Integrand size = 29, antiderivative size = 112 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{4 a^2 d}-\frac {2 \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{a^2 d}-\frac {\cot ^5(c+d x)}{5 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{4 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d} \]
[Out]
Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2948, 2836, 3852, 3853, 3855} \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{4 a^2 d}-\frac {\cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^3(c+d x)}{a^2 d}-\frac {2 \cot (c+d x)}{a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{4 a^2 d} \]
[In]
[Out]
Rule 2836
Rule 2948
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \csc ^6(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4} \\ & = \frac {\int \left (a^2 \csc ^4(c+d x)-2 a^2 \csc ^5(c+d x)+a^2 \csc ^6(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \csc ^4(c+d x) \, dx}{a^2}+\frac {\int \csc ^6(c+d x) \, dx}{a^2}-\frac {2 \int \csc ^5(c+d x) \, dx}{a^2} \\ & = \frac {\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}-\frac {3 \int \csc ^3(c+d x) \, dx}{2 a^2}-\frac {\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}-\frac {\text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d} \\ & = -\frac {2 \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{a^2 d}-\frac {\cot ^5(c+d x)}{5 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{4 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}-\frac {3 \int \csc (c+d x) \, dx}{4 a^2} \\ & = \frac {3 \text {arctanh}(\cos (c+d x))}{4 a^2 d}-\frac {2 \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{a^2 d}-\frac {\cot ^5(c+d x)}{5 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{4 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d} \\ \end{align*}
Time = 1.40 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.69 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\csc ^5(c+d x) \left (-160 \cos (c+d x)+120 \cos (3 (c+d x))-24 \cos (5 (c+d x))+150 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-150 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+140 \sin (2 (c+d x))-75 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+75 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-30 \sin (4 (c+d x))+15 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-15 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))\right )}{320 a^2 d} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.30
method | result | size |
risch | \(-\frac {15 \,{\mathrm e}^{9 i \left (d x +c \right )}-40 i {\mathrm e}^{6 i \left (d x +c \right )}-70 \,{\mathrm e}^{7 i \left (d x +c \right )}+200 i {\mathrm e}^{4 i \left (d x +c \right )}-120 i {\mathrm e}^{2 i \left (d x +c \right )}+70 \,{\mathrm e}^{3 i \left (d x +c \right )}+24 i-15 \,{\mathrm e}^{i \left (d x +c \right )}}{10 a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{4 d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{4 d \,a^{2}}\) | \(146\) |
parallelrisch | \(\frac {-\left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )+5 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+40 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-110 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-120 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+110 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d \,a^{2}}\) | \(146\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {22}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{32 d \,a^{2}}\) | \(148\) |
default | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {22}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{32 d \,a^{2}}\) | \(148\) |
norman | \(\frac {-\frac {1}{160 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{80 d a}-\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d a}+\frac {9 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d a}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}+\frac {3 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}-\frac {9 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d a}+\frac {3 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d a}-\frac {\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )}{80 d a}+\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d a}-\frac {45 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}-\frac {171 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}-\frac {99 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{2}}\) | \(283\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.60 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {48 \, \cos \left (d x + c\right )^{5} - 120 \, \cos \left (d x + c\right )^{3} - 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 10 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 80 \, \cos \left (d x + c\right )}{40 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )} \sin \left (d x + c\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (104) = 208\).
Time = 0.33 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.08 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {110 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {40 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {5 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{2}} - \frac {120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {{\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {110 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{2} \sin \left (d x + c\right )^{5}}}{160 \, d} \]
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.66 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {274 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - \frac {a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 15 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 110 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{10}}}{160 \, d} \]
[In]
[Out]
Time = 10.43 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.58 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-110\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+110\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+120\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
[In]
[Out]