Integrand size = 12, antiderivative size = 112 \[ \int \frac {1}{(a+a \cos (c+d x))^4} \, dx=\frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {3 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {2 \sin (c+d x)}{35 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {2 \sin (c+d x)}{35 d \left (a^4+a^4 \cos (c+d x)\right )} \]
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Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2729, 2727} \[ \int \frac {1}{(a+a \cos (c+d x))^4} \, dx=\frac {2 \sin (c+d x)}{35 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac {2 \sin (c+d x)}{35 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac {3 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rule 2727
Rule 2729
Rubi steps \begin{align*} \text {integral}& = \frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {3 \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 {3 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {6 \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 {3 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {2 \sin (c+d x)}{35 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {2 \int \frac {1}{a+a \cos (c+d x)} \, dx}{35 a^3} \\ & = \frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {3 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {2 \sin (c+d x)}{35 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {2 \sin (c+d x)}{35 d \left (a^4+a^4 \cos (c+d x)\right )} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(a+a \cos (c+d x))^4} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \left (35 \sin \left (\frac {1}{2} (c+d x)\right )+21 \sin \left (\frac {3}{2} (c+d x)\right )+7 \sin \left (\frac {5}{2} (c+d x)\right )+\sin \left (\frac {7}{2} (c+d x)\right )\right )}{70 a^4 d (1+\cos (c+d x))^4} \]
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Time = 0.63 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.50
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}\) | \(56\) |
default | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}\) | \(56\) |
risch | \(\frac {4 i \left (35 \,{\mathrm e}^{3 i \left (d x +c \right )}+21 \,{\mathrm e}^{2 i \left (d x +c \right )}+7 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{35 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(58\) |
parallelrisch | \(\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{280 a^{4} d}\) | \(60\) |
norman | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d a}+\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{56 d a}}{a^{3}}\) | \(80\) |
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Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(a+a \cos (c+d x))^4} \, dx=\frac {{\left (2 \, \cos \left (d x + c\right )^{3} + 8 \, \cos \left (d x + c\right )^{2} + 13 \, \cos \left (d x + c\right ) + 12\right )} \sin \left (d x + c\right )}{35 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
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Time = 1.43 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(a+a \cos (c+d x))^4} \, dx=\begin {cases} \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} + \frac {3 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} + \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{280 \, a^{4} d} \]
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Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.53 \[ \int \frac {1}{(a+a \cos (c+d x))^4} \, dx=\frac {5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{280 \, a^{4} d} \]
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Time = 15.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(a+a \cos (c+d x))^4} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+35\right )}{280\,a^4\,d} \]
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