Integrand size = 29, antiderivative size = 209 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=b^2 x-\frac {3 a b \text {arctanh}(\cos (c+d x))}{4 d}-\frac {\left (3 a^4-14 a^2 b^2+b^4\right ) \cot (c+d x)}{15 a^2 d}+\frac {b \left (27 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{60 a d}+\frac {\left (12 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{30 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 a d} \]
[Out]
Time = 0.37 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2972, 3126, 3110, 3100, 2814, 3855} \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {b \left (27 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{60 a d}+\frac {\left (12 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{30 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\left (3 a^4-14 a^2 b^2+b^4\right ) \cot (c+d x)}{15 a^2 d}-\frac {3 a b \text {arctanh}(\cos (c+d x))}{4 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 a d}+b^2 x \]
[In]
[Out]
Rule 2814
Rule 2972
Rule 3100
Rule 3110
Rule 3126
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 a d}-\frac {\int \csc ^4(c+d x) (a+b \sin (c+d x))^2 \left (2 \left (12 a^2-b^2\right )+2 a b \sin (c+d x)-20 a^2 \sin ^2(c+d x)\right ) \, dx}{20 a^2} \\ & = \frac {\left (12 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{30 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 a d}-\frac {\int \csc ^3(c+d x) (a+b \sin (c+d x)) \left (2 b \left (27 a^2-2 b^2\right )-2 a \left (6 a^2-b^2\right ) \sin (c+d x)-60 a^2 b \sin ^2(c+d x)\right ) \, dx}{60 a^2} \\ & = \frac {b \left (27 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{60 a d}+\frac {\left (12 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{30 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 a d}+\frac {\int \csc ^2(c+d x) \left (8 \left (3 a^4-14 a^2 b^2+b^4\right )+90 a^3 b \sin (c+d x)+120 a^2 b^2 \sin ^2(c+d x)\right ) \, dx}{120 a^2} \\ & = -\frac {\left (3 a^4-14 a^2 b^2+b^4\right ) \cot (c+d x)}{15 a^2 d}+\frac {b \left (27 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{60 a d}+\frac {\left (12 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{30 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 a d}+\frac {\int \csc (c+d x) \left (90 a^3 b+120 a^2 b^2 \sin (c+d x)\right ) \, dx}{120 a^2} \\ & = b^2 x-\frac {\left (3 a^4-14 a^2 b^2+b^4\right ) \cot (c+d x)}{15 a^2 d}+\frac {b \left (27 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{60 a d}+\frac {\left (12 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{30 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 a d}+\frac {1}{4} (3 a b) \int \csc (c+d x) \, dx \\ & = b^2 x-\frac {3 a b \text {arctanh}(\cos (c+d x))}{4 d}-\frac {\left (3 a^4-14 a^2 b^2+b^4\right ) \cot (c+d x)}{15 a^2 d}+\frac {b \left (27 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{60 a d}+\frac {\left (12 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{30 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 a d} \\ \end{align*}
Time = 1.79 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.36 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {960 b^2 c+960 b^2 d x+\left (-96 a^2+640 b^2\right ) \cot \left (\frac {1}{2} (c+d x)\right )+300 a b \csc ^2\left (\frac {1}{2} (c+d x)\right )-720 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+720 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-300 a b \sec ^2\left (\frac {1}{2} (c+d x)\right )+30 a b \sec ^4\left (\frac {1}{2} (c+d x)\right )-336 a^2 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+320 b^2 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+192 a^2 \csc ^5(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )-3 a^2 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+\csc ^4\left (\frac {1}{2} (c+d x)\right ) \left (-30 a b+\left (21 a^2-20 b^2\right ) \sin (c+d x)\right )+96 a^2 \tan \left (\frac {1}{2} (c+d x)\right )-640 b^2 \tan \left (\frac {1}{2} (c+d x)\right )}{960 d} \]
[In]
[Out]
Time = 0.38 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.62
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+2 a b \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+b^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(129\) |
default | \(\frac {-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+2 a b \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+b^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(129\) |
risch | \(b^{2} x -\frac {60 i a^{2} {\mathrm e}^{8 i \left (d x +c \right )}-120 i b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+75 a b \,{\mathrm e}^{9 i \left (d x +c \right )}+360 i b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-30 a b \,{\mathrm e}^{7 i \left (d x +c \right )}+120 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-440 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+280 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+30 a b \,{\mathrm e}^{3 i \left (d x +c \right )}+12 i a^{2}-80 i b^{2}-75 a b \,{\mathrm e}^{i \left (d x +c \right )}}{30 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {3 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{4 d}-\frac {3 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{4 d}\) | \(218\) |
parallelrisch | \(\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}-3 a^{2} \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -15 a b \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}+20 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}+15 a^{2} \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 b^{2} \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b +120 a b \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+480 b^{2} d x +30 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}-300 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}+360 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -30 a^{2} \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+300 b^{2} \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{480 d}\) | \(240\) |
norman | \(\frac {b^{2} x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b^{2} x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {a^{2}}{160 d}+\frac {a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}+2 b^{2} x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (3 a^{2}-260 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}+\frac {\left (3 a^{2}-260 b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}-\frac {\left (3 a^{2}-56 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}+\frac {\left (3 a^{2}-56 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}+\frac {\left (9 a^{2}-20 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}-\frac {\left (9 a^{2}-20 b^{2}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}+\frac {15 a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {15 a b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d}+\frac {3 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {3 a b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {a b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {3 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}\) | \(391\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.15 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {8 \, {\left (3 \, a^{2} - 20 \, b^{2}\right )} \cos \left (d x + c\right )^{5} + 280 \, b^{2} \cos \left (d x + c\right )^{3} - 120 \, b^{2} \cos \left (d x + c\right ) + 45 \, {\left (a b \cos \left (d x + c\right )^{4} - 2 \, a b \cos \left (d x + c\right )^{2} + a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 45 \, {\left (a b \cos \left (d x + c\right )^{4} - 2 \, a b \cos \left (d x + c\right )^{2} + a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 30 \, {\left (4 \, b^{2} d x \cos \left (d x + c\right )^{4} - 8 \, b^{2} d x \cos \left (d x + c\right )^{2} - 5 \, a b \cos \left (d x + c\right )^{3} + 4 \, b^{2} d x + 3 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
[In]
[Out]
Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.40 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.59 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {40 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} b^{2} - 15 \, a b {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {24 \, a^{2}}{\tan \left (d x + c\right )^{5}}}{120 \, d} \]
[In]
[Out]
none
Time = 0.54 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.26 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 480 \, {\left (d x + c\right )} b^{2} + 360 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 30 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 300 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {822 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 30 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 300 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 20 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
[In]
[Out]
Time = 10.95 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.66 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {b^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}+\frac {5\,b^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {5\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {2\,b^2\,\mathrm {atan}\left (\frac {4\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+3\,a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3\,a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-4\,b\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {a\,b\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}-\frac {a\,b\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}+\frac {3\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d} \]
[In]
[Out]