Integrand size = 21, antiderivative size = 114 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {7 x}{2 a^2}-\frac {16 \sin (c+d x)}{3 a^2 d}+\frac {7 \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac {8 \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {\cos ^3(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2} \]
[Out]
Time = 0.18 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2844, 3056, 2813} \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx=-\frac {16 \sin (c+d x)}{3 a^2 d}-\frac {8 \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac {7 \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac {7 x}{2 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
[In]
[Out]
Rule 2813
Rule 2844
Rule 3056
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^3(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) (3 a-5 a \cos (c+d x))}{a+a \cos (c+d x)} \, dx}{3 a^2} \\ & = -\frac {8 \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {\cos ^3(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {\int \cos (c+d x) \left (16 a^2-21 a^2 \cos (c+d x)\right ) \, dx}{3 a^4} \\ & = \frac {7 x}{2 a^2}-\frac {16 \sin (c+d x)}{3 a^2 d}+\frac {7 \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac {8 \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {\cos ^3(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\sin (c+d x) \left (-84 \arcsin (\cos (c+d x)) \cos ^4\left (\frac {1}{2} (c+d x)\right )+\left (-32-43 \cos (c+d x)-6 \cos ^2(c+d x)+3 \cos ^3(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{6 a^2 d \sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^{5/2}} \]
[In]
[Out]
Time = 0.79 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.58
method | result | size |
parallelrisch | \(\frac {-163 \left (\cos \left (d x +c \right )+\frac {12 \cos \left (2 d x +2 c \right )}{163}-\frac {3 \cos \left (3 d x +3 c \right )}{163}+\frac {140}{163}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+168 d x}{48 a^{2} d}\) | \(66\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-10 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+14 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(88\) |
default | \(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-10 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+14 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(88\) |
risch | \(\frac {7 x}{2 a^{2}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{2}}+\frac {i {\mathrm e}^{i \left (d x +c \right )}}{a^{2} d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{a^{2} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{2}}-\frac {2 i \left (12 \,{\mathrm e}^{2 i \left (d x +c \right )}+21 \,{\mathrm e}^{i \left (d x +c \right )}+11\right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\) | \(126\) |
norman | \(\frac {\frac {7 x}{2 a}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {149 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}-\frac {100 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {18 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {17 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{6 d a}+\frac {14 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {21 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {14 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {7 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(207\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.87 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {21 \, d x \cos \left (d x + c\right )^{2} + 42 \, d x \cos \left (d x + c\right ) + 21 \, d x + {\left (3 \, \cos \left (d x + c\right )^{3} - 6 \, \cos \left (d x + c\right )^{2} - 43 \, \cos \left (d x + c\right ) - 32\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (107) = 214\).
Time = 1.77 (sec) , antiderivative size = 413, normalized size of antiderivative = 3.62 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\begin {cases} \frac {21 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} + \frac {42 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} + \frac {21 d x}{6 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} + \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} - \frac {19 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} - \frac {71 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} - \frac {39 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{4}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.44 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx=-\frac {\frac {6 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {42 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{6 \, d} \]
[In]
[Out]
none
Time = 0.45 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\frac {21 \, {\left (d x + c\right )}}{a^{2}} - \frac {6 \, {\left (5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{2}} + \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
[In]
[Out]
Time = 14.58 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.99 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-22\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-30\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+12\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (c+d\,x\right )}{6\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3} \]
[In]
[Out]