Integrand size = 27, antiderivative size = 89 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=-2 a b x+\frac {\left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 d}+\frac {3 b^2 \cos (c+d x)}{2 d}-\frac {a b \cot (c+d x)}{d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d} \]
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Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2968, 3127, 3110, 3102, 2814, 3855} \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a b \cot (c+d x)}{d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-2 a b x+\frac {3 b^2 \cos (c+d x)}{2 d} \]
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Rule 2814
Rule 2968
Rule 3102
Rule 3110
Rule 3127
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \csc ^3(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 (c+d x) (a+b \sin (c+d x))^2}{2 d}+\frac {1}{2} \int \csc ^2(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)}{d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac {1}{2} \int \csc (c+d x) \left (a^2-2 b^2+4 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right ) \, dx \\ & = \frac {3 b^2 \cos (c+d x)}{2 d}-\frac {a b \cot (c+d x)}{d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac {1}{2} \int \csc (c+d x) \left (a^2-2 b^2+4 a b \sin (c+d x)\right ) \, dx \\ & = -2 a b x+\frac {3 b^2 \cos (c+d x)}{2 d}-\frac {a b \cot (c+d x)}{d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac {1}{2} \left (a^2-2 b^2\right ) \int \csc (c+d x) \, dx \\ & = -2 a b x+\frac {\left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 d}+\frac {3 b^2 \cos (c+d x)}{2 d}-\frac {a b \cot (c+d x)}{d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d} \\ \end{align*}
Time = 1.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.74 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {-16 a b c-16 a b d x+8 b^2 \cos (c+d x)-8 a b \cot \left (\frac {1}{2} (c+d x)\right )-a^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+4 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-8 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )+8 a b \tan \left (\frac {1}{2} (c+d x)\right )}{8 d} \]
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Time = 0.38 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.15
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+2 a b \left (-\cot \left (d x +c \right )-d x -c \right )+b^{2} \left (\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(102\) |
default | \(\frac {a^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+2 a b \left (-\cot \left (d x +c \right )-d x -c \right )+b^{2} \left (\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(102\) |
parallelrisch | \(\frac {4 \left (-a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a^{2} \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \csc \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) a b -a^{2} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 a b x d -16 a b \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+8 \cos \left (d x +c \right ) b^{2}-8 b^{2}}{8 d}\) | \(131\) |
risch | \(-2 a b x +\frac {{\mathrm e}^{i \left (d x +c \right )} b^{2}}{2 d}+\frac {{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{2 d}-\frac {i a \left (i a \,{\mathrm e}^{3 i \left (d x +c \right )}+i a \,{\mathrm e}^{i \left (d x +c \right )}+4 b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{d}\) | \(182\) |
norman | \(\frac {\frac {a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{2}}{8 d}+\frac {a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (a^{2}-4 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {\left (a^{2}-4 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-2 a b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a b x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a b x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {\left (a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(247\) |
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (83) = 166\).
Time = 0.30 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.89 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {8 \, a b d x \cos \left (d x + c\right )^{2} - 4 \, b^{2} \cos \left (d x + c\right )^{3} - 8 \, a b d x - 8 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right ) - {\left ({\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + 2 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + 2 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
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\[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right )^{2} \cos ^{2}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {8 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a b - a^{2} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 2 \, b^{2} {\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{4 \, d} \]
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Time = 0.45 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.66 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, {\left (d x + c\right )} a b + 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, {\left (a^{2} - 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {16 \, b^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
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Time = 11.12 (sec) , antiderivative size = 397, normalized size of antiderivative = 4.46 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\cos \left (c+d\,x\right )\,\left (\frac {a^2}{2}-\frac {b^2}{4}\right )-\frac {b^2}{2}+\frac {a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4}-\frac {b^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+\frac {b^2\,\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {b^2\,\cos \left (3\,c+3\,d\,x\right )}{4}-2\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{-\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )-\frac {a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )}{4}+\frac {b^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )}{2}+a\,b\,\sin \left (2\,c+2\,d\,x\right )+2\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{-\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\cos \left (2\,c+2\,d\,x\right )}{d\,\left ({\cos \left (c+d\,x\right )}^2-1\right )} \]
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