Integrand size = 29, antiderivative size = 91 \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=-\frac {7 a A \text {arctanh}(\cos (c+d x))}{8 d}-\frac {2 a A \cot (c+d x)}{d}-\frac {2 a A \cot ^3(c+d x)}{3 d}-\frac {7 a A \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d} \]
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Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {21, 3873, 3852, 4131, 3853, 3855} \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=-\frac {7 a A \text {arctanh}(\cos (c+d x))}{8 d}-\frac {2 a A \cot ^3(c+d x)}{3 d}-\frac {2 a A \cot (c+d x)}{d}-\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {7 a A \cot (c+d x) \csc (c+d x)}{8 d} \]
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Rule 21
Rule 3852
Rule 3853
Rule 3855
Rule 3873
Rule 4131
Rubi steps \begin{align*} \text {integral}& = \frac {A \int \csc ^3(c+d x) (a+a \csc (c+d x))^2 \, dx}{a} \\ & = \frac {A \int \csc ^3(c+d x) \left (a^2+a^2 \csc ^2(c+d x)\right ) \, dx}{a}+(2 a A) \int \csc ^4(c+d x) \, dx \\ & = -\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{4} (7 a A) \int \csc ^3(c+d x) \, dx-\frac {(2 a A) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d} \\ & = -\frac {2 a A \cot (c+d x)}{d}-\frac {2 a A \cot ^3(c+d x)}{3 d}-\frac {7 a A \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{8} (7 a A) \int \csc (c+d x) \, dx \\ & = -\frac {7 a A \text {arctanh}(\cos (c+d x))}{8 d}-\frac {2 a A \cot (c+d x)}{d}-\frac {2 a A \cot ^3(c+d x)}{3 d}-\frac {7 a A \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.79 \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=-\frac {4 a A \cot (c+d x)}{3 d}-\frac {7 a A \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a A \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {2 a A \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac {7 a A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {7 a A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {7 a A \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a A \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d} \]
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Time = 1.44 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.23
method | result | size |
derivativedivides | \(\frac {A a \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )+2 A a \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )+A a \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}\) | \(112\) |
default | \(\frac {A a \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )+2 A a \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )+A a \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}\) | \(112\) |
parts | \(\frac {A a \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}+\frac {A a \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )}{d}+\frac {2 A a \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )}{d}\) | \(117\) |
parallelrisch | \(-\frac {a \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {16 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+16 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+48 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-56 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) A}{64 d}\) | \(119\) |
risch | \(\frac {A a \left (21 \,{\mathrm e}^{7 i \left (d x +c \right )}-45 \,{\mathrm e}^{5 i \left (d x +c \right )}-45 \,{\mathrm e}^{3 i \left (d x +c \right )}+96 i {\mathrm e}^{4 i \left (d x +c \right )}+21 \,{\mathrm e}^{i \left (d x +c \right )}-128 i {\mathrm e}^{2 i \left (d x +c \right )}+32 i\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {7 A a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {7 A a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}\) | \(131\) |
norman | \(\frac {-\frac {A a}{64 d}-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d}-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 d}-\frac {3 A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 d}+\frac {3 A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}+\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 d}+\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{12 d}+\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {7 A a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(163\) |
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Time = 0.25 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.77 \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=\frac {42 \, A a \cos \left (d x + c\right )^{3} - 54 \, A a \cos \left (d x + c\right ) - 21 \, {\left (A a \cos \left (d x + c\right )^{4} - 2 \, A a \cos \left (d x + c\right )^{2} + A a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 21 \, {\left (A a \cos \left (d x + c\right )^{4} - 2 \, A a \cos \left (d x + c\right )^{2} + A a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 32 \, {\left (2 \, A a \cos \left (d x + c\right )^{3} - 3 \, A a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
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\[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=A a \left (\int \csc ^{3}{\left (c + d x \right )}\, dx + \int 2 \csc ^{4}{\left (c + d x \right )}\, dx + \int \csc ^{5}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.24 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.59 \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=\frac {3 \, A a {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12 \, A a {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {32 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a}{\tan \left (d x + c\right )^{3}}}{48 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.70 \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=\frac {3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 168 \, A a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 144 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {350 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 144 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 16 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
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Time = 18.94 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.66 \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=\frac {A\,a\,\left (3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+16\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+144\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-144\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+168\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}{192\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4} \]
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