Integrand size = 12, antiderivative size = 138 \[ \int \frac {1}{(3-5 \cos (c+d x))^4} \, dx=-\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac {995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))} \]
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Time = 0.18 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2743, 2833, 12, 2738, 213} \[ \int \frac {1}{(3-5 \cos (c+d x))^4} \, dx=-\frac {995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}-\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {279 \log \left (2 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d} \]
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Rule 12
Rule 213
Rule 2738
Rule 2743
Rule 2833
Rubi steps \begin{align*} \text {integral}& = -\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {1}{48} \int \frac {-9-10 \cos (c+d x)}{(3-5 \cos (c+d x))^3} \, dx \\ & = -\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}+\frac {\int \frac {154+75 \cos (c+d x)}{(3-5 \cos (c+d x))^2} \, dx}{1536} \\ & = -\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac {995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac {\int -\frac {837}{3-5 \cos (c+d x)} \, dx}{24576} \\ & = -\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac {995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}-\frac {279 \int \frac {1}{3-5 \cos (c+d x)} \, dx}{8192} \\ & = -\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac {995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}-\frac {279 \text {Subst}\left (\int \frac {1}{-2+8 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{4096 d} \\ & = -\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac {995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(288\) vs. \(2(138)=276\).
Time = 0.36 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.09 \[ \int \frac {1}{(3-5 \cos (c+d x))^4} \, dx=\frac {467046 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )-104625 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )-765855 \cos (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+376650 \cos (2 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-467046 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )+104625 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )+226140 \sin (c+d x)-190800 \sin (2 (c+d x))+99500 \sin (3 (c+d x))}{393216 d (-3+5 \cos (c+d x))^3} \]
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Time = 0.73 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.76
method | result | size |
norman | \(\frac {-\frac {745 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8192 d}+\frac {265 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768 d}-\frac {295 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512 d}}{{\left (4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}^{3}}-\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32768 d}+\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768 d}\) | \(105\) |
risch | \(-\frac {i \left (20925 \,{\mathrm e}^{5 i \left (d x +c \right )}-62775 \,{\mathrm e}^{4 i \left (d x +c \right )}+111042 \,{\mathrm e}^{3 i \left (d x +c \right )}-119310 \,{\mathrm e}^{2 i \left (d x +c \right )}+68625 \,{\mathrm e}^{i \left (d x +c \right )}-24875\right )}{12288 d \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}-6 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )^{3}}+\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {3}{5}+\frac {4 i}{5}\right )}{32768 d}-\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {3}{5}-\frac {4 i}{5}\right )}{32768 d}\) | \(129\) |
derivativedivides | \(\frac {-\frac {125}{49152 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {25}{8192 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {295}{32768 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32768}-\frac {125}{49152 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {25}{8192 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {295}{32768 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768}}{d}\) | \(140\) |
default | \(\frac {-\frac {125}{49152 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {25}{8192 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {295}{32768 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32768}-\frac {125}{49152 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {25}{8192 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {295}{32768 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768}}{d}\) | \(140\) |
parallelrisch | \(\frac {\left (-765855 \cos \left (d x +c \right )+376650 \cos \left (2 d x +2 c \right )-104625 \cos \left (3 d x +3 c \right )+467046\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )+\left (765855 \cos \left (d x +c \right )-376650 \cos \left (2 d x +2 c \right )+104625 \cos \left (3 d x +3 c \right )-467046\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{2}\right )+226140 \sin \left (d x +c \right )-190800 \sin \left (2 d x +2 c \right )+99500 \sin \left (3 d x +3 c \right )}{98304 d \left (-558+125 \cos \left (3 d x +3 c \right )-450 \cos \left (2 d x +2 c \right )+915 \cos \left (d x +c \right )\right )}\) | \(161\) |
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Time = 0.28 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(3-5 \cos (c+d x))^4} \, dx=\frac {837 \, {\left (125 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) - 27\right )} \log \left (-\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) - 837 \, {\left (125 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) - 27\right )} \log \left (-\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 40 \, {\left (4975 \, \cos \left (d x + c\right )^{2} - 4770 \, \cos \left (d x + c\right ) + 1583\right )} \sin \left (d x + c\right )}{196608 \, {\left (125 \, d \cos \left (d x + c\right )^{3} - 225 \, d \cos \left (d x + c\right )^{2} + 135 \, d \cos \left (d x + c\right ) - 27 \, d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 831 vs. \(2 (126) = 252\).
Time = 2.47 (sec) , antiderivative size = 831, normalized size of antiderivative = 6.02 \[ \int \frac {1}{(3-5 \cos (c+d x))^4} \, dx=\text {Too large to display} \]
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Time = 0.24 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.28 \[ \int \frac {1}{(3-5 \cos (c+d x))^4} \, dx=-\frac {\frac {20 \, {\left (\frac {447 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1696 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2832 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {48 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {64 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 1} - 837 \, \log \left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) + 837 \, \log \left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{98304 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(3-5 \cos (c+d x))^4} \, dx=-\frac {\frac {20 \, {\left (2832 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1696 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 447 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}} - 837 \, \log \left ({\left | 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 837 \, \log \left ({\left | 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{98304 \, d} \]
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Time = 15.70 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(3-5 \cos (c+d x))^4} \, dx=\frac {279\,\mathrm {atanh}\left (2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16384\,d}-\frac {\frac {295\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32768}-\frac {265\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{49152}+\frac {745\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{524288}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16}-\frac {1}{64}\right )} \]
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