Integrand size = 21, antiderivative size = 98 \[ \int (a+a \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{d}+\frac {14 a^3 \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d} \]
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Time = 0.24 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2842, 3060, 2852, 212} \[ \int (a+a \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}+\frac {14 a^3 \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d} \]
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Rule 212
Rule 2842
Rule 2852
Rule 3060
Rubi steps \begin{align*} \text {integral}& = \frac {2 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2}{3} \int \sqrt {a+a \cos (c+d x)} \left (\frac {3 a^2}{2}+\frac {7}{2} a^2 \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {14 a^3 \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d}+a^2 \int \sqrt {a+a \cos (c+d x)} \sec (c+d x) \, dx \\ & = \frac {14 a^3 \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d}-\frac {\left (2 a^3\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} \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{d}+\frac {14 a^3 \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.91 \[ \int (a+a \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\frac {2 a^2 \sqrt {a (1+\cos (c+d x))} \left (3 \text {arctanh}\left (\sqrt {1-\cos (c+d x)}\right )+\sqrt {1-\cos (c+d x)} (8+\cos (c+d x))\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{3 d \sqrt {1-\cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(245\) vs. \(2(84)=168\).
Time = 1.97 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.51
method | result | size |
default | \(\frac {a^{\frac {3}{2}} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-4 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}+3 \ln \left (\frac {4 \sqrt {2}\, a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+4 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}+8 a}{2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {2}}\right ) a +3 \ln \left (-\frac {4 \left (\sqrt {2}\, a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}-2 a \right )}{2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {2}}\right ) a \right )}{3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(246\) |
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Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.50 \[ \int (a+a \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\frac {3 \, {\left (a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + 4 \, {\left (a^{2} \cos \left (d x + c\right ) + 8 \, a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+a \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \]
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Time = 0.39 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.22 \[ \int (a+a \cos (c+d x))^{5/2} \sec (c+d x) \, dx=-\frac {\sqrt {2} {\left (8 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, \sqrt {2} a^{2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 36 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{6 \, d} \]
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Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \sec (c+d x) \, dx=\int \frac {{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}}{\cos \left (c+d\,x\right )} \,d x \]
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