Integrand size = 29, antiderivative size = 114 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x)}{a d}-\frac {2 \cot ^3(c+d x)}{3 a d}-\frac {\cot ^5(c+d x)}{5 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2918, 3852, 3853, 3855} \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}-\frac {\cot ^5(c+d x)}{5 a d}-\frac {2 \cot ^3(c+d x)}{3 a d}-\frac {\cot (c+d x)}{a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d} \]
[In]
[Out]
Rule 2918
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \csc ^5(c+d x) \, dx}{a}+\frac {\int \csc ^6(c+d x) \, dx}{a} \\ & = \frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {3 \int \csc ^3(c+d x) \, dx}{4 a}-\frac {\text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a d} \\ & = -\frac {\cot (c+d x)}{a d}-\frac {2 \cot ^3(c+d x)}{3 a d}-\frac {\cot ^5(c+d x)}{5 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {3 \int \csc (c+d x) \, dx}{8 a} \\ & = \frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x)}{a d}-\frac {2 \cot ^3(c+d x)}{3 a d}-\frac {\cot ^5(c+d x)}{5 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \\ \end{align*}
Time = 1.24 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.66 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^5(c+d x) \left (-640 \cos (c+d x)+320 \cos (3 (c+d x))-64 \cos (5 (c+d x))+450 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-450 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+420 \sin (2 (c+d x))-225 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+225 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-90 \sin (4 (c+d x))+45 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-45 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))\right )}{1920 a d} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.18
method | result | size |
risch | \(-\frac {45 \,{\mathrm e}^{9 i \left (d x +c \right )}-210 \,{\mathrm e}^{7 i \left (d x +c \right )}+640 i {\mathrm e}^{4 i \left (d x +c \right )}-320 i {\mathrm e}^{2 i \left (d x +c \right )}+210 \,{\mathrm e}^{3 i \left (d x +c \right )}+64 i-45 \,{\mathrm e}^{i \left (d x +c \right )}}{60 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}\) | \(134\) |
parallelrisch | \(\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+50 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-50 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+300 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-300 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-360 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{960 d a}\) | \(148\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {5}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{32 d a}\) | \(150\) |
default | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {5}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{32 d a}\) | \(150\) |
norman | \(\frac {-\frac {1}{160 a d}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{320 d a}-\frac {7 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}+\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}+\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}-\frac {7 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {7 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {3 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d a}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d a}-\frac {5 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}\) | \(242\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.52 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {128 \, \cos \left (d x + c\right )^{5} - 320 \, \cos \left (d x + c\right )^{3} - 45 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 45 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 30 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 240 \, \cos \left (d x + c\right )}{240 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )} \sin \left (d x + c\right )} \]
[In]
[Out]
\[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\cos ^{2}{\left (c + d x \right )} \csc ^{6}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (104) = 208\).
Time = 0.19 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.05 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {300 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {120 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {50 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {6 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a} - \frac {360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {{\left (\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {50 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {120 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {300 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 6\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a \sin \left (d x + c\right )^{5}}}{960 \, d} \]
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.64 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {360 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 50 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 300 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{5}} - \frac {822 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 300 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 50 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]
[In]
[Out]
Time = 10.24 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.55 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {6\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-50\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-300\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+300\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+50\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+360\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{960\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
[In]
[Out]