Integrand size = 8, antiderivative size = 42 \[ \int \csc ^6(a+b x) \, dx=-\frac {\cot (a+b x)}{b}-\frac {2 \cot ^3(a+b x)}{3 b}-\frac {\cot ^5(a+b x)}{5 b} \]
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Time = 0.02 (sec) , antiderivative size = 42, 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} \[ \int \csc ^6(a+b x) \, dx=-\frac {\cot ^5(a+b x)}{5 b}-\frac {2 \cot ^3(a+b x)}{3 b}-\frac {\cot (a+b x)}{b} \]
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Rule 3852
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (a+b x)\right )}{b} \\ & = -\frac {\cot (a+b x)}{b}-\frac {2 \cot ^3(a+b x)}{3 b}-\frac {\cot ^5(a+b x)}{5 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.33 \[ \int \csc ^6(a+b x) \, dx=-\frac {8 \cot (a+b x)}{15 b}-\frac {4 \cot (a+b x) \csc ^2(a+b x)}{15 b}-\frac {\cot (a+b x) \csc ^4(a+b x)}{5 b} \]
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Time = 0.52 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\left (-\frac {8}{15}-\frac {\csc \left (x b +a \right )^{4}}{5}-\frac {4 \csc \left (x b +a \right )^{2}}{15}\right ) \cot \left (x b +a \right )}{b}\) | \(33\) |
default | \(\frac {\left (-\frac {8}{15}-\frac {\csc \left (x b +a \right )^{4}}{5}-\frac {4 \csc \left (x b +a \right )^{2}}{15}\right ) \cot \left (x b +a \right )}{b}\) | \(33\) |
risch | \(-\frac {16 i \left (10 \,{\mathrm e}^{4 i \left (x b +a \right )}-5 \,{\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{15 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{5}}\) | \(44\) |
parallelrisch | \(\frac {-3 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}+3 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}-25 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}+25 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}-150 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )+150 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{480 b}\) | \(81\) |
norman | \(\frac {-\frac {1}{160 b}-\frac {5 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{96 b}-\frac {5 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{16 b}+\frac {5 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{16 b}+\frac {5 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{96 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{10}}{160 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}\) | \(99\) |
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Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.52 \[ \int \csc ^6(a+b x) \, dx=-\frac {8 \, \cos \left (b x + a\right )^{5} - 20 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )}{15 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )} \sin \left (b x + a\right )} \]
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\[ \int \csc ^6(a+b x) \, dx=\int \csc ^{6}{\left (a + b x \right )}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83 \[ \int \csc ^6(a+b x) \, dx=-\frac {15 \, \tan \left (b x + a\right )^{4} + 10 \, \tan \left (b x + a\right )^{2} + 3}{15 \, b \tan \left (b x + a\right )^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83 \[ \int \csc ^6(a+b x) \, dx=-\frac {15 \, \tan \left (b x + a\right )^{4} + 10 \, \tan \left (b x + a\right )^{2} + 3}{15 \, b \tan \left (b x + a\right )^{5}} \]
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Time = 20.95 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90 \[ \int \csc ^6(a+b x) \, dx=-\frac {\mathrm {cot}\left (a+b\,x\right )}{b}-\frac {2\,{\mathrm {cot}\left (a+b\,x\right )}^3}{3\,b}-\frac {{\mathrm {cot}\left (a+b\,x\right )}^5}{5\,b} \]
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