Integrand size = 30, antiderivative size = 61 \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=-\frac {a A \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a A \cot (c+d x) \csc (c+d x)}{8 d}+\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d} \]
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Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4047, 2691, 3853, 3855} \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=-\frac {a A \text {arctanh}(\cos (c+d x))}{8 d}+\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {a A \cot (c+d x) \csc (c+d x)}{8 d} \]
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Rule 2691
Rule 3853
Rule 3855
Rule 4047
Rubi steps \begin{align*} \text {integral}& = -\left ((a A) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx\right ) \\ & = \frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{4} (a A) \int \csc ^3(c+d x) \, dx \\ & = -\frac {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} (a A) \int \csc (c+d x) \, dx \\ & = -\frac {a A \text {arctanh}(\cos (c+d x))}{8 d}-\frac {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 = 0.02 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.92 \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=-a A \left (\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}\right ) \]
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Time = 0.89 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(-\frac {A a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-1\right )}{64 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}\) | \(55\) |
norman | \(\frac {\frac {A a}{64 d}-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {A a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(57\) |
derivativedivides | \(\frac {-A a \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )+A a \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}\) | \(91\) |
default | \(\frac {-A a \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )+A a \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}\) | \(91\) |
parts | \(\frac {A a \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}-\frac {A a \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )}{d}\) | \(93\) |
risch | \(\frac {A a \left ({\mathrm e}^{7 i \left (d x +c \right )}+7 \,{\mathrm e}^{5 i \left (d x +c \right )}+7 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {A a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {A a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}\) | \(101\) |
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Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (55) = 110\).
Time = 0.25 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.11 \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=\frac {2 \, A a \cos \left (d x + c\right )^{3} + 2 \, A a \cos \left (d x + c\right ) - {\left (A a \cos \left (d x + c\right )^{4} - 2 \, A a \cos \left (d x + c\right )^{2} + A a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (A a \cos \left (d x + c\right )^{4} - 2 \, A a \cos \left (d x + c\right )^{2} + A a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{16 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
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\[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=- A a \left (\int \left (- \csc ^{3}{\left (c + d x \right )}\right )\, dx + \int \csc ^{5}{\left (c + d x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (55) = 110\).
Time = 0.23 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.97 \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=-\frac {A a {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, A a {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{16 \, d} \]
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Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.75 \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=\frac {4 \, A a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - \frac {A a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {{\left (A a - \frac {2 \, A a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}}{64 \, d} \]
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Time = 18.77 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=\frac {A\,a\,\left (8\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+1\right )}{64\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4} \]
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