Integrand size = 29, antiderivative size = 96 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=-b^2 x+\frac {a b \text {arctanh}(\cos (c+d x))}{d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}-\frac {a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d} \]
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Time = 0.28 (sec) , antiderivative size = 96, 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 = {2968, 3127, 3110, 3100, 2814, 3855} \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}+\frac {a b \text {arctanh}(\cos (c+d x))}{d}-\frac {a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}+b^2 (-x) \]
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Rule 2814
Rule 2968
Rule 3100
Rule 3110
Rule 3127
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \csc ^4(c+d x) (a+b \sin (c+d x))^2 \left (1-\sin ^2(c+d x)\right ) \, dx \\ & = -\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}+\frac {1}{3} \int \csc ^3(c+d x) (a+b \sin (c+d x)) \left (2 b-a \sin (c+d x)-3 b \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}-\frac {1}{6} \int \csc ^2(c+d x) \left (2 \left (a^2-2 b^2\right )+6 a b \sin (c+d x)+6 b^2 \sin ^2(c+d x)\right ) \, dx \\ & = \frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}-\frac {a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}-\frac {1}{6} \int \csc (c+d x) \left (6 a b+6 b^2 \sin (c+d x)\right ) \, dx \\ & = -b^2 x+\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}-\frac {a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}-(a b) \int \csc (c+d x) \, dx \\ & = -b^2 x+\frac {a b \text {arctanh}(\cos (c+d x))}{d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}-\frac {a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(538\) vs. \(2(96)=192\).
Time = 6.71 (sec) , antiderivative size = 538, normalized size of antiderivative = 5.60 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {b^2 (c+d x) (b+a \csc (c+d x))^2 \sin ^2(c+d x)}{d (a+b \sin (c+d x))^2}+\frac {\left (a^2 \cos \left (\frac {1}{2} (c+d x)\right )-3 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right ) (b+a \csc (c+d x))^2 \sin ^2(c+d x)}{6 d (a+b \sin (c+d x))^2}-\frac {a b \csc ^2\left (\frac {1}{2} (c+d x)\right ) (b+a \csc (c+d x))^2 \sin ^2(c+d x)}{4 d (a+b \sin (c+d x))^2}-\frac {a^2 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (b+a \csc (c+d x))^2 \sin ^2(c+d x)}{24 d (a+b \sin (c+d x))^2}+\frac {a b (b+a \csc (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin ^2(c+d x)}{d (a+b \sin (c+d x))^2}-\frac {a b (b+a \csc (c+d x))^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin ^2(c+d x)}{d (a+b \sin (c+d x))^2}+\frac {a b (b+a \csc (c+d x))^2 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2(c+d x)}{4 d (a+b \sin (c+d x))^2}+\frac {(b+a \csc (c+d x))^2 \sec \left (\frac {1}{2} (c+d x)\right ) \left (-a^2 \sin \left (\frac {1}{2} (c+d x)\right )+3 b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin ^2(c+d x)}{6 d (a+b \sin (c+d x))^2}+\frac {a^2 (b+a \csc (c+d x))^2 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{24 d (a+b \sin (c+d x))^2} \]
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Time = 0.32 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )^{3}}+2 a b \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(96\) |
default | \(\frac {-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )^{3}}+2 a b \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(96\) |
parallelrisch | \(\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}-a^{2} \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -6 a b \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 b^{2} d x -3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}+3 a^{2} \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-12 b^{2} \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b}{24 d}\) | \(145\) |
risch | \(-b^{2} x +\frac {2 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-2 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+2 a b \,{\mathrm e}^{5 i \left (d x +c \right )}+4 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a^{2}}{3}-2 i b^{2}-2 a b \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}-\frac {a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}\) | \(147\) |
norman | \(\frac {\frac {a b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{2}}{24 d}+\frac {a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-b^{2} x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b^{2} x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b^{2} x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (a^{2}-12 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {\left (a^{2}-12 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {\left (a^{2}-6 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {\left (a^{2}-6 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(293\) |
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Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.74 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {2 \, {\left (a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 6 \, b^{2} \cos \left (d x + c\right ) + 3 \, {\left (a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, {\left (a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 6 \, {\left (b^{2} d x \cos \left (d x + c\right )^{2} - b^{2} d x - a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {6 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} b^{2} - 3 \, a b {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {2 \, a^{2}}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]
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Time = 0.66 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.74 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, {\left (d x + c\right )} b^{2} - 24 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {44 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
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Time = 10.01 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.41 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}+\frac {a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {b^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {2\,b^2\,\mathrm {atan}\left (\frac {b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-b\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a\,b\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}-\frac {a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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