Integrand size = 23, antiderivative size = 105 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=-\frac {7 \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {2 \sin (c+d x)}{a d \sqrt {a+a \cos (c+d x)}} \]
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Time = 0.18 (sec) , antiderivative size = 105, 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 = {2837, 2830, 2728, 212} \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=-\frac {7 \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {2 \sin (c+d x)}{a d \sqrt {a \cos (c+d x)+a}}+\frac {\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}} \]
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Rule 212
Rule 2728
Rule 2830
Rule 2837
Rubi steps \begin{align*} \text {integral}& = \frac {\sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {\int \frac {-\frac {3 a}{2}+2 a \cos (c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{2 a^2} \\ & = \frac {\sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {2 \sin (c+d x)}{a d \sqrt {a+a \cos (c+d x)}}-\frac {7 \int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx}{4 a} \\ & = \frac {\sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {2 \sin (c+d x)}{a d \sqrt {a+a \cos (c+d x)}}+\frac {7 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{2 a d} \\ & = -\frac {7 \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {2 \sin (c+d x)}{a d \sqrt {a+a \cos (c+d x)}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {\left (-7 \sqrt {2} \text {arctanh}\left (\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}\right ) (1+\cos (c+d x))+2 \sqrt {1-\cos (c+d x)} (5+4 \cos (c+d x))\right ) \sin (c+d x)}{4 d \sqrt {1-\cos (c+d x)} (a (1+\cos (c+d x)))^{3/2}} \]
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Time = 1.56 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.65
method | result | size |
default | \(\frac {\sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-7 \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {2}\, \sqrt {a}+\sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\right )}{4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{\frac {5}{2}} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(173\) |
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Time = 0.25 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.56 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {7 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 4 \, \sqrt {a \cos \left (d x + c\right ) + a} {\left (4 \, \cos \left (d x + c\right ) + 5\right )} \sin \left (d x + c\right )}{8 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
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\[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )}}{\left (a \left (\cos {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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Timed out. \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Time = 0.46 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.38 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=-\frac {\frac {7 \, \sqrt {2} \log \left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {7 \, \sqrt {2} \log \left (-\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {16 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {2 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{8 \, d} \]
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Timed out. \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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