Integrand size = 23, antiderivative size = 107 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx=\frac {19 \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {\sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {13 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}} \]
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Time = 0.17 (sec) , antiderivative size = 107, 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, 2829, 2728, 212} \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx=\frac {19 \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {13 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac {\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}} \]
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Rule 212
Rule 2728
Rule 2829
Rule 2837
Rubi steps \begin{align*} \text {integral}& = \frac {\sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac {\int \frac {-\frac {5 a}{2}+4 a \cos (c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx}{4 a^2} \\ & = \frac {\sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {13 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac {19 \int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx}{32 a^2} \\ & = \frac {\sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {13 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac {19 \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 )}{16 a^2 d} \\ & = \frac {19 \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {\sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {13 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx=\frac {\left (76 \sqrt {2} \text {arctanh}\left (\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}\right ) \cos ^4\left (\frac {1}{2} (c+d x)\right )-2 \sqrt {1-\cos (c+d x)} (9+13 \cos (c+d x))\right ) \sin (c+d x)}{32 d \sqrt {1-\cos (c+d x)} (a (1+\cos (c+d x)))^{5/2}} \]
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Time = 1.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.63
method | result | size |
default | \(\frac {\sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (19 \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-13 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {2}\, \sqrt {a}+2 \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\right )}{32 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{\frac {7}{2}} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(174\) |
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Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (88) = 176\).
Time = 0.26 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.76 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx=\frac {19 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) - 4 \, \sqrt {a \cos \left (d x + c\right ) + a} {\left (13 \, \cos \left (d x + c\right ) + 9\right )} \sin \left (d x + c\right )}{64 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
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Timed out. \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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none
Time = 1.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.29 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx=\frac {\frac {19 \, \sqrt {2} \log \left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {19 \, \sqrt {2} \log \left (-\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {2 \, \sqrt {2} {\left (13 \, \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 11 \, \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{64 \, d} \]
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Timed out. \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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