Integrand size = 33, antiderivative size = 170 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {2 a^{5/2} A \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{d}+\frac {2 a^3 (49 A+32 C) \sin (c+d x)}{21 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (7 A+8 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \]
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Time = 0.86 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3125, 3055, 3060, 2852, 212} \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {2 a^{5/2} A \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}+\frac {2 a^3 (49 A+32 C) \sin (c+d x)}{21 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (7 A+8 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{21 d}+\frac {2 a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
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Rule 212
Rule 2852
Rule 3055
Rule 3060
Rule 3125
Rubi steps \begin{align*} \text {integral}& = \frac {2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {2 \int (a+a \cos (c+d x))^{5/2} \left (\frac {7 a A}{2}+\frac {5}{2} a C \cos (c+d x)\right ) \sec (c+d x) \, dx}{7 a} \\ & = \frac {2 a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {4 \int (a+a \cos (c+d x))^{3/2} \left (\frac {35 a^2 A}{4}+\frac {5}{4} a^2 (7 A+8 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{35 a} \\ & = \frac {2 a^2 (7 A+8 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {8 \int \sqrt {a+a \cos (c+d x)} \left (\frac {105 a^3 A}{8}+\frac {5}{8} a^3 (49 A+32 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{105 a} \\ & = \frac {2 a^3 (49 A+32 C) \sin (c+d x)}{21 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (7 A+8 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\left (a^2 A\right ) \int \sqrt {a+a \cos (c+d x)} \sec (c+d x) \, dx \\ & = \frac {2 a^3 (49 A+32 C) \sin (c+d x)}{21 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (7 A+8 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}-\frac {\left (2 a^3 A\right ) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{d} \\ & = \frac {2 a^{5/2} A \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{d}+\frac {2 a^3 (49 A+32 C) \sin (c+d x)}{21 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (7 A+8 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.68 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (84 \sqrt {2} A \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )+2 (224 A+208 C+(28 A+101 C) \cos (c+d x)+24 C \cos (2 (c+d x))+3 C \cos (3 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{84 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(333\) vs. \(2(148)=296\).
Time = 14.13 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.96
method | result | size |
parts | \(\frac {A \,a^{\frac {3}{2}} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-4 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}+3 \ln \left (\frac {4 \sqrt {2}\, a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+4 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}+8 a}{2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {2}}\right ) a +3 \ln \left (-\frac {4 \left (\sqrt {2}\, a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}-2 a \right )}{2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {2}}\right ) a \right )}{3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}+\frac {8 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (6 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8\right ) \sqrt {2}}{21 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(334\) |
default | \(\frac {a^{\frac {3}{2}} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-48 C \sqrt {a}\, \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+168 C \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-28 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\, \left (A +8 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+126 A \sqrt {a}\, \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+21 A \ln \left (\frac {4 \sqrt {2}\, a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+4 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}+8 a}{2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {2}}\right ) a +21 A \ln \left (-\frac {4 \left (\sqrt {2}\, a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}-2 a \right )}{2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {2}}\right ) a +168 C \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\right )}{21 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(348\) |
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Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.13 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {21 \, {\left (A a^{2} \cos \left (d x + c\right ) + A a^{2}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + 4 \, {\left (3 \, C a^{2} \cos \left (d x + c\right )^{3} + 12 \, C a^{2} \cos \left (d x + c\right )^{2} + {\left (7 \, A + 23 \, C\right )} a^{2} \cos \left (d x + c\right ) + 2 \, {\left (28 \, A + 23 \, C\right )} a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{42 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Timed out} \]
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Time = 0.41 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.46 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {{\left (3 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 21 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 77 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 315 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{84 \, d} \]
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Time = 1.39 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.35 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=-\frac {\sqrt {2} {\left (96 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 56 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 448 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, \sqrt {2} A a^{2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 252 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 336 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{42 \, d} \]
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Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}}{\cos \left (c+d\,x\right )} \,d x \]
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