Integrand size = 29, antiderivative size = 94 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{a}-\frac {3 \text {arctanh}(\cos (c+d x))}{2 a d}+\frac {3 \cos (c+d x)}{2 a d}+\frac {\cot (c+d x)}{a d}+\frac {\cos (c+d x) \cot ^2(c+d x)}{2 a d}-\frac {\cot ^3(c+d x)}{3 a d} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2918, 3554, 8, 2672, 294, 327, 212} \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 \text {arctanh}(\cos (c+d x))}{2 a d}+\frac {3 \cos (c+d x)}{2 a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot (c+d x)}{a d}+\frac {\cos (c+d x) \cot ^2(c+d x)}{2 a d}+\frac {x}{a} \]
[In]
[Out]
Rule 8
Rule 212
Rule 294
Rule 327
Rule 2672
Rule 2918
Rule 3554
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cos (c+d x) \cot ^3(c+d x) \, dx}{a}+\frac {\int \cot ^4(c+d x) \, dx}{a} \\ & = -\frac {\cot ^3(c+d x)}{3 a d}-\frac {\int \cot ^2(c+d x) \, dx}{a}+\frac {\text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {\cot (c+d x)}{a d}+\frac {\cos (c+d x) \cot ^2(c+d x)}{2 a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\int 1 \, dx}{a}-\frac {3 \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 a d} \\ & = \frac {x}{a}+\frac {3 \cos (c+d x)}{2 a d}+\frac {\cot (c+d x)}{a d}+\frac {\cos (c+d x) \cot ^2(c+d x)}{2 a d}-\frac {\cot ^3(c+d x)}{3 a d}-\frac {3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 a d} \\ & = \frac {x}{a}-\frac {3 \text {arctanh}(\cos (c+d x))}{2 a d}+\frac {3 \cos (c+d x)}{2 a d}+\frac {\cot (c+d x)}{a d}+\frac {\cos (c+d x) \cot ^2(c+d x)}{2 a d}-\frac {\cot ^3(c+d x)}{3 a d} \\ \end{align*}
Time = 1.33 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.47 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (12 \left (2 c+2 d x-3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin ^3(c+d x)-2 \cos (3 (c+d x)) (4+3 \sin (c+d x))+9 \sin (2 (c+d x))\right )}{192 a d (1+\sin (c+d x))} \]
[In]
[Out]
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {16}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+16 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}\) | \(125\) |
default | \(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {16}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+16 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}\) | \(125\) |
parallelrisch | \(\frac {36 \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )-3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (4 d x -9\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 d x +3 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+14 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{24 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(147\) |
risch | \(\frac {x}{a}+\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 a d}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 a d}-\frac {-12 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}+12 i {\mathrm e}^{2 i \left (d x +c \right )}-8 i-3 \,{\mathrm e}^{i \left (d x +c \right )}}{3 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d a}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d a}\) | \(151\) |
norman | \(\frac {\frac {x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {1}{24 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d a}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {2 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d a}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}+\frac {2 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {2 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {29 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {25 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {43 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}-\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d}\) | \(355\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.57 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {16 \, \cos \left (d x + c\right )^{3} - 9 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 9 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 6 \, {\left (2 \, d x \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, d x - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 12 \, \cos \left (d x + c\right )}{12 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (86) = 172\).
Time = 0.31 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.55 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a} - \frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {14 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {51 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 1}{\frac {a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} - \frac {48 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {36 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{24 \, d} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.67 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {24 \, {\left (d x + c\right )}}{a} + \frac {36 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} + \frac {48}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a} - \frac {66 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
[In]
[Out]
Time = 10.10 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.26 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}-\frac {2\,\mathrm {atan}\left (\frac {6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-6}+\frac {4}{4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-6}\right )}{a\,d}+\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a\,d}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{3}}{d\,\left (8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,d} \]
[In]
[Out]