Integrand size = 8, antiderivative size = 55 \[ \int \csc ^8(a+b x) \, dx=-\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{b}-\frac {3 \cot ^5(a+b x)}{5 b}-\frac {\cot ^7(a+b x)}{7 b} \]
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Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3852} \[ \int \csc ^8(a+b x) \, dx=-\frac {\cot ^7(a+b x)}{7 b}-\frac {3 \cot ^5(a+b x)}{5 b}-\frac {\cot ^3(a+b x)}{b}-\frac {\cot (a+b x)}{b} \]
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Rule 3852
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (a+b x)\right )}{b} \\ & = -\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{b}-\frac {3 \cot ^5(a+b x)}{5 b}-\frac {\cot ^7(a+b x)}{7 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.40 \[ \int \csc ^8(a+b x) \, dx=-\frac {16 \cot (a+b x)}{35 b}-\frac {8 \cot (a+b x) \csc ^2(a+b x)}{35 b}-\frac {6 \cot (a+b x) \csc ^4(a+b x)}{35 b}-\frac {\cot (a+b x) \csc ^6(a+b x)}{7 b} \]
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Time = 0.66 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\left (-\frac {16}{35}-\frac {\csc \left (x b +a \right )^{6}}{7}-\frac {6 \csc \left (x b +a \right )^{4}}{35}-\frac {8 \csc \left (x b +a \right )^{2}}{35}\right ) \cot \left (x b +a \right )}{b}\) | \(43\) |
default | \(\frac {\left (-\frac {16}{35}-\frac {\csc \left (x b +a \right )^{6}}{7}-\frac {6 \csc \left (x b +a \right )^{4}}{35}-\frac {8 \csc \left (x b +a \right )^{2}}{35}\right ) \cot \left (x b +a \right )}{b}\) | \(43\) |
risch | \(\frac {32 i \left (35 \,{\mathrm e}^{6 i \left (x b +a \right )}-21 \,{\mathrm e}^{4 i \left (x b +a \right )}+7 \,{\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{35 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{7}}\) | \(55\) |
parallelrisch | \(\frac {5 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}-5 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}+49 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}-49 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}+245 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}-245 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}+1225 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )-1225 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )}{4480 b}\) | \(107\) |
norman | \(\frac {-\frac {1}{896 b}-\frac {7 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{640 b}-\frac {7 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{128 b}-\frac {35 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{128 b}+\frac {35 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{128 b}+\frac {7 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{10}}{128 b}+\frac {7 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{12}}{640 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{14}}{896 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}}\) | \(131\) |
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Time = 0.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.58 \[ \int \csc ^8(a+b x) \, dx=-\frac {16 \, \cos \left (b x + a\right )^{7} - 56 \, \cos \left (b x + a\right )^{5} + 70 \, \cos \left (b x + a\right )^{3} - 35 \, \cos \left (b x + a\right )}{35 \, {\left (b \cos \left (b x + a\right )^{6} - 3 \, b \cos \left (b x + a\right )^{4} + 3 \, b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
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\[ \int \csc ^8(a+b x) \, dx=\int \csc ^{8}{\left (a + b x \right )}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82 \[ \int \csc ^8(a+b x) \, dx=-\frac {35 \, \tan \left (b x + a\right )^{6} + 35 \, \tan \left (b x + a\right )^{4} + 21 \, \tan \left (b x + a\right )^{2} + 5}{35 \, b \tan \left (b x + a\right )^{7}} \]
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Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82 \[ \int \csc ^8(a+b x) \, dx=-\frac {35 \, \tan \left (b x + a\right )^{6} + 35 \, \tan \left (b x + a\right )^{4} + 21 \, \tan \left (b x + a\right )^{2} + 5}{35 \, b \tan \left (b x + a\right )^{7}} \]
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Time = 20.41 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.75 \[ \int \csc ^8(a+b x) \, dx=-\frac {{\mathrm {tan}\left (a+b\,x\right )}^6+{\mathrm {tan}\left (a+b\,x\right )}^4+\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^2}{5}+\frac {1}{7}}{b\,{\mathrm {tan}\left (a+b\,x\right )}^7} \]
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