6 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {10 a \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{21} (5 a) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {6 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {10 a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {10 a \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.09 (sec) , antiderivative size = 490, normalized size of antiderivative = 4.41 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x)) \, dx=a \left (\sqrt {\cos (c+d x)} (1+\cos (c+d x)) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-\frac {3 \cot (c)}{5 d}+\frac {23 \cos (d x) \sin (c)}{84 d}+\frac {\cos (2 d x) \sin (2 c)}{10 d}+\frac {\cos (3 d x) \sin (3 c)}{28 d}+\frac {23 \cos (c) \sin (d x)}{84 d}+\frac {\cos (2 c) \sin (2 d x)}{10 d}+\frac {\cos (3 c) \sin (3 d x)}{28 d}\right )-\frac {5 (1+\cos (c+d x)) \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{21 d \sqrt {1+\cot ^2(c)}}-\frac {3 (1+\cos (c+d x)) \csc (c) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{10 d}\right ) \]
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Time = 9.44 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.43
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a \left (240 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-528 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+448 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-122 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+25 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-63 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\right )}{105 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(270\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.33 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x)) \, dx=\frac {2 \, {\left (15 \, a \cos \left (d x + c\right )^{2} + 21 \, a \cos \left (d x + c\right ) + 25 \, a\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 25 i \, \sqrt {2} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 25 i \, \sqrt {2} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 63 i \, \sqrt {2} a {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 63 i \, \sqrt {2} a {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{105 \, d} \]
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Timed out. \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x)) \, dx=\text {Timed out} \]
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\[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x)) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
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\[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x)) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
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Time = 13.78 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.78 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {2\,a\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,a\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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