Integrand size = 21, antiderivative size = 66 \[ \int (a+a \cos (c+d x))^{3/2} \sec (c+d x) \, dx=\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{d}+\frac {2 a^2 \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}} \]
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Time = 0.13 (sec) , antiderivative size = 66, 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, 21, 2852, 212} \[ \int (a+a \cos (c+d x))^{3/2} \sec (c+d x) \, dx=\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}+\frac {2 a^2 \sin (c+d x)}{d \sqrt {a \cos (c+d x)+a}} \]
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Rule 21
Rule 212
Rule 2842
Rule 2852
Rubi steps \begin{align*} \text {integral}& = \frac {2 a^2 \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}}+2 \int \frac {\left (\frac {a^2}{2}+\frac {1}{2} a^2 \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx \\ & = \frac {2 a^2 \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}}+a \int \sqrt {a+a \cos (c+d x)} \sec (c+d x) \, dx \\ & = \frac {2 a^2 \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}}-\frac {\left (2 a^2\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^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{d}+\frac {2 a^2 \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98 \[ \int (a+a \cos (c+d x))^{3/2} \sec (c+d x) \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (\sqrt {2} \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(208\) vs. \(2(58)=116\).
Time = 1.34 (sec) , antiderivative size = 209, normalized size of antiderivative = 3.17
method | result | size |
default | \(\frac {\sqrt {a}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (2 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}+\ln \left (\frac {4 \sqrt {2}\, a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+4 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}+8 a}{2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {2}}\right ) a +\ln \left (-\frac {4 \left (\sqrt {2}\, a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}-2 a \right )}{2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {2}}\right ) a \right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(209\) |
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Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (58) = 116\).
Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.92 \[ \int (a+a \cos (c+d x))^{3/2} \sec (c+d x) \, dx=\frac {{\left (a \cos \left (d x + c\right ) + a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + 4 \, \sqrt {a \cos \left (d x + c\right ) + a} a \sin \left (d x + c\right )}{2 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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\[ \int (a+a \cos (c+d x))^{3/2} \sec (c+d x) \, dx=\int \left (a \left (\cos {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \sec {\left (c + d x \right )}\, dx \]
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\[ \int (a+a \cos (c+d x))^{3/2} \sec (c+d x) \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right ) \,d x } \]
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none
Time = 0.31 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.35 \[ \int (a+a \cos (c+d x))^{3/2} \sec (c+d x) \, dx=-\frac {\sqrt {2} {\left (\sqrt {2} a \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 4 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{2 \, d} \]
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Timed out. \[ \int (a+a \cos (c+d x))^{3/2} \sec (c+d x) \, dx=\int \frac {{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}}{\cos \left (c+d\,x\right )} \,d x \]
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