Integrand size = 29, antiderivative size = 152 \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=-b^3 x+\frac {a \left (a^2+12 b^2\right ) \text {arctanh}(\cos (c+d x))}{8 d}+\frac {b \left (2 a^2-b^2\right ) \cot (c+d x)}{2 d}+\frac {a \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d} \]
[Out]
Time = 0.35 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2968, 3127, 3126, 3110, 3100, 2814, 3855} \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a \left (a^2+12 b^2\right ) \text {arctanh}(\cos (c+d x))}{8 d}+\frac {b \left (2 a^2-b^2\right ) \cot (c+d x)}{2 d}+\frac {a \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}+b^3 (-x) \]
[In]
[Out]
Rule 2814
Rule 2968
Rule 3100
Rule 3110
Rule 3126
Rule 3127
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \csc ^5(c+d x) (a+b \sin (c+d x))^3 \left (1-\sin ^2(c+d x)\right ) \, dx \\ & = -\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}+\frac {1}{4} \int \csc ^4(c+d x) (a+b \sin (c+d x))^2 \left (3 b-a \sin (c+d x)-4 b \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}+\frac {1}{12} \int \csc ^3(c+d x) (a+b \sin (c+d x)) \left (-3 \left (a^2-2 b^2\right )-9 a b \sin (c+d x)-12 b^2 \sin ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac {1}{24} \int \csc ^2(c+d x) \left (12 b \left (2 a^2-b^2\right )+3 a \left (a^2+12 b^2\right ) \sin (c+d x)+24 b^3 \sin ^2(c+d x)\right ) \, dx \\ & = \frac {b \left (2 a^2-b^2\right ) \cot (c+d x)}{2 d}+\frac {a \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac {1}{24} \int \csc (c+d x) \left (3 a \left (a^2+12 b^2\right )+24 b^3 \sin (c+d x)\right ) \, dx \\ & = -b^3 x+\frac {b \left (2 a^2-b^2\right ) \cot (c+d x)}{2 d}+\frac {a \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac {1}{8} \left (a \left (a^2+12 b^2\right )\right ) \int \csc (c+d x) \, dx \\ & = -b^3 x+\frac {a \left (a^2+12 b^2\right ) \text {arctanh}(\cos (c+d x))}{8 d}+\frac {b \left (2 a^2-b^2\right ) \cot (c+d x)}{2 d}+\frac {a \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(690\) vs. \(2(152)=304\).
Time = 6.81 (sec) , antiderivative size = 690, normalized size of antiderivative = 4.54 \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {b^3 (c+d x) (b+a \csc (c+d x))^3 \sin ^3(c+d x)}{d (a+b \sin (c+d x))^3}+\frac {\left (a^2 b \cos \left (\frac {1}{2} (c+d x)\right )-b^3 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right ) (b+a \csc (c+d x))^3 \sin ^3(c+d x)}{2 d (a+b \sin (c+d x))^3}+\frac {\left (a^3-12 a b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (b+a \csc (c+d x))^3 \sin ^3(c+d x)}{32 d (a+b \sin (c+d x))^3}-\frac {a^2 b \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (b+a \csc (c+d x))^3 \sin ^3(c+d x)}{8 d (a+b \sin (c+d x))^3}-\frac {a^3 \csc ^4\left (\frac {1}{2} (c+d x)\right ) (b+a \csc (c+d x))^3 \sin ^3(c+d x)}{64 d (a+b \sin (c+d x))^3}+\frac {\left (a^3+12 a b^2\right ) (b+a \csc (c+d x))^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin ^3(c+d x)}{8 d (a+b \sin (c+d x))^3}+\frac {\left (-a^3-12 a b^2\right ) (b+a \csc (c+d x))^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin ^3(c+d x)}{8 d (a+b \sin (c+d x))^3}+\frac {\left (-a^3+12 a b^2\right ) (b+a \csc (c+d x))^3 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^3(c+d x)}{32 d (a+b \sin (c+d x))^3}+\frac {a^3 (b+a \csc (c+d x))^3 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sin ^3(c+d x)}{64 d (a+b \sin (c+d x))^3}+\frac {(b+a \csc (c+d x))^3 \sec \left (\frac {1}{2} (c+d x)\right ) \left (-a^2 b \sin \left (\frac {1}{2} (c+d x)\right )+b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin ^3(c+d x)}{2 d (a+b \sin (c+d x))^3}+\frac {a^2 b (b+a \csc (c+d x))^3 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^3(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{8 d (a+b \sin (c+d x))^3} \]
[In]
[Out]
Time = 0.40 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {a^{2} b \left (\cos ^{3}\left (d x +c \right )\right )}{\sin \left (d x +c \right )^{3}}+3 a \,b^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+b^{3} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(166\) |
default | \(\frac {a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {a^{2} b \left (\cos ^{3}\left (d x +c \right )\right )}{\sin \left (d x +c \right )^{3}}+3 a \,b^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+b^{3} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(166\) |
parallelrisch | \(\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-a^{3} \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -8 a^{2} b \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-24 a \,b^{2} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-64 b^{3} d x -24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b +32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}-8 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-96 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+24 a^{2} b \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-32 b^{3} \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}\) | \(202\) |
risch | \(-b^{3} x -\frac {i \left (12 i a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-7 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-i a^{3} {\mathrm e}^{i \left (d x +c \right )}-24 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+8 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-7 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-12 i a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+24 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-24 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-12 i a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-8 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+24 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+8 a^{2} b -8 b^{3}\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{2 d}\) | \(332\) |
norman | \(\frac {-\frac {a^{3}}{64 d}+\frac {a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {b^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {b^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-b^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 b^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 b^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b^{3} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (a^{3}+21 a \,b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (3 a^{3}+72 a \,b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {\left (5 a^{3}+132 a \,b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {3 a \left (a^{2}+8 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {3 a \left (a^{2}+8 b^{2}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a^{2} b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {b \left (3 a^{2}-8 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {b \left (3 a^{2}-8 b^{2}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {a \left (a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(416\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.74 \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {16 \, b^{3} d x \cos \left (d x + c\right )^{4} - 32 \, b^{3} d x \cos \left (d x + c\right )^{2} + 16 \, b^{3} d x + 2 \, {\left (a^{3} - 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right ) - {\left ({\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} + 12 \, a b^{2} - 2 \, {\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} + 12 \, a b^{2} - 2 \, {\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 16 \, {\left (b^{3} \cos \left (d x + c\right ) + {\left (a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{16 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
[In]
[Out]
Timed out. \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.42 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.98 \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {16 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} b^{3} + a^{3} {\left (\frac {2 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 12 \, a b^{2} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {16 \, a^{2} b}{\tan \left (d x + c\right )^{3}}}{16 \, d} \]
[In]
[Out]
none
Time = 0.54 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.54 \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 192 \, {\left (d x + c\right )} b^{3} - 72 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, {\left (a^{3} + 12 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {50 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 600 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 72 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
[In]
[Out]
Time = 10.49 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.29 \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{8\,d}-\frac {b^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}+\frac {b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {2\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3+12\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^2+8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^3}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3+12\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^2-8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^3}\right )}{d}-\frac {3\,a\,b^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2\,d}+\frac {3\,a^2\,b\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {3\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {3\,a\,b^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {a^2\,b\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d}+\frac {3\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d} \]
[In]
[Out]