Integrand size = 19, antiderivative size = 112 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {4 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {8 \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {8 \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2829, 2729, 2727} \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {8 \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac {8 \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac {4 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
[In]
[Out]
Rule 2727
Rule 2729
Rule 2829
Rubi steps \begin{align*} \text {integral}& = -\frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {4 \int \frac {1}{(a+a \cos (c+d x))^3} \, dx}{7 a} \\ & = -\frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {4 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {8 \int \frac {1}{(a+a \cos (c+d x))^2} \, dx}{35 a^2} \\ & = -\frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {4 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {8 \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {8 \int \frac {1}{a+a \cos (c+d x)} \, dx}{105 a^3} \\ & = -\frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {4 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {8 \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {8 \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.50 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\left (13+52 \cos (c+d x)+32 \cos ^2(c+d x)+8 \cos ^3(c+d x)\right ) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^4} \]
[In]
[Out]
Time = 0.78 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.51
method | result | size |
parallelrisch | \(-\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {7 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {7 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-7\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{56 a^{4} d}\) | \(57\) |
derivativedivides | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}\) | \(58\) |
default | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}\) | \(58\) |
risch | \(\frac {8 i \left (35 \,{\mathrm e}^{4 i \left (d x +c \right )}+35 \,{\mathrm e}^{3 i \left (d x +c \right )}+42 \,{\mathrm e}^{2 i \left (d x +c \right )}+14 \,{\mathrm e}^{i \left (d x +c \right )}+2\right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(69\) |
norman | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{6 d a}+\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{60 d a}-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{70 d a}-\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{56 d a}}{a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(114\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {{\left (8 \, \cos \left (d x + c\right )^{3} + 32 \, \cos \left (d x + c\right )^{2} + 52 \, \cos \left (d x + c\right ) + 13\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
[In]
[Out]
Time = 1.58 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.76 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\begin {cases} - \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} + \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \cos {\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.78 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{840 \, a^{4} d} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.53 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{840 \, a^{4} d} \]
[In]
[Out]
Time = 14.94 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.52 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+105\right )}{840\,a^4\,d} \]
[In]
[Out]