Integrand size = 12, antiderivative size = 140 \[ \int \frac {1}{(-3-5 \cos (c+d x))^4} \, dx=\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {5 \sin (c+d x)}{48 d (3+5 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (3+5 \cos (c+d x))^2}+\frac {995 \sin (c+d x)}{24576 d (3+5 \cos (c+d x))} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 140, 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} \[ \int \frac {1}{(-3-5 \cos (c+d x))^4} \, dx=\frac {995 \sin (c+d x)}{24576 d (5 \cos (c+d x)+3)}-\frac {25 \sin (c+d x)}{512 d (5 \cos (c+d x)+3)^2}+\frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}+\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d} \]
[In]
[Out]
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}{-8+2 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{4096 d} \\ & = \frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\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. \(296\) vs. \(2(140)=280\).
Time = 0.01 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.11 \[ \int \frac {1}{(-3-5 \cos (c+d x))^4} \, dx=\frac {467046 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+104625 \cos (3 (c+d x)) \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+765855 \cos (c+d x) \left (\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+376650 \cos (2 (c+d x)) \left (\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-467046 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-104625 \cos (3 (c+d x)) \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+226140 \sin (c+d x)+190800 \sin (2 (c+d x))+99500 \sin (3 (c+d x))}{393216 d (3+5 \cos (c+d x))^3} \]
[In]
[Out]
Time = 0.68 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.71
method | result | size |
norman | \(\frac {-\frac {295 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512 d}+\frac {265 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768 d}-\frac {745 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8192 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-4\right )^{3}}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768 d}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768 d}\) | \(99\) |
derivativedivides | \(\frac {-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}-\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}+\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}}{d}\) | \(124\) |
default | \(\frac {-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}-\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}+\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}}{d}\) | \(124\) |
risch | \(\frac {i \left (20925 \,{\mathrm e}^{5 i \left (d x +c \right )}+62775 \,{\mathrm e}^{4 i \left (d x +c \right )}+111042 \,{\mathrm e}^{3 i \left (d x +c \right )}+119310 \,{\mathrm e}^{2 i \left (d x +c \right )}+68625 \,{\mathrm e}^{i \left (d x +c \right )}+24875\right )}{12288 d \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )^{3}}+\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}-\frac {4 i}{5}\right )}{32768 d}-\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}+\frac {4 i}{5}\right )}{32768 d}\) | \(129\) |
parallelrisch | \(\frac {\left (765855 \cos \left (d x +c \right )+376650 \cos \left (2 d x +2 c \right )+104625 \cos \left (3 d x +3 c \right )+467046\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )+\left (-765855 \cos \left (d x +c \right )-376650 \cos \left (2 d x +2 c \right )-104625 \cos \left (3 d x +3 c \right )-467046\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )+226140 \sin \left (d x +c \right )+190800 \sin \left (2 d x +2 c \right )+99500 \sin \left (3 d x +3 c \right )}{98304 d \left (558+125 \cos \left (3 d x +3 c \right )+915 \cos \left (d x +c \right )+450 \cos \left (2 d x +2 c \right )\right )}\) | \(161\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(-3-5 \cos (c+d x))^4} \, dx=-\frac {837 \, {\left (125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) + 27\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) - 837 \, {\left (125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) + 27\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) - 40 \, {\left (4975 \, \cos \left (d x + c\right )^{2} + 4770 \, \cos \left (d x + c\right ) + 1583\right )} \sin \left (d x + c\right )}{196608 \, {\left (125 \, d \cos \left (d x + c\right )^{3} + 225 \, d \cos \left (d x + c\right )^{2} + 135 \, d \cos \left (d x + c\right ) + 27 \, d\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 816 vs. \(2 (126) = 252\).
Time = 2.44 (sec) , antiderivative size = 816, normalized size of antiderivative = 5.83 \[ \int \frac {1}{(-3-5 \cos (c+d x))^4} \, dx=\text {Too large to display} \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(-3-5 \cos (c+d x))^4} \, dx=-\frac {\frac {20 \, {\left (\frac {2832 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1696 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {447 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {12 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 64} + 837 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - 837 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{98304 \, d} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(-3-5 \cos (c+d x))^4} \, dx=-\frac {\frac {20 \, {\left (447 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1696 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2832 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4\right )}^{3}} + 837 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) - 837 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{98304 \, d} \]
[In]
[Out]
Time = 0.00 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(-3-5 \cos (c+d x))^4} \, dx=-\frac {279\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{16384\,d}-\frac {\frac {745\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8192}-\frac {265\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{768}+\frac {295\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-64\right )} \]
[In]
[Out]